thermodynamics of mesoscopic quantum systems: from a single qubit to light harvesting complexes

Diss. ETH No. 23461

Thermodynamics of Mesoscopic Quantum Systems:from a single qubit to light harvesting complexes

A dissertation submitted toETH ZÜRICH, DEPARTEMENT MATERIALWISSENSCHAFT

to attain the degree ofDOKTOR DER WISSENSCHAFTEN

Presented byJérôme Flakowski

MSc in Physics, EPFLMSc in Biology, UNIGEborn November 8, 1979

citizen of Regensdorf (ZH)

accepted on the recommendation of

Prof. Dr. Hans Christian ÖttingerProf. Dr. Renato RennerProf. Dr. Fausto Rossi

Dr. David Taj

2016

AcknowledgementsFirst of all, I would like to thank my supervisor Hans Christian Öttinger for having given methe opportunity to work in his group as well as for his patience and understanding. It wasa difficult but great journey which has greatly broadened my scientific horizon. During myPhD I worked intensively with David Taj, my co-supervisor. I would like to thank him warmlyfor having invested so much energy into our work. I also highly appreciated the numerousadvices of Patrick Ilg helping me to find my way at the beginning of my PhD and in my firstteaching duties in conjunction with Andrei Gusev. I also thank Thierry Savin to ease pressuresinevitably linked to any PhD studies by playing table soccer with him and the other membersof the group. I really appreciated to work with Martin Kröger as a teaching assistant for thePolymer Physics course. I thank him also for his periodic help just about everything rangingfrom math to computing tricks as well as sharing a bit of his coffee addiction. I also warmlythank Patricia Horn without whom I would have been lost in the administrative maze andHarald Lehmann for regularly fixing computer issues. I am grateful to Renato Renner andFausto Rossi for having accepted to be external experts in the thesis committee.I really enjoyed to work with Marco Schweizer on our paper on nonlinear stochastic processes,for which we developed our famous boost procedure, as well as for his taste for crappy moviesand Korean bullfights. I thank Maksym Osmanov for his support and for our endless scien-tific discussions about our different projects. I am grateful to Ingo Füreder for our Matlab-Mathematica exchanges and for pushing me to go to the various Department events so thatI could think about other things than work. I particularly enjoyed to drink tea after lunchwith Alan Luo who was regularly able to motivate me to go to the sport center. In fact, thegroup addiction for sport started with Carl Zinner who convinced me one day to go to themuscle pump course. I also enjoyed to give the Rheology Praktikum with Alberto Montefuscowithout having a single clue about Rheology beforehand still hoping that the students did notrealize it. I would also like to thank the remaining past and present members of the group fortheir kindness, namely Orit Peleg, Majid Mosayebi, Monirosadat Sadati, Aparna Sreekumari,Ahmad Moghimi-kheirabadi and Meisam Pourali hoping I did not forget anyone.Finally, I would like to thank my parents for their invaluable support as well as for alwaysbeing there for me. Without them, all this would have been impossible. I am also grateful tomy friends from the “Welschland” for their constant encouragement and allowing me to forgetmy worries during the weekends.

Zurich, 2016 J. F.

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AbstractOver the past few years, the exploration of mesoscopic quantum systems progressed drasticallywith experiments one could not imagine twenty years ago. New measurements, such as the de-tection of entanglement in large-scale quantum systems, require novel modeling tools. The openquantum system formalism, which took off in the 70s but was popularized at the turn of thecentury, played a central role in the analysis of these data. The existence of so-called non-trivialquantum effects got increasingly manifest with the measurement of quantum coherence at roomtemperature in bacterial light-harvesting complexes and two-dimensional nanostructures. Thereexists actually a plethora of techniques to discuss such kind of effects. However, since open sys-tems are out-of-equilibrium it is crucial to ensure their thermodynamical consistency. To producetractable models, we have decided in this work to rely on the modular dynamical semigroup ap-proach. This framework permits to postulate Markovian equations leading to a time-evolutionpreserving the positivity of the density matrix. Moreover, it ensures that the second law of ther-modynamics and the convergence to the correct steady-state hold.In order to compare these equations with experiments, we adapt the fluctuation dissipation the-orem and the Green-Kubo formula to open quantum system. Thereby, we examine the linearresponse of a single qubit connected to a heat reservoir and find that the decay of the coher-ence depends on the initial state’s phase in presence of non-secular terms. Besides, we showthe emergence of a biexponential decay of the coherence, associated to two distinct decoherencetimes and non-Lorentzian susceptibility profiles. Both phenomena were measured in numeroussetups; they could be exploited to shorten or prolong the coherence time to contribute, for exam-ple, to fulfill one of DiVincenzo’s criteria for quantum computing. These findings remain true fortwo qubits, either connected to independent reservoirs or to a single common reservoir, allowingto accelerate or refrain the decay of entanglement. In the same vein, we explain how to createcoherence and entanglement in a transitory or steady manner through the action of a commonreservoir. On the other hand, we prove that one can fix the problem of the artificial position diffu-sion taking place in quantum Brownian motion within the presented framework. Furthermore,we provide a scheme for the nonlinear response going beyond Mukamel’s formulation. Takenaltogether, our results support the idea that the postulated master equations can be substitutedto the derived ones to fix their drawbacks. Finally, we explain how coherence can be used totune the energy transfer in light-harvesting complexes, near and far-away from equilibrium, bymanipulating the system Hamiltonian as well as the interaction with the environment.

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ZusammenfassungDie Erforschung von mesoskopischen Quantensystemen entwickelte sich erst vor kurzem auf beein-druckende Weise und ermöglichte die Durchführung von Messungen, wie man sie sich vor zwanzigJahren kaum vorstellen konnte. Neue experimentelle Ergebnisse, wie z.B. die Erkennung von gros-sen Quantensystemen Verschränkungen, erfordern neue Modellierungsmethoden. Offene Quanten-systeme, die insbesonders entworfen wurden um Systeme in verrauschten Umgebungen zu erfor-schen, nahmen in den 70er Jahren zu, wurden aber erst zur Jahrhundertwende populär. Die nichttriviale Rolle von Quanteneffekten wurde aufrichtig ernst genommen durch den Nachweis vonQuantenkohärenz bei Raumtemperatur in bakteriellen lichtsammelnden Komplexen und zweidi-mensionalen Nanostrukturen unterstützt. Es gibt tatsächlich ein Fülle von Methoden, um solcheAuswirkungen zu diskutieren. Da jedoch offene Systeme nicht im Gleichgewicht sind, ist es von ent-scheidender Bedeutung die thermodynamische Konsistenz dieser Methoden zu gewährleisten. Umhandhabbare Modelle zu produzieren, werden wir uns auf der modularen dynamische Halbgruppebedienen. Dieser Rahmen ermöglicht es Markovsche Gleichungen zu postulieren, die zu einer Zeit-entwicklung führen, welche die Positivität des Dichteoperators erhält sowie den zweiten Hauptsatzder Thermodynamik erfüllt und zudem den richtigen stationären Zustand garantiert.Um diese Gleichungen mit Experimenten zu vergleichen, passen wir das Fluktuations -Dissipations-Theorem und die Green-Kubo-Formel für offene Quantensystem an. Wir beginnen mit der Untersu-chung der linearen Reaktion eines Qubits welches mit einem Wärmebad verbunden ist. Wir beobachtendass der Abfall der Kohärenz von der Startphase abhängt in Gegenwart von nichtsäkularen Beiträgen.Nebenbei zeigen wir die Entstehung von einem biexponentiellen Abfall der Kohärenz welche mit zweiunterschiedlichen Dekohärenzzeiten und nicht-Lorentzischen Suszeptibilitäten verbunden ist. Diesebeiden Phänomene wurden in zahlreichen Experimenten gemessen und könnten genutzt werden umdie Kohärenzzeit zu verkürzen oder zu verlängern um z.B. dazu beitragen eines der DiVincenzo Krite-rien für Quantenrechner zu erfüllen. Diese Ergebnisse bleiben für zwei Qubits gültig, die entweder mitunabhängigen Reservoirs oder einem einzigen gemeinsamen Reservoir verbunden sind, und ermögli-chen den Zerfall der Verschränkungen zu beschleunigen oder zu verlangsamen. Zugleich erklären wirwie man Kohärenz und Verschränkungen in einer vorübergehenden oder stetigen Weise mit Hilfe ei-nes gemeinsamen Reservoir erschaffen kann. Andererseits, beweisen wir dass man ebenso das Problemder künstlichen Positions-Diffusion der Standardformulierung der Quanten-Brownschen Bewegung be-hebt innerhalb des vorgegebenen Rahmens. Zusätzlich liefern wir ein Schema für die nichtlineare Reak-tion welches über Mukamels Formulierung hinaus geht. Insgesamt, unterstützen unsere Ergebnisse dieIdee, dass die postulierten die abgeleiteten Mastergleichungen ersetzten könnten um ihre Nachteile zubeheben. Schließlich erklären wir wie Kohärenz genutzt werden kann um die Energieübertragung inlichtwertenden Komplexen abzustimmen, nahe und weit weg vom Gleichgewicht, durch Manipulationdes Systems Hamilton-Operator sowie der Wechselwirkung mit der Umgebung.

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ContentsAcknowledgements i

Abstract (English/German) iii

List of figures xi

List of tables xvii

List of symbols and abbreviations xix

1 Introduction 11.1 Coarse-graining techniques for open quantum systems . . . . . . . . . . 21.2 A renewed thermodynamical perspective . . . . . . . . . . . . . . . . . . 31.3 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Traditional master equations to study open quantum systems 52.1 The microscopic approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Exact integro-differential master equations . . . . . . . . . . . . . 62.1.2 Time scales separation . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.3 Derivation under a weak system-reservoir coupling . . . . . . . . 92.1.4 Alternative microscopic derivations . . . . . . . . . . . . . . . . . 182.1.5 Ergodicity property . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Axiomatic formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.1 Lindblad master equation . . . . . . . . . . . . . . . . . . . . . . . 232.2.2 The Schrödinger and Heisenberg picture: a duality relation . . . 242.2.3 Linear dynamical maps . . . . . . . . . . . . . . . . . . . . . . . . 252.2.4 Quantum detailed balance . . . . . . . . . . . . . . . . . . . . . . . 252.2.5 Complete positivity . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Thermodynamic reformulation: nonlinear open quantum systems 293.1 Modular dynamical semigroup (MDS) in a nutshell . . . . . . . . . . . . 30

3.1.1 Time-evolution of the system of interest . . . . . . . . . . . . . . . 303.1.2 MDS master equation for the system of interest . . . . . . . . . . 323.1.3 Entropy production . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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3.1.4 Positivity argument for the master equation . . . . . . . . . . . . 353.2 From the MDS to the thermodynamic master equations . . . . . . . . . . 36

3.2.1 The Lindblad-Davies master equation . . . . . . . . . . . . . . . . 373.2.2 The dynamically time-averaged NTME . . . . . . . . . . . . . . . 403.2.3 The pre-averaged NTME . . . . . . . . . . . . . . . . . . . . . . . 423.2.4 The modular-free NTME . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Practical implementation of the nonlinear master equations 474.1 Discret formulation of the dynamically averaged NTME . . . . . . . . . 484.2 How to use the dynamical coarse-graining time? . . . . . . . . . . . . . . 49

4.2.1 Role of the dynamical coarse-graining time . . . . . . . . . . . . . 494.2.2 Redefinition of the standard timescale separations . . . . . . . . . 504.2.3 Matching a given time-evolution . . . . . . . . . . . . . . . . . . . 524.2.4 Time-averaging the pre-averaged NTME . . . . . . . . . . . . . . 52

4.3 Why two NTMEs to explore the WCR? . . . . . . . . . . . . . . . . . . . . 534.4 Linearization around equilibrium . . . . . . . . . . . . . . . . . . . . . . . 54

4.4.1 General procedure: small perturbation around the Gibbs state . . 544.4.2 Linearization of the dynamically time-averaged NTME . . . . . . 564.4.3 Linearization of the pre-averaged NTME . . . . . . . . . . . . . . 564.4.4 Linearization of the modular-free NTME . . . . . . . . . . . . . . 57

4.5 Explicit formulation of the NTMEs . . . . . . . . . . . . . . . . . . . . . . 574.5.1 The dynamically time-averaged NTME . . . . . . . . . . . . . . . 574.5.2 The pre-averaged NTME . . . . . . . . . . . . . . . . . . . . . . . 604.5.3 The modular-free NTME . . . . . . . . . . . . . . . . . . . . . . . . 61

5 Response theory 635.1 Essential tools to work in the linear response regime . . . . . . . . . . . . 64

5.1.1 Linearized entropy production . . . . . . . . . . . . . . . . . . . . 645.1.2 Purely dissipative nature of the linearized evolution superoperator 645.1.3 Quantum detailed balance revisited . . . . . . . . . . . . . . . . . 65

5.2 Fluctuation-dissipation theorem (FDT) . . . . . . . . . . . . . . . . . . . . 665.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2.2 Standard mistake for open quantum systems . . . . . . . . . . . . 675.2.3 Thermodynamical reformulation . . . . . . . . . . . . . . . . . . . 685.2.4 Perturbation of the MDS scattering operators . . . . . . . . . . . . 71

5.3 Green-Kubo formula (GKF) . . . . . . . . . . . . . . . . . . . . . . . . . . 725.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.3.2 Onsager coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3.3 Strategy to deal with the singular nature of the integrated GKF . 765.3.4 The Onsager reciprocity relation . . . . . . . . . . . . . . . . . . . 77

5.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.4.1 The FDT to study the decoherence process of a single qubit . . . 785.4.2 Linear response of a simplified light-harvesting complex . . . . . 92

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5.4.3 The GKF to study the transport process through a single qubit . . 935.4.4 Implications of the linear response for the modular free NTME . 99

5.5 Beyond the FDT: the nonlinear response regime . . . . . . . . . . . . . . 1055.5.1 Polarization and nonlinear response function . . . . . . . . . . . . 1055.5.2 Third-order susceptibility . . . . . . . . . . . . . . . . . . . . . . . 106

6 Quantum superposition and correlations in open quantum systems 1116.1 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.1.1 Coherence of a single qubit . . . . . . . . . . . . . . . . . . . . . . 1136.1.2 Different measures of coherence . . . . . . . . . . . . . . . . . . . 1146.1.3 Creation of coherence . . . . . . . . . . . . . . . . . . . . . . . . . 1156.1.4 Three types of decoherence processes . . . . . . . . . . . . . . . . 116

6.2 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.2.1 Bipartite system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.2.2 Quantum operations and entanglement . . . . . . . . . . . . . . . 1186.2.3 Different measures of entanglement . . . . . . . . . . . . . . . . . 1196.2.4 Creation of entanglement . . . . . . . . . . . . . . . . . . . . . . . 1216.2.5 Entanglement and non-CP time evolution . . . . . . . . . . . . . . 121

6.3 Interaction of multipartite system with a reservoir . . . . . . . . . . . . . 1226.3.1 Uncovering the eigenoperators . . . . . . . . . . . . . . . . . . . . 1236.3.2 Enhancing the eigenoperators’ action with non-secular terms . . 127

6.4 Analysis of two qubit systems . . . . . . . . . . . . . . . . . . . . . . . . . 1286.4.1 Two non-interacting qubits connected to independent reservoirs 1286.4.2 Equal qubits connected to a common reservoir . . . . . . . . . . . 1296.4.3 Unequal qubits connected to a common reservoir . . . . . . . . . 1316.4.4 Sudden-death of the concurrence for a common reservoir . . . . . 1356.4.5 Decay rates of two qubit systems: local and common reservoirs . 1386.4.6 Impact of the coherence on entanglement for a common reservoir 1396.4.7 Concurrence’s decay in a common reservoir: initial phases’ role . 1416.4.8 Concurrence for a common reservoir: strong damping limit . . . 142

7 Light harvesting complexes 1457.1 FMO subunit: the perfect “toy model” . . . . . . . . . . . . . . . . . . . . 1467.2 FMO subunit described as an open quantum system . . . . . . . . . . . . 147

7.2.1 Relevant and irrelevant DOF . . . . . . . . . . . . . . . . . . . . . 1477.2.2 The excitonic Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 1487.2.3 The reservoir and interaction Hamiltonians . . . . . . . . . . . . . 1507.2.4 Time and energy scales for the system . . . . . . . . . . . . . . . . 1527.2.5 The initial state and the creation of coherence . . . . . . . . . . . . 1527.2.6 Coherence and entanglement in a FMO subunit . . . . . . . . . . 153

7.3 Energy transfer through the FMO subunit . . . . . . . . . . . . . . . . . . 1537.3.1 The impact of coherence on the energy transfer efficiency . . . . . 1547.3.2 Optimization of the FMO subunit’s thermal conductance . . . . . 158

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7.3.3 Role of the steady-state coherence . . . . . . . . . . . . . . . . . . 160

8 Conclusions and outlook 1618.1 Why phenomenological nonlinear master equation? . . . . . . . . . . . . 1618.2 Did we get new insights by using the NTMEs? . . . . . . . . . . . . . . . 1628.3 Future perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

9 Appendix 1659.1 Incompatibility of the usual Lindblad eigenoperators with continuous

energy spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1659.2 Generalized Gibbs state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1669.3 The zero temperature limit and the NTMEs . . . . . . . . . . . . . . . . . 1689.4 Seemingly nonlinear commutation relation . . . . . . . . . . . . . . . . . 1699.5 Extended eigenvalue equation . . . . . . . . . . . . . . . . . . . . . . . . . 1709.6 Dynamical time-averaging of the first NTME . . . . . . . . . . . . . . . . 1719.7 Seemingly nonlinear commutator identity . . . . . . . . . . . . . . . . . . 1749.8 Time-averaging procedure applied to the third NTME . . . . . . . . . . . 1749.9 Linearization of the perturbed density matrix . . . . . . . . . . . . . . . . 1769.10 From the linearized NTME to the LDME for Tg → ∞ . . . . . . . . . . . . 177

9.10.1 Main steps of the proof . . . . . . . . . . . . . . . . . . . . . . . . . 1779.10.2 Detailed calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 178

9.11 Linearized equation: decomposition for a N-level system . . . . . . . . . 1799.12 Generalized perturbation schemes . . . . . . . . . . . . . . . . . . . . . . 1809.13 Explicit expression of Kubo’s superoperator . . . . . . . . . . . . . . . . . 1809.14 Kubo’s property and the reversible time evolution . . . . . . . . . . . . . 1819.15 Perturbed propagator calculation . . . . . . . . . . . . . . . . . . . . . . . 1829.16 Derivative of the instantaneous Gibbs state . . . . . . . . . . . . . . . . . 1829.17 Numerical extraction of the susceptibility . . . . . . . . . . . . . . . . . . 1839.18 Modified Spohn and Lebowitz Lemma calculation . . . . . . . . . . . . . 1859.19 Decomposition of the Kubo superoperator . . . . . . . . . . . . . . . . . . 1869.20 Small momentum exchange limit . . . . . . . . . . . . . . . . . . . . . . . 187

Bibliography 189

Curriculum Vitae 229

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List of Figures4.1 Feynman diagram of the collision process between the particle of interest

and the reservoir modeled according to the dynamically time-averagedNTME (3.53): the heavy system’s particle, which is carrying a momentump′ and energy ε(p′), collides with one of the multiple lighter reservoir’sparticles having a momentum q and absorbs during the process an energyν (note that we use h=1) so that it back-scatters with a momentum p andenergy ε(p). The disk with a radius Eg =ωg represents the Gaussian (3.48)of the modified eigenoperators (3.45). For an infinite Tg we have ωg =0such that the Gaussian δTg(ε) is replaced by a Dirac-delta and we recovera “sharp” collision process with exact energy conservation (i.e. the sphereshrinks to a single point). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1 Comparison of the imaginary part of the susceptibility χDD(ω) for theLDME (3.38) in black and the modular-free NTME (3.66) in red for aweak-coupling (γ = 0.1) at low and high temperatures (β = 5 on theleft, β = 1/6 on the right) using a bosonic spectral function (2.25) withJ(ω) = γω (including an exponential cut-off). The full lines have beencomputed with the FDT in the frequency domain while the black dots/redsquares have been obtained with the numerical scheme presented in theAppendix 9.17. Clearly, at low temperature the modular-free NTME isunphysical for small couplings since its susceptibility diverges (implyingthat the ratio T2/(2T1)→∞). . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2 Comparison of the imaginary part of the susceptibility χDD(ω) for theLDME (3.38), BRME (2.35) and 2nd NTME (4.30) at low temperature (β=6) using two different absorptions: a(∆) = 0.001∆ on the left (beforethe threshold) and a(∆) = 0.1∆ on the right (after the threshold) whileaBRME

thr =0.005∆ and a2nd NTMEthr =0.016∆. . . . . . . . . . . . . . . . . . . . . . 84

5.3 Time evolution before (left panel, absorption a(∆) = 0.0075∆) and af-ter (right panel, absorption a(∆)=0.075∆) the threshold aNTME

thr =0.065∆ inabsence of pure dephasing and at low temperature (β=4). We compareinitial states with distinct orientations φ(1)

0 = π5 and φ(2)

0 = φ(1)0 + π

2 whiler(1, 2)

0 =0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

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List of Figures

5.4 Bifurcation for a(∆) > aBRMEthr = 0.036∆ and a2ndNTME

thr = 0.065∆ (left panel,same parameters as for Fig. 5.3) and temperature-dependence of thethresholds (right panel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.5 On the left panel we compare numerically the threshold of the Tg-dependentor dynamically time-averaged NTME (4.28) with the Tg-independent orpre-averaged NTME (4.30) and the BRME (2.35). To this end, we use abosonic spectral function (2.25) with J(ω)=γω (including an exponentialcut-off) and a Tg =0.1 to improve the match with the microscopic picturegiven by the BRME. On the right panel, we compare the threshold fordifferent Tg values: as Tg increases, the threshold ath goes towards infinityleading to the WCL where no bifurcation takes place. . . . . . . . . . . . 89

5.6 From left to right: we increase the pure dephasing |Q11 − Q22| = q =

0.1, 0.5, 1.0, 2.0. To perform the simulations, we have used a bosonic spec-tral function (2.25) for J(ω)=γω (including an exponential cut-off) withβ=4 for different γ’s to obtain the corresponding absorption a(∆). Bottomplot: Comparison of the upright plot with the LDME with the same set ofparameters and |Q11 −Q22|=2.0. Clearly, the LDME displays no bifurcati-on even in the T1T2T3 model and no coupling between the populationsand coherence is observed. Note that in the WCL the two equations agreeas it should be. As previously stated, we describe the bifurcation pointphenomenologically. An intermediate to a large pure dephasing impliesthat the ratios obtained from the pre-averaged NTME deviate from theLDME prior to the bifurcation contrary to the zero pure dephasing regimedescribed in Fig. 5.4 associated to the T1T2 model. This effect is producedby the coupling of the populations and coherences. . . . . . . . . . . . . 91

5.7 Left plot: Comparison of the thermal conductance Gκ given by (5.91)using the GKF (5.52) and a direct simulation to compute the transportcoefficient L12 (the simulation parameters are provided in the text). Rightplot: Comparison of the thermal conductance Gκ obtained from the GKFfor the LDME (3.38), the BRME (2.35), the Tg-dependent or dynamicallytime-averaged NTME (3.53) and Tg-independent or pre-averaged NT-ME (3.58) using the same parameters as for the left plot apart from thecoupling strength γ1=γ2=0.1. . . . . . . . . . . . . . . . . . . . . . . . . 97

5.8 Examination on the impact of difference of coupling strength δγ12 =

γ1 − γ2 of the two reservoirs: a small difference implies a large increaseof the thermal conductance. Note that this increase becomes less and lessimportant for an increasingly larger difference δγ12. . . . . . . . . . . . . 98

5.9 Left plot: Comparison of the thermal conductance obtained from themodular-free NTME with the LDME and pre-averaged NTME (sameparameters as Fig. 5.8 apart from γ1 = 0.05 and γ2 = 0.15). Right plot:Check the dependence on the friction of the thermal conductance for themodular-free NTME (fixed γ1= 0.05 and vary γ2). . . . . . . . . . . . . . 98

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List of Figures

6.1 Decoherence process simulated with the LDME and displayed using theBloch sphere representation of a two-level system. The initial state is pureand located on the shell whereas the final state located in the center isfully mixed. On the axis, we have the magnetizations mi = tr(ρσi) with σi

the ith Pauli matrix and i= x, y, z. . . . . . . . . . . . . . . . . . . . . . . 114

6.2 Left plot: asymptotic entanglement of two identical qubits connectedsymmetrically to a common reservoir under the action of the LDME atlow temperature and friction (β=5 and γ=0.1; ∆ is equal to the largestenergy gap of the system). The EP (2.48) is broken since ∃X 6= c1 suchthat [Aω, X] = [A−ω, X] = 0. Right plot: breaking the symmetry of thecoupling (6.28) restores the EP (2.48) and the convergence to the Gibbsstate eliminating the created entanglement at long times. . . . . . . . . . 130

6.3 Left plots: In absence of pure dephasing, the LDME generates no entangle-ment whereas the BRME and the pre-averaged NTME yield a very similarentanglement creation pattern (γ=0.3, β=5). Left plots: In presence ofpure dephasing, the BRME fails to preserve positivity (see the smallesteigenvalue of ρ in the lower plot) so that one cannot trust its concurrence;on the other hand, the positivity of the NTME is not jeopardized (sameparameters as for the left plot). . . . . . . . . . . . . . . . . . . . . . . . . 132

6.4 Comparison of the concurrence for the pre-averaged NTME for settingswere the BRME does not fail; the two equations produce a very similartime evolution. Simulations performed for the frictions γ= 0.1 and 0.4as well as the Hamiltonian (6.27) and coupling operator (6.28); all otherparameters are the same as for Fig. 6.3. . . . . . . . . . . . . . . . . . . . . 132

6.5 Comparison of the concurrence produced by the pre-averaged NTME forthe non-symmetric independent coupling operator (6.28), represented bythe red curve with pure-dephasing and the yellow curve without, and itscollective version represented by the blue curve (left plot for γ=0.1 andright plot for γ=0.4). We have used a log-scale to distinguish the creationof entanglement at short and long times. . . . . . . . . . . . . . . . . . . . 134

6.6 Creation of entanglement for the LDME for a collective Q, i.e. whoseanti-diagonal contains non-zero elements, in presence of an initial cohe-rence (ρ12 of each individual qubit has to be non-zero). We use a non-degenerted Hamiltonian as well as a friction γ = 0.1 and an inversetemperature β=5.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.7 Creation of entanglement and its subsequent destruction at differentreservoir temperatures T using the pre-averaged NTME with the sameparameters as in Fig. 6.5 but only for a friction γ=0.1. . . . . . . . . . . . 135

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List of Figures

6.8 Left plot: comparison of the entanglement sudden-death (ESD) betweenthe LDME, the BRME and the pre-averaged NTME. The non-secular termslead to a delayed ESD; the NTME should be used instead of the BRMEwhose density matrix displays negative eigenvalues at short time. Rightplot: ESD for the pre-averged NTME for different frictions γ. For thelargest γ a revival of entanglement takes place. . . . . . . . . . . . . . . . 136

6.9 Decay rate ratios in black for the LDME and in red for the pre-averagedNTME; we plotted them as a function of the absorption rate a(∆max) ofthe two qubit system’s largest energy gap ∆max. Contrary to the singlequbit case, we have two bifurcation events associated to different thres-holds (athr = 0.0023 and 0.0031) leading to a more complicated decohe-rence pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.10 Upper plot: Comparison of the correlation between the radius r0 charac-terizing the initial state and the maximal concurrence corr(r0, Cmax) aswell as the integrated concurrence over time corr(r0, Cint) as a functionof the two initial angles θ0 and φ0. Lower plot: Values for the angles θ0

and φ0. The numbers on the x-axis represent the index of the different(θ0, φ0) pairs for a series of r0 comprised in [0,1) used to compute the quan-tities corr(r0, Cmax) and corr(r0, Cint). We only represented a portion ofthe screened states since the phase φ0 has a small impact on the dynamicsfor the selected settings (we come back to the role of phases later for anentangled X-state). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.11 Comparison of the l1 norm and concurrence C for selected radius r0 withthe angles θ0 = 0 (left plot) and θ0 = 2.6 (right plot). In the left plot thecoherence enhances the creation of entanglement while it has exactly theopposite effect in the right plot. . . . . . . . . . . . . . . . . . . . . . . . . 141

6.12 Comparison of the normalized integrated concurrence Cint starting froman entangled mixed X-state for two non-interacting qubit connected to acommon reservoir. Left panel: in absence of pure dephasing. Right panel:with pure dephasing (q = 1.5) we observe a displacement/deformationof the “optimal islands” (the red regions). We used φ0, ψ0 ∈ [0, 4π] (andnot 2π) to better display the periodic pattern of Cint. . . . . . . . . . . . . 142

6.13 Left plot: comparison of the energy relaxation of the 3rd NTME (3.66),also denominated as the modular free NTME, with the LSME (2.44) goingtowards the fully mixed state and the NRSME (2.45) ending in an ap-proximate Gibbs state (β = 1, γ = 2.0). Middle plot: ESD for the threeequations (β = 1, γ = 2.0). The NRSME displays negative eigenvalues.Right plot: Creation of coherence for the 3rd NTME for different γ (β=1).A higher friction leads to a more durable creation of coherence. . . . . . 143

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List of Figures

6.14 Left plot: comparison of the entanglement sudden birth of the 3rd NTME(3.66), the LSME (2.44) and the NRSME (2.45) for γ=0.5 and β=0.2 (thesame system Hamiltonian and coupling operator as for the Fig. 6.6 hasbeen used). In black is indicated when the density matrix produced bythe NRSME carries negative eigenvalues. Right plot: Sudden birth ofentanglement for the 3rd NTME for increasing frictions. . . . . . . . . . . 143

7.1 (a) Structure of a single FMO subunit: in yellow the protein scaffold andin green the chromophores, (b) Sketch of the LHC from the chlorosomeantenna to the RC. (c) Arrangement of the seven chromophores in theFMO subunit highlighted in black. Picture and description reproducedfrom Cheng, Y.-C. and Fleming, G. R., Annu. Rev. Phys. Chem., vol. 60,no. 1, pp. 241–262 (2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.2 Histogram for the normalized integrated total tangle 〈τT〉 obtained from(7.12) extracted from 10′000 simulations of the pre-averaged NTME forrandomly generated coupling operators Q at β = 1 and γ = 0.5. They-axis provides the number of Q operator which are associated to a given〈τT〉 value. Note that the LDME does not generate any coherence (orentanglement). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.3 Comparison for the pathway 1→ 2→ 3 of the excitonic energy transferefficiency at a given site (red curve), given by the trapping time differencesδ〈ttrap,n〉 defined from (7.11) and (7.13), and the degree of coherence (bluecurve) defined by the normalized averaged total tangles 〈τT〉 obtainedfrom (7.10) and (7.12). On the x-axis is indicated the index of the 10′000different coupling operators Q. . . . . . . . . . . . . . . . . . . . . . . . . 156

7.4 Top left picture: Sketch of the one-exciton Hamiltonian (7.4) where thedipole-dipole interaction involved in the dominant coherent energy trans-fer pathway 1→2→3 is highlighted in red. Top right picture: Illustrationof the FMO subunit with the seven BChl (in green) where the pathway isalso in red. Reproduced from G. D. Scholes et al., Nat. Chem., vol. 3, pp.763–774 (2011). Bottom picture: Optimization of the energy transfer effi-ciency at site 3, which is given by the trapping time differences δ〈ttrap,3〉,starting from the an average Q obtained in Fig. 7.3. The yellow dot andlines represent the pair of dipole-dipole interactions obtained from expe-rimental data for H(1)

e, FMO :=HS so that HS,12=−0.445 and HS,23=0.143. Onthe other hand, the red dot and lines represent the best modified pair ofdipole-dipole interactions is given by HS,12=−0.462 and HS,23=0.025. . 157

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List of Figures

7.5 Top left plot: Histogram counting the repartition of the randomly genera-ted pair of coupling operators (Q1, Q2), abbreviated Qi, according to thereference temperature Tref using the same parameters as in Fig. 7.2. Topright plot: Tref associated to each pair of coupling operators Qi. Bottomleft plot: peak of conductance Gκ for each pair of coupling operators Qi,the largest value being highlighted with a red circle. Bottom right plot:selection of conductances Gκ whose peak is close to Tref = 1; the largestconductance Gκ, highlighted again in red, displays a peak at Tref = 0.55. 159

7.6 Role of the steady-state coherence at the NESS. Left plot: δJQ,c = JNTQ,c− JLD

Q,cis the difference of heat flux JQ,c towards the cold reservoir betweenthe pre-averaged NTME and the LDME, Right plot: the correspondingamount of coherence in the NESS quantified with the coherence l1 norm.Clearly, an intermediate amount of coherence leads to the largest heatflux (i.e. the red curves). The BRME leads to similar results. . . . . . . . . 160

xvi

List of Tables3.1 Summary of the master equations; we use a unique reservoir and coup-

ling operator. The equations contain either a spectral function h(ω) asgiven in (2.25), a coarse-graining matrix hτ(ω1, ω2) as provided in (2.65),a friction γ for a reservoir or η for an arbitrary environment. Besides,they depend either on eigenoperators Aω or Aν as provided in (2.12)–(2.13) or (3.45), a Lindblad operator L or a coupling operator Q (can betaken as position operator X for a harmonic oscillator or a Brownianparticle). For the steady-state ρ(∞) the equations converge either to theGibbs state π, only partially to the Gibbs state ∼ π or to the fully mi-xed state 1. We indicate if a master equation can afford a continuousenergy spectrum (cont. spectrum). Abbreviations: Bloch-Redfield masterequation (BRME), Lindblad-Davies master equation (LDME), Lindbladsingular master equation (LSME), non-rotating-wave singular masterequation (NRSME), Caldeira-Leggett master equation (CLME), Lindblad-Kossakowski master equation (LKME), coarse-grained master equati-on (CGME), 1st or dynamically time-averaged nonlinear thermodynamicmaster equation (1st NTME), 2nd or pre-averaged nonlinear thermody-namic master equation (2nd NTME) and 3rd or modular free nonlinearthermodynamic master equation (3rd NTME). The Massieu operator Ψis given by (3.11) while the frequency-dependent Kubo superoperatorK ω

ρ comes from (3.34) for the 1st/2nd NTME. The 3rd NTME dependson the standard Kubo superoperator Kρ given in (4.18); the second ver-sion (3rd NTME*) is for a harmonic oscillator/Brownian particle . Diffe-rent regimes: weak-coupling limit (WCL), weak-coupling regime (WCR),Brownian motion limit (BML), strong-coupling regime (SCR) for a whitenoise (classical) reservoir, high temperature regime (HTR) or unspecified.Finally, we indicate if the equation was obtained through a microscopicderivation (micro), by phenomenological means (pheno) or a mixture ofthem (hybrid). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

xvii

List of symbols and abreviationsSymbol Units Descriptionh 6.626× 10−34 J s Planck constant (set to 1)kB 1.381× 10−23 J K−1 Boltzmann constant (set to 1)

Units DescriptioneV 1.602× 10−19 J electon-voltcm−1 1× 10−2 m wavenumber (spectroscopy unit)Hz 1× 10−1 s Herz (unit of frequency)

Symbol Descriptionρ or ρS density matrix of the reduced/open systemρ(I) ... in the interaction picturepn and |pn〉 eigenvalues and orthonormal eigenvectors of ρ

ρ+ nonequilibrium steady-state of the reduced/open systemρδA perturbed Gibbs state by the force operator δAπ or ρβ Gibbs state of the reduced/open system (for single reservoir)πk or ρβk

local Gibbs state of the reduced/open system (for multiple reservoirs)pk,n and | pk,n〉 equilibrium eigenvalues and orthonormal eigenvectors of πk

σ density matrix of the reservoirσ(I) ... in the interaction pictureσβk

local Gibbs state of the kth reservoir$ density matrix of the total system$(I) ... in the interaction picture$β Gibbs state of the total systemHTot total HamiltonianH0 free HamiltonianHS system HamiltonianHR reservoir HamiltonianHSR interaction HamiltonianQk or Q system coupling operators (kth reservoir or single reservoir)Φk or Φ reservoir coupling operatorsFk,ω or Fω Lindblad eigenoperators (non-diagonal friction matrix)

xix

List of symbols and abreviations

Ak,ω or Aω Lindblad eigenoperators (diagonal friction matrix)Ak,ν or Aν modified/Gaussified Lindblad eigenoperatorsQk or Q coarse-grained coupling operatorsLk or L Lindblad operatorsδTg(t) or δωg(ν) normalized Gaussianω frequency (in general for discrete spectrum)ν frequency (in general for continuous spectrum)Tg dynamical coarse-graining timeωg inverse of the dynamical coarse-graining timeδKr(ω) Kronecker-delta functionδD(ν) Dirac-delta functionc(k)(ω) or c(ω) friction or spectral matrixc(ω, |rk − rl |) element (k,l) of a spectral matrix with spatial dependencehk(ω) or h(ω) spectral function (from diagonalized friction matrix)tR time scale of the reservoir correlationtH Heisenberg time scale of the free evolutiontS time scale of the systemtc integration cut-off (of the time-dependent spectral function)gk or g system-reservoir coupling strengthγk or γ frictionKk or K evolution superoperator (nonlinear dynamics)Lk or L linear generatorL∗k or L∗ dual of the linear generatorΛk,t or Λt linear dynamical mapsΛ∗k,t or Λ∗t dual of the linear dynamical mapsHt(·) Hamiltonian contribution to the phenomenological time-evolutionSt(·) Entropic contribution to the phenomenological time-evolution[[·, ·]]βk

ρ,αkmodular dissipative bilinear brackets

λ modular parameterαk index specifying the nature of the dissipative interactionXk,λ

αkmodular scattering operator

Dk,λα′kα′′k

modular friction matrixYα′k

interaction operatorhk,λ(ω) modular spectral functionRk orR irreversible contribution (Redfield)Dk or D irreversible contribution (Davies)Jk or J irreversible contribution (general, mainly used for the NTMEs)Kρ(·) Kubo-Mori superoperatorKk,ω

ρ (·) or Kωρ (·) frequency-dependent Kubo-Mori superoperator

Kπk(·) or Kπ(·) equilibrium Kubo-Mori superoperatorΨk or Ψ Massieu superoperator∆Sk or ∆S relative modular operator

xx

listSymbols

S(ρ) or Sρ von Neumann entropy operatorσS(ρ) entropy productionσS(ρ) linearized entropy production (around equilibrium)χ(1)

AB(t) linear susceptibility (time domain)χ(1)

AB(ω) linear susceptibility (frequency domain)T1 energy relaxation time (inverse of the rate Γ1)T2 decoherence time (inverse of the rate Γ2)T2,+ and T2,− short/long decoherence time (biexponential decay)T3 decay time due to the coupling of the coherences and populationsµ dipole operator (lead to the polarization)P(t) and P(j)(t) polarization and polarization of order jS(j) susceptibility of order j (associated to the polarization)S(3)(ω3, τ2, ω1) third order susceptibility for 2D IR spectroscopyGt Green function superoperatorV×0 (·) reversible interaction superoperatorV×1 (·) extended interaction superoperatorρε

t perturbed density matrixWε

t perturbed evolution operatorLε

t perturbed linear generatorKε

t perturbed nonlinear evolution superoperatorLkl(β) Onsager’s transport or kinetic coefficientJ

k,βQ (t) flux operator J

k,βQ evolved in the Heisenberg picture

xk or yk thermodynamical driving force or macroscopic affinityJ k,(1)Q,+ first order contribution to the heat flux Jk

Q at the NESSκ thermal conductivityGκ thermal conductanceC(ρ) or EC(ρ) concurrence (entanglement measureCint integrated concurrence over the time-evolutionCmax maximal value of the concurrence over the time-evolution(r0, θ0, φ0) spherical coordinates to parametrize the state of a single qubitρ(0) or ρ(0) initial density matrix (ρ(0)

A initial state of the qubit A)ρBS Bell stateρX X-stateHe excitonic HamiltonianH(1)

el one-exciton (single manifold) HamiltonianH(1)

e, FMO ... for the FMO sububitτT, l1 total tangle (entanglement measure), l1 norm (coherence measure)〈ttrap,n〉 trapping time of the site n (energy transfer measure)δ〈ttrap,n〉 difference between two trapping times〈τT〉 integration of τT over the full decay time (normalized version 〈τT〉)

xxi

List of symbols and abreviations

Abreviations DescriptionAK-QDB Alicki-Kossakowski quantum detailed balanceAM-QDB Agarwal-Majewski quantum detailed balanceBChl bacteriochlorophyllBML Brownian motion limitBRME Bloch-Redfield master equationCGME coarse-grained master equationCLME Caldeira-Leggett master equationCP completely positive (or complete positivity)CPTP completely positive and trace preserving (by CP we implicitly mean CPTP)DOF degrees of freedomEP ergodicity propertyESD entanglement sudden-deathEET electronic energy transferFDT fluctuation-dissipation theoremG-QDB Grabert quantum detailed balanceGKF Green-Kubo formulah.c. Hermitian conjugateKMB Kubo-Mori-BogoliubovKMS Kubo-Martin-SchwingerKMS-DB KMS detailed-balanceLDME Lindblad-Davies master equationLKME Lindblad-Kossakowski master equationLSME Lindblad singular master equationLSTME Lindblad stochastic master equationLHC light-harvesting complexL.S. Lamb-ShiftMDS modular dynamical semigroupNESS non-equilibrium steady-stateNRSME non-rotating-wave singular master equationOQS open quantum systemsPES potential energy surfaceQD quantum dot (QDs for quantum dots)RC reaction centerRCS reaction coordinatesSCL singular coupling limitSTL stochastic limitTFC total flux conservationWCL weak-coupling limitWCR weak-coupling regime (sometimes designed as the Born-Markov regime)ZPL zero-phonon line

xxii

1 Introduction

Living organisms rely essentially on harvesting energy, carrying out chemical reactionsand processing information to survive [1, 2, 3]. Prior to the 19th century, the scientificcommunity believed that all these processes could be described using classical physics.However, the development of the wave theory of light, the black body radiation problemand discrete spectra naturally led at the beginning of the 20th century to the hypothe-sis that the energy can be absorbed as well as emitted in discrete packet of energy orquanta. After this first step, quantum mechanics developed rapidly to reach around1930 its actual mathematical form [4]. It became rapidly obvious that to understandnatural phenomena one has to take into account the microscopic structure of the mat-ter. Nowadays, quantum thermodynamics [5] as well as quantum chemistry [6] andquantum information theory [7] are crucial to describe and design nanometric systems.The most prominent example is the development of computers which are more andmore miniaturized to improve their performance and reach the size where quantumphysics come into play with the promise of renewed computationally capabilities [8, 9].However, despite all the progress since then, it is still unclear to which extend quantumeffects really matter on the macroscopic level, not only to hold together the elementarybuilding blocks of the matter, but as a real resource. Moreover, the gap between classicaland quantum mechanics, i.e. their deep connections, is still a grey area which needs tobe clarified [10]. It is commonly believed that quantum effects can be measured andexploited only for small systems at very low temperatures. However, in the recentyears it became manifest that quantum effects can equally take place in large quantumsystems embedded in hot and wet environments [11, 12, 13, 14, 15]. In this respect, themost striking example is the discovery of the non-trivial role of the coherent energytransfer in light harvesting complexes [16, 17, 18] whose role is to transform the sunlightradiations into chemical energy during the photosynthesis. To describe such a highlycomplex open system, numerous tools primarily relying on coarse-graining strategiesare required [19, 20, 21, 22, 23, 24] in order to withhold its key features.

1

Chapter 1. Introduction

1.1 Coarse-graining techniques for open quantum systems

Identifying the microscopic Hamiltonian governing the dynamics of a closed many-body quantum system is an extremely challenging task. Nowadays, its spectrum canbe partially reconstructed by combining various experimental inputs and appropriateprocessing schemes [25, 26, 27, 28, 29]. However, the equation of motion generated bysuch a Hamiltonian is intractable, even numerically, except for few purely academicproblems. This difficulty is typically avoided by focusing on a well-defined portion ofthe system by ”tracing over“ its irrelevant degrees of freedom (commonly designed asthe heat bath or reservoir). The resulting reduced system, on which all measurements ofinterest are performed, is governed by the exact Nakajima-Zwanzig integro-differentialmaster equation [30, 31]. In most cases, this equation can unfortunately not be solved.One has to further simplify the description by performing the Born-Markov approxi-mation in the weak-coupling limit which consists in taking the coupling constant of thesystem-bath interaction towards zero after a time rescaling (i.e. the secular approxima-tion). Such a procedure yields the celebrated Lindblad-Davies master equation for therelevant degrees of freedom [32, 33, 34] generating the time-evolution of reduced densitymatrix ρ of the open quantum system [19, 20, 21, 22, 23, 24]. This irreversible semigroupdynamics correctly describes the projected, but fully reversible, exact Hamiltoniandynamics [35, 36] and is perfectly thermodynamically consistent [37].

Although Davies’ equation accurately models numerous measurements, it fails to pre-dict certain phenomena since the experimental settings do not necessarily agree with theinvoked assumptions. In the first place is the fact that the equation is related to the under-lying Hamiltonian description only in the scaling limit, not for small but finite couplingconstants. For instance, its structural properties naturally lead to an exponential decayof the populations and coherences [38], along with Lorentzian profiles for the suscepti-bility. However, quantum dots can display a biexponential decay of the polarizationsassociated to homogeneous non-Lorentzian susceptibility profile [39, 40, 41, 42]. More-over, it is unable to generate entanglement revival between two non-directly interactingqubits, connected to a common heat bath, having different frequencies [43, 44, 45]. Therapid oscillations responsible for this phenomenon are removed by the secular approxi-mation eliminating the exchanges taking place between the coherences (off-diagonalelements of ρ) as well as disconnecting the coherences from the populations (diagonalelements of ρ). There are also many other situations where the secular approximationand rotating-wave approximation, neglecting the counter-rotating terms, are not justi-fied [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61] forbidding to use Davies’approach. In addition, when studying the diffusion phenomenon this standard time-averaging produces a term, required to insure the positivity of the equation, whichinduces an artificial position diffusion [62, 63, 64]. This drawback is very similar to theposition diffusion’s fictitious effect observed at the classical level when deriving theSmoluchowski equation using a coarse-graining in time [65] but absent when avoiding

2

1.2. A renewed thermodynamical perspective

it and producing Kramer’s equation [66]. Moreover, in the weak-coupling limit theharmonic oscillator does not exhibit a periodic modulation of the energy loss in thephase space contrary to a classical Fokker-Planck equation where the position and themomentum interact with the friction in different ways. Thus, removing the counter-rotating terms destroy the quantum-classical correspondence [63]. Finally, Davies him-self pointed out that his generator is only valid for discrete energy spectra [32, 67].

Thus, seeking for an exact microscopic derivation is not necessarily the best strategyto produce a reliable model. Instead, one could use a phenomenological approach anddirectly postulate the Markovian time-evolution. This idea is far from being unusual: theLindblad equation [33, 34, 68] was obtained using the principle of linearity and completepositivity without referring to any underlying microscopic description, under somelimiting procedure, and is thus phenomenological in contrast to Davies’ equation [21].Therefore, it can even afford strong couplings [69]. Thus, only a specific choice ofthe eigenoperators modeling the incoherent transitions of the system gives back theDavies’ equation [21]. The Lindbladian can, for example, be constructed ”by hand“to reconnect the populations and coherences, enforce a detailed balance as well asinsuring the convergence to the correct steady-state [70]. Alternatively, extracting theeigenoperators from experimental data using quantum process tomography is possiblebut non-trivial [71, 72, 73, 74].

1.2 A renewed thermodynamical perspective

In the present work, we have chosen to neither rely on a first principle derivation nor oncompletely positive maps so that we have to impose other guiding principles. The mostlogical choice is to use a set of sensible thermodynamical and statistical arguments toguarantee a proper connection with the macroscopic world [37] in analogy to classicalnonequilibrium systems [75, 76]. To this end, we demand the convergence at longtimes to a unique steady-state, a quantum detailed balance, a non-negative entropyproduction as well as Kubo’s fluctuation-dissipation theorem and Onsager’s reciprocityrelations close to equilibrium. All these properties are gathered by non-necessarilylinear master equations generating a modular dynamical semigroup [77, 78]. SinceDavies’ equation [32, 67] fulfill the aforementioned properties, it naturally leads to sucha semigroup. Rather, the Bloch-Redfield master equation [20], commonly used wherethe secular approximation does not hold [79, 80, 81], does not generate such a modulardynamics because it fails, in general, to preserve the positivity of the density matrix [82]and of the entropy production [83, 84]. Numerous physical inconsistencies present inmicroscopically derived master equations can be fixed under thermodynamical guidance.This permits to consider non-secular contributions in a consistent manner to describe, forexample, the dependence of the time evolution on the initial phases, the biexponentialdecay of a single qubit’s coherences, the creation of quantum correlations between twoqubits or the appearance of steady-state coherence in light-harvesting complexes.

3

Chapter 1. Introduction

1.3 Organization of the thesis

In order to serenely face the aforementioned modular dynamical semigroup (MDS)approach, we present in Chapter 2 the microscopic derivation leading under the Born-Markov approximation to the Bloch-Redfield master equation (BRME) in the weak-coupling regime (WCR), i.e. for a finite but small system-reservoir coupling, and to theLindblad-Davies master equation (LDME) in the weak-coupling limit (WCL). We alsodiscuss a series of other microscopic derivations such as the coarse-grained masterequation (CGME) and the ergodicity property. Thereafter, we summarize the axiomaticformulation of open quantum system giving the Lindblad-Kossakowski master equa-tion (LKME) and the notion of quantum detailed balance. With all these tools, weaddress in Chapter 3 the MDS formalism and expound how to recover the LDME as wellas three different nonlinear thermodynamic master equations (NTMEs). In Chapter 4 weexplain how to discretize, implement as well as linearize these nonlinear equations. Wealso discuss the extension of the standard Born-Markov time-scale separation based ona dynamical coarse-graining time, permitting a continuous transition from the WCR tothe WCL. In Chapter 5 we explore the fluctuation-dissipation theorem (FDT) as well asthe Green-Kubo Formula (GKF) from the renewed perspective provided by the MDS.We apply these two theorems to study the linear response of a single qubit and checkat the same time the validity of the postulated NTMEs. We begin with the FDT andshow that beyond a temperature dependent threshold one can generate an ultralongcoherence. This effect arises through the emergence of two distinct decoherence timesassociated to a biexponential decay of the coherence and a non-Lorentzian susceptibilityprofile. Thereafter, we use the GKF to compute the transport coefficients using Fourier’slaw. Using these results, we demonstrate that it is generally not enough that a masterequation generates a MDS to ensure that it correctly describes a given physical setup.In fact, one of the NTME cannot be used in the WCL at low temperatures. We explainwhy and provide the appropriate regime for this equation. By the same token, we showthat it can be used to avoid the position diffusion’s fictitious effect present in Lindbladequations modeling the Brownian motion of a free particle. We end the chapter byproviding an extension of the standard formulation of nonlinear susceptibilities. InChapter 6 we study the notion of coherence and entanglement for a two qubit system. Weshow that one can entangle a bipartite system initially in a separable state in the WCLand in the WCR as well as produce an asymptotic entanglement for systems composedof two identical atoms. We also provide general strategies to create coherence and entan-glement in the WCR as well as to prolong the survival of an initially entangled bipartitesystem. Afterwards, we discuss the interrelation between coherence and entanglementand show that the initial phases affect their decay. Finally, we study in Chapter 7 theenergy transport through a light-harvesting complex (LHC) near and far-away fromequilibrium. We show that the dynamical and steady coherence can either enhance orhinder the energy transport of natural and artificial systems depending on the choice oftheir underlying parameters.

4

2 Traditional master equations tostudy open quantum systems

To discuss open quantum systems (OQS), two main strategies can be adopted [20]:perform a microscopic derivation “from first principles” or use an axiomatic formulation.To begin with, we would like to go through the microscopic derivation of the BRMEmodeling the dynamics of a small quantum system weakly coupled to a large reservoir.To do so, one usually projects out the degrees of freedom of the reservoir and appliesa perturbative expansion followed by successive temporal coarse-graining. We willbriefly go through these steps in such a way that the BRME can be connected to theLDME when the system-reservoir coupling strength vanishes in the long time limit.Our aim is not to redo all the calculations but provide a sufficient microscopic contextto afterwards address the MDS formalism [77, 78] using the elements presented inthis section (coupling operators, spectral functions, eigenoperators,. . . ). We also leaveaside the mathematical technicalities of the various calculations (but provide referencesfor them when possible) and focus on the physics more relevant in our case since weultimately tend towards a “thermodynamical modeling” of open quantum systems. Wealso take advantage of this “excursion” to point out three physical approximations:

• the system-reservoir factorization,

• the local-time assumption,

• the time-averaging procedure.

They all depend on the fact that the system is weakly coupled to the reservoir. Onlythe first two are necessary to get BRME, which is obtained in the WCR, whereas allof them are required to reach the LDME arising in the WCL. On a pragmatic basis,the WCR is said to hold for small but finite system-reservoir couplings whereas theWCL requires a vanishing system-reservoir coupling. Indeed, only the WCL providesmaster equations whose density matrix remains positive for all times. Other microscopicderivations can lead to Lindblad equations, such as the singular coupling limit (SCL) and

5

Chapter 2. Traditional master equations to study open quantum systems

stochastic limit (STL), or to non-Lindblad equations. Because these three approximationsare interdependent, applying one of them without the others can lead to unphysicalresults (i.e. non-positive density matrices) for certain choices of parameters, typically theinitial state. Thereafter, we summarize an axiomatic formulation of the Lindblad equationand briefly explain the concepts of complete positivity and quantum detailed balance.All these notions will be of importance later on so we decided to present them here in acoherent manner to be as self-contained as possible. A summary of the different masterequations studied in the present work is given in the Table 3.1.

To obtain the various master equations we mainly follow the textbooks [20, 21, 23] aswell as the tutorial [85] whereas we rely on works [37, 86] to consider multiple reservoirs.Finally, to present the notion of complete positivity, we mainly use the two reviews[87, 88]. An expert in open quantum system could get the impression that we misssome topics in this overview: we do indeed, but our aim is to only provide the toolsrequired for our forthcoming developments as already mentioned. For more elegantcalculations (but yet more tedious) to reach the LDME, one could directly look at Davieswork [32, 19]. For a brief review of Davies method, as well as some refinements, onemay look at the Secs. I-II of [35] as well as the paper [89] or Sec. 6.3 of the book [23] bothdiscussing dynamical coarse-graining kind of procedures.

2.1 The microscopic approaches

2.1.1 Exact integro-differential master equations

To model the dynamics of a spatially confined quantum system S connected to aninfinitely extended quantum environment taken here as an inexhaustible reservoirR,let use define the following Hamiltonian for the total system

HTot =H0 + HSR (2.1)

with

H0 = HS ⊗ 1R + 1S ⊗ HR (2.2)

HSR =N

∑k

gk (Qk ⊗Φk) . (2.3)

The free Hamiltonian H0 is composed of the Hamiltonian HS of the system and theHamiltonian HR of the reservoir. On the other hand, the interaction Hamiltonian HSRis expressed in terms of N coupling operator Qk and Φk as well as the adimensionalparameter gk = ggk which is provided in terms of the overall coupling strength g tolater take the WCL. If Hermitian, the coupling operator Qk can be seen as a coordinate

6

2.1. The microscopic approaches

of the system monitored by a field Φk of the reservoirs. To describe the state of thetotal quantum system, we consider a density matrix $(t) := $t on a discrete Hilbertspace HTot =HS ⊗HR whose dimension is equal to dSR = dSdR with di =dim(Hi)<∞.The density matrix $t belongs to the space of bounded positive semi-definite operatorswith unit trace which we design as D(HTot) 1. On the other hand, the Hamiltonians HS ,HR and HTot belong respectively to the space of bounded operators B(HS), B(HR) andB(HTot) 2. Hereafter, we work in the natural units kB= h=1.

To deduce the time-evolution of the reduced system S , obtained by eliminating thedegree of freedom of the reservoir, one usually rewrites the von Neumann equation$t =−i[HTot, $t] for the full system into the following exact non-local integro-differentialform in the interaction picture 3

$(I)t = −i[H(I)

SR,t, $(I)t0]−

∫ t

t0

ds Σt,s($(I)s ) (2.4)

whose memory kernel reads

Σt,s(·) = [H(I)SR,t, [H

(I)SR,s, · ]], (2.5)

where $(I)t = eiH0t$te−iH0t is the density matrix of the full system in the interaction picture

depending on the free Hamiltonian H0 given in (2.2). To consider M independent reservoirs,we have to decompose the reservoir Hamiltonian into HR=∑M

j HRjso that HSR becomes

in the interaction picture H(I)SR,t =∑M

j H(I)SRj ,t

with 4

H(I)SRj ,t

=N

∑k

gk (eiHS tQke−iHS t)︸︷︷︸Qk,t

⊗ (eiHRjtΦke−iHRjt)︸︷︷︸Φ(j)

k,t

. (2.6)

The exact formulation (2.4)–(2.6) is better obtained from the Nakajima-Zwanzig-Mori-Argyres time-independent projection operator technique [30, 31, 91, 92]; for a detailed ex-planation see Sec. 1.1 of [93] or Sec. 5.1–2 of [23]. In general, the projection operator Pis chosen so that it projects the state of the total system $ onto a tensor product free ofany correlation between the system and the reservoir, i.e. P$= trR($)⊗ σ with σ a fixedreservoir state (typically a Gibbs state) and trR(·) the partial trace over the reservoir.This ansatz is, however, only true if these correlations are small and else they have to

1It corresponds to a Banach space of trace class operators associated to a trace norm ||A||1 which is givenby the sum of the eigenvalues of

√AA† (see e.g. Sec.1.1 of [23] or Sec. II A of [90]).

2It is a Banach space composed of all linear and bounded operators associated to an operator norm||A||∞ given by the largest eigenvalue of

√AA† (see e.g. Sec.1.2.1 of [21] or Appendix A of [90]).

3To obtain it, one formally integrates the von Neumann equation in the interaction picture for the fullsystem $

(I)t =−i[H(I)

SR,t, $(I)t ] leading to the solution $

(I)t = $

(I)t0− i∫ t

t0ds[H(I)

SR,s, $(I)s ] which has to be inserted

back into the von Neumann equation. We follow here the calculation presented in Sec. 3.3.1 of [20], but wecould have also used the more accurate derivation proposed in the tutorial [85].

4Qk and Φk can always be taken as self-adjoint without loss of generality, see Sec. 5.2.1 of [23].

7


be taken into account, for example, with a correlated projection superoperator [94] or astochastic Liouville equation [95, 96, 97]. Moreover, to treat a reservoir changing on asimilar time-scale as the system one could use the Picard-Willis time-dependent projectionoperator leading to coupled system-environment equations for factorized [98, 99, 100] orcorrelated [101] system-reservoir states.

2.1.2 Time scales separation

Before we go any further, let us look at the main timescales which will be involved inthe forthcoming calculations. The first one is the Heisenberg time scale tH ticking for thedynamics of the free system (i.e. in absence of an interaction with the reservoir). Thesecond one is the reservoir correlation time scale tR indicating the time range on which thereservoir correlation function decays. The third one is the system time scale tS associatedto the dynamics of the system interacting with the reservoir. It is sometimes designed asa dissipative time scale because the system displays an irreversible dynamics (althoughthe dynamics of the total closed system is reversible). In this sense, tS is associated to theoverall lifetime of the system 5. On the other hand, one also refers to tS as a macroscopicor observation time scale because it is the time interval over which one can follow thetime-evolution of the macroscopic observable of interest.

To obtain the LDME for the reduced system S , mainly two time scales separation arerequired [20]

tR � tS, tH � tS. (2.7)

The first condition is necessary to perform the Markov approximation. It indicates thatthe excitation of the reservoir decays on the time scale tR which cannot be resolvedon the time scale of the system tS. In this regime, the back-action of the reservoironto the system is negligible (but this doesn’t mean that the system cannot createan excitation in the reservoir) and the reservoir is in a steady-state from the point ofview of the system. This explains why one can perform a temporal coarse-grainingon the memory kernel to eliminate the non-Markovian effects. The second conditionis required to apply the time-averaging procedure, or the secular approximation (SA),eliminating the fast oscillating contributions taking place at the time scale tH. Fora discrete energy spectrum tH is roughly given by the inverse of the largest energysplitting ∆max. Note that a system with closely spaced energy levels weakly coupled to areservoir 6 possesses a tH ∝1/∆max∼ tS ∝1/g2 contradicting the second condition in (2.7)

5For a two level system it is associated to two decay times, namely the energy relaxation time T1 (i.e.decayof the population) and the decoherence time T2 (i.e. decay of the phase coherence between states) fulfillingthe relation T2 ≤ 2T1 in the WCL [102].

6The irreversible contribution obtained through a perturbative scheme is, in general, of the order g2 (allcontributions of higher order are ignored) so that the associated time scale tS is proportional to 1/g2.

8


implying that the usual SA, as well as the rotating-wave approximation (RWA) 7, areinvalid in this regime [104]. This issue is even more pronounced for continuous energyspectrum displaying a vanishing energy splitting ∆max∼0 as pointed out by DeRoeckand Spehner [105]. In the next section, we use the two conditions (2.7) to perform thesuccessive time coarse-grainings.

2.1.3 Derivation under a weak system-reservoir coupling

We take for granted that it is possible to prepare at time t0 an initial state displaying nocorrelations between the system and the reservoir 8, i.e. they are unentangled, so that$(I)

t0=ρ(I)

t0⊗ σ(I)

t0with ρ(I)

t0and σ(I)

t0the initial density matrix of the system and the reservoir,

respectively. However, correlations will build-up as time goes on through the interactionHSR since

$(I)t = ρ(I)

t ⊗ σ(I)∞ + ξ(t), (2.8)

where ρ(I)t ∈D(HS) is the density matrix of the system in the interaction picture while

σ(I)∞ =σ(I)

β1⊗. . .⊗σ(I)

βM∈D(HR1

⊗ . . .⊗HRM) is the product state in the interaction pictureof the M independent reservoirs’ steady-state σ(I)

β j, reached in the long time limit, at

the inverse temperature β j for j = 1, . . . , M. We can simplify (2.8) by noting that theindividual steady-states can be rewritten as σ(I)

β j= eiHRjtσβ j

e−iHRjt =σβ jsince

σβ j= e−β j HRj /tr(e−β j HRj ) (2.9)

and [σβ j, HRj

] = 0 implying that the steady state of the reservoir becomes σ(I)∞ = σ∞.

For a non-thermal reservoir, such a simplification would not necessarily be possibleand would jeopardize the upcoming calculations (mainly at the level of the two-timecorrelation function). The correlations matrix ξ(t)=O(H(I)

SR,t) in the decomposition (2.8)is an inhomogeneous contribution arising from the system-reservoir correlations 9. Weare now ready to proceed towards the Born-Markov approximation (BMA) .

7More precisely defined, the time-averaging is equivalent to the SA but it is different from the RWA.Yet, the RWA and SA both eliminate fast oscillations but at two different levels of description: the RWA isapplied on the interaction Hamiltonian of the full system, i.e. before eliminating the degrees of freedom ofthe reservoir, while the SA as well as the time-averaging is applied in the interaction picture of the reducedsystem, i.e. after having performed the traced over the reservoir. The RWA/SA and their link with thenon-Markovian regime is discussed in [103]; see e.g. the Hamiltonian (2) therein where the RWA consistsin neglecting the phases leading to the master equation (6) which in the WCL is similar (but not exactlyidentical) to the master equation (7) obtained in the SA.

8During an experiment, a reservoir which remains unchanged after every preparation is not necessarilya reasonable general requirement as pointed-out in [106]. However, for many relevant physical situationsthe factorization assumption is fully consistent [87].

9The choice of the initial states ρ(I)t0

and σ(I)t0

dictates the form of ξ(t) [107]. For example, if the system isinitially in a pure state ξ(t) should be zero to produce a physical time-evolution, i.e. the inhomogeneousterm restricts the states which the system can take initially. For further explanations, see e.g. [108, 109].

9


First step: the factorization assumption We apply the system-reservoir factorizationby only taking into account terms up to the second order when replacing (2.8) in the mas-ter equation (2.4)–(2.5). One can then trace over the reservoirs and use trRj

(σβ jH(I)SRj ,t

)=0yielding after a change of variable s→ s− t and few calculations 10

ρ(I)t = −

M

∑j

∫ tc

0ds trRj

([H(I)SRj ,t

, [H(I)SRj ,t−s, ρ(I)

t−s ⊗ σβ j)]])

, (2.10)

where we shifted the boundary of the integral by −t0 to define the integration cut-offtc = t− t0. In the Born regime, the independent reservoirs remain for the system in thesteady-state σβ j

since all time dependency is included in the higher-order contributionsO(g3) which can be neglected in the WCR (i.e. a small but finite coupling strength g)for a reservoir with an infinite number of degrees of freedom. Thus, the system and thereservoir remain factorized for all t > 0. The method we employed to reach (2.10) is aneffective one and, to formally get it, projection operators should be used (see e.g. [93]).The non-Markovian equation (2.10) is non-local in time, affecting the short as well as thelong time dynamics, since to solve it one has to compute the integral over s requiring alldensity matrices from s=0 to s= tc.

Second step: the local-time assumption This assumption permits to get rid of thementioned retardation effects by imposing ρ(I)

t−s→ρ(I)t requiring that the density matrix

does not significantly change over the integration time in (2.10). This implies that thehistory of the system dynamics stored in the bath is no longer accessible on the timescale tS of the system. This makes sense when the main contribution to the integral takesplace in the interval 0≤ s . tR whose upper bound is much smaller than the time-scaleof the system tS according to our first time-scale separation (2.7). This approximationleads to the so-called Redfield master equation 11

ρ(I)t = −

M

∑j

∫ tc

0ds trRj

([H(I)SRj ,t

, [H(I)SRj ,t−s, ρ(I)

t ⊗ σβ j]])

. (2.11)

This equation corresponds, for example, to the equation (5.20) obtained more formally in[23] by means of projection operators under the same approximations. Note that we didnot fully apply the first time-scale separation provided in (2.7) but rather tR < tc � tS

with tc ∼ ∆max/g2 so that it still contains short-time memory, i.e. the system and thereservoir interact coherently up to the time tc, and is therefore non-Markovian but local

10We can disregard the term∫ t

t0ds trRj ([H

(I)SRj ,t

, ρ(I)t−s ⊗ σβ j )]) by assuming a null mean value for the

reservoir coupling operator trRj (σβ j Φ(j)k, t)=0 since Qk,ttrR(Φ

(j)k, tσβ j )ρ

(I)t − ρ

(I)t Qk,ttrRj (σβ j Φ

(j)k, t) vanishes. This

is clearly not a restrictive statement because we can always rescale the interaction Hamiltonian withcentered coupling operators ∑k Qk ⊗ (Φk − trRj (Φkσβ j )). Moreover, the first term in (2.4) depending on theinitial state vanishes too for the same reasons if the reservoir is initially in an equilibrium state.

11The Bloch-Redfield-Wangsness theory was originally proposed in [110, 111, 112]. For the “modern”formulation which we use hereafter see [113] or Sec. 3.3.1 and equation (3.117) of [20].

10


in time [114]. Note that we encountered so far three kinds of non-Markovianity which mustnot be confused: the first one associated to non-factorized system-reservoir states (2.8),the second one attached to an evolution being non-local in time (2.10) and the last oneexclusively linked to short-time memory effects (2.11).

To clearly separate the system from the reservoir contributions in (2.11), we use theinteraction Hamiltonian (2.6) in conjunction with the spectral decomposition of the timeevolved system’s coupling operator

Qk,t = ∑ω

e−iωt Fk,ω, (2.12)

where the sum runs over the set of energy differences or Bohr frequencies ω of the Hamil-tonian system HS , given in terms of the operator-valued Fourier transform coefficients

Fk,ω = ∑Em,En

En−Em=ω

(Qk)mn |Em〉〈En| , (2.13)

where (Qk)mn = 〈Em|Qk |En〉 are the matrix elements of Qk, depending on the systemHamiltonian eigenstates |Em〉 and eigenvalues Em. In order that all Fk,ω are bounded itis necessary that all system coupling operators Qk are bounded too according to (2.13).Note that Fk,ω is an eigenoperator of the reversible generator −i[HS , ·] so that

[HS , Fk,ω] = −ωFk,ω (2.14)

and F†k,ω = Fk,−ω according to the definition (2.13). Consequently, the Redfield master

equation (2.11) becomes

ρ(I)t = −

M

∑j

∫ tc

0ds trRj

(H(I)SRj ,t

H(I)SRj ,t−s(ρ

(I)t ⊗ σβ j

)− H(I)SRj ,t−s(ρ

(I)t ⊗ σβ j

)H(I)SRj ,t

+ h.c.)

= −M

∑j

N

∑k,l

∫ tc

0ds gkgl

(Qk,tQl,t−sρ

(I)t trRj

(σβ jΦ(j)

k,t Φ(j)l,t−s)︸︷︷︸

C(j)k,l (t,t−s)

−Ql,t−sρ(I)t Qk,t trRj

(Φ(j)l,t−sσβ j

Φ(j)k,t )︸︷︷︸

C(j)k,l (t,t−s)

+h.c.)

= −M

∑j

N

∑k,l

∑ω1,ω2

e−i(ω1+ω2)t C(j)k,l (tc,−ω2)

(Fk,ω1 Fl,ω2 ρ(I)

t − Fl,ω2 ρ(I)t Fk,ω1

)+ h.c.,

(2.15)

where the shorthand notation h.c. stands for the Hermitian conjugate. In the previouscalculations we have used the two-time correlation functions

C(j)k,l (t, t− s) = gkgl trRj

(σβ jΦ(j)

k,t Φ(j)l,t−s) = gkgl trRj

(σβ jΦ(j)

k, 0Φ(j)l,−s) = C(j)

k,l (−s). (2.16)

11


The simplification C(j)k,l (t, t− s) = C(j)

k,l (−s) is possible because the correlation function ishomogeneous in time since σβ j

, which commutes with e±iHRjt, is the steady-state of thejth reservoir. We can then compute the Fourier transform

C(j)k,l (tc,−ω) =

∫ tc

0ds eiω(−s) gkgl trRj

(σβ jΦ(j)

k, 0Φ(j)l, s) =

∫ tc

0ds ei(−ω)s C(j)

k,l (s), (2.17)

where we used the change of variable −s→ s. We now decompose the Fourier trans-formed two-time correlation functions

C(j)k,l (tc, ω) =

12

c(j)k,l (tc, ω) + is(j)

k,l (tc, ω) (2.18)

by using

c(j)k,l (tc, ω) = C(j)

k,l (tc, ω) + C(j),∗k,l (tc, ω) (2.19)

s(j)k,l (tc, ω) =

12i

(C(j)

k,l (tc, ω)− C(j),∗k,l (tc, ω)

)(2.20)

to produce two Hermitian time-dependent spectral matrices c(j)(tc, ω) = (c(j)k,l (tc, ω))

and s(j)(tc, ω) = (s(j)k,l (tc, ω)). The term c(j)(tc, ω) contributes to the irreversible time-

evolution and is often designed as a damping or friction matrix. On the other hand,s(j)(tc, ω) participates to the reversible one, solely rescaling the system Hamiltonian,and is known as the Lamb-shift (L.S.). The time-dependent spectral function c(j)

k, l(tc, ω)

associated to the jth reservoir leads to a richer relaxation and dephasing behavior thanits time-independent counterpart, obtained for tc→∞, and influences the coherence andpopulation beating of photosynthetic complexes at short-times [115]. Going back to thelast line of (2.15), we change the sign of the second Bohr frequency ω2 and explicitlywrite down the Hermitian conjugate as well as interchanging the summation indicesk↔ l to regroup the different terms and dropping the L.S. so that finally

ρ(I)t = −

M

∑j

N

∑k,l

∑ω1,ω2

e−i(ω1−ω2)t c(j)k,l (tc, ω2)

[Fk,ω1 F†

l,ω2ρ(I)

t − F†l,ω2

ρ(I)t Fk,ω1

+ ρ(I)t Fk,ω2 F†

l,ω1− F†

l,ω1ρ(I)

t Fk,ω2

] (2.21)

To get an explicit expression for the spectral function one has to specify the nature of thereservoir. For a bosonic bath, one can show that the time-dependent spectral function

12


reads 12

c(j)k,l (tc, ω) =

4π

δk,l

∫ ∞

0dω Jk(ω)

(δtc(ω−ω) (nβ j ,ω + 1) + δtc(ω+ω) nβ j ,ω

)(2.22)

provided in terms of an antisymmetric spectral density Jk(ω), which has to be specified,as well as the bosonic distribution nβ j ,ω =1/(eβ jω − 1) and the sinc function

δtc(ω±ω) =sin((ω± ω)tc)

ω± ω= sinc((ω±ω)tc). (2.23)

The index k of the spectral density Jk(ω) indicates that it carries the coupling strength gk

appearing in the interaction Hamiltonian (2.3). Note that the time-dependent spectralfunction (2.22) can take negative values at short times which can spoil the positivityof the density matrix. However, in the long-time limit the spectral function (2.22)behaves like a time-independent one and is no longer negative. The third step of theBM approximation requires a fast decay of the bath correlations, so that the system and thereservoir only interact in an incoherent manner. This means that for tR � tc = t− t0

the cut-off in the integral (2.11) has to be sent to infinity using t= τ/g2, recalling thatgk = ggk in the spectral function (2.17), with τ fixed and g→0 leading to a delta-Dirac

limtc→∞

sinc((ω±ω)tc) =π

2δD(ω± ω). (2.24)

Accordingly, we get with nβ j ,ω =1/(eβ jω − 1) the standard bosonic spectral function

c(j)k,l (ω) = δk,l 2(nβ j ,ω + 1) Jk(ω) (2.25)

so that we have now a purely Markovian dynamics. Moreover, it is necessary that allelements of the spectral matrix are bounded |c(j)

k, l(ω)|<∞ 13. In practice, this is achievedby including a cut-off for large frequencies (see e.g. Sec. III of [117] for the case of abosonic bath). Note that, even without specifying the exact nature of the reservoirs, onecan show that the spectral function

c(j)k,l (ω) =

∫ ∞

−∞dt eiωt gkgl trRj

(Φ(j)k, tΦ

(j)l, 0σβ j

) (2.26)

12To obtain this spectral function, one can follow the Sec. III of [115] using the identity∫ tc

0 dt ei(ω±ω)t =sin((ω ± ω)tc)/(ω ± ω). Note that such an integral is ill-defined and for a meaningful (but not mathe-matically exact) treatment one should look at Sec. 5.2.4 of [23]. The origin of the factor δk,l for a bosonicreservoir (i.e. radiation modes) comes from the integration over the solid angle when computing the frictionmatrix as shown in Eq. (3.200)–(3.201) of [20]. For a non-diagonal friction matrix, see the simple modelproposed in the Sec. 5.2.5 of [23] or for a more concrete microscopic model, including a spatial dependencein the off-diagonal elements, see the equation (36) in [116]. The prefactor 4/π ensures that in the limittc =∞ we recover the usual time-independent spectral function (2.25) for a bosonic bath using (2.24).

13More precisely, one should check Davies theorem requiring the existence of∫ ∞

0 dt|c(j)k, l(t)|(1− t)ε <∞

for some ε>0 as pointed out in Sec. 5.2.3 of [23]. Moreover, it is important to ensure that all higher ordercorrelation functions can be rewritten as sums of two-point correlation functions, i.e. the reservoir has to be“quasi-free”, as highlighted in Remark 3.7 of [87].

13


observes the microscopic detailed balance

c(j)k,l (−ω) = eβ jω c(j)

k,l (ω), (2.27)

with β j the inverse temperature of jth reservoir, arising from the Kubo-Martin-Schwinger(KMS) property [118, 119]. The time-domain version of the KMS property reads

trRj(Φ(j)

k, tΦ(j)l, 0σβ j

) = trRj

(Φ(j)

k, 0Φ(j)l, t + iβ j

σβ j

). (2.28)

The two previous properties are computed in the equations (5.38)–(5.39) of [23] or,more formally, in the Appendix D of [120]. They are crucial to ensure that the systemconverges to the correct steady-state. For long times, the KMS condition is also en-sured for the time-dependent version of the spectral function (2.22) converging to thecorrect steady-state. Note also that for a single reservoir, or multiple reservoirs withequal t inverse temperatures β j = β, the system ends up in the β-KMS or Gibbs stateρβ = e−βHS/tr(e−βHS ). On the other hand, for multiple reservoirs at different inversetemperatures β j 6= β the system converges to a non-equilibrium steady-state (NESS),which we design hereafter as ρ+, depending on the nature of the relaxation processcontrary to the Gibbs state ρβ. Indeed, each reservoir drives the system towards his“own” equilibrium state so that the final stationary state is a complicated mixture oftheir mutual action (i.e. it depends on the individual spectral functions c(j)

k,l (ω), the eigen-operators Fk,ω and the structure of the master equation). In general, the NESS is uniqueand independent of the initial condition thanks to the KMS condition 14.

We can now go back to the Schrödinger picture by computing e−iHS tρ(I)t eiHS t with the

help of the properties

eiHS t Fω,k e−iHS t = e−iωtFω,k (2.29)

eiHS t F−ω,l e−iHS t = eiωtF−ω,l (2.30)

leading for the first term in the last line of (2.21) to

e−iHS t((e−iω1tFk,ω1)(e

iω2tFl,−ω2)ρ(I)t

)eiHS t

= (e−iHS teiHS t)Fk,ω1(e−iHS teiHS t)Fl,−ω2(e

−iHS tρ(I)t eiHS t)

= Fk,ω1 Fk,−ω2 ρt = Fk,ω1 F†k,ω2

ρt

(2.31)

and doing the same for the other terms. Note that the reversible contribution naturally

14At least in the WCL, leading to the LDME, and assuming that the ergodic property holds [37, 86]. TheLDME is later given in equation (2.38) while the ergodic property is discussed in Sec. 2.1.5.

14


arises when going back to the Schrödinger picture with e−iHS tρ(I)t eiHS t leading to

e−iHS t(

∂t(eiHS tρte−iHS t))

eiHS t = ρt + i[HS , ρt]. (2.32)

Finally, one obtains the Markovian Bloch-Redfield master equation [112, 91, 20], laterabbreviated with BRME, in the form ρ = −i[HS , ρ] + ∑M

j ∑Nk,lR

(j)k,l ρ whose individual

irreversible contributions in the eigenoperator formulation read

R(j)k,l ρ = −1

2 ∑ω1,ω2

c(j)k, l(ω2)

(Fk,ω1 F†

l,ω2ρ− F†

l,ω2ρFk,ω1 + ρFk,ω2 F†

l,ω1− F†

l,ω1ρFk,ω2

). (2.33)

By construction, the friction matrix c(j)(ω) = (c(j)k, l(ω)) associated to the jth reservoir

is positive definite so that we can diagonalize it to simplify the previous expression.Therefore, we use an appropriate unitary transformation W, i.e. composed from theeigenvectors of the friction matrix c(j)(ω). Thus, Wc(j)(ω)W† =diag(h(j)

1 (ω), . . . , h(j)N (ω))

leads to the diagonal matrix h(j)(ω)=(h(j)i (ω)) given by the non-negative eigenvalues

h(j)i (ω) of the friction matrix c(j)(ω). Thereafter, we introduce following Sec. IV.A of [121]

a new set of eigenoperators 15 rescaled by the aforementioned unitary transformation16

Ai,ω = ∑k

WikFk,ω (2.34)

which satisfies an eigenoperator equation of the type (2.14). Altogether, the equations(2.33)–(2.34) lead to the more common formulation ρ=−i[HS , ρ] + ∑M

j ∑Ni R

(j)i ρ with

R(j)i ρ = −1

2 ∑ω1,ω2

h(j)i (ω2)

(Ai,ω1 A†

i,ω2ρ−A†

i,ω2ρAi,ω1 + h.c.

). (2.35)

The BRME for a harmonic oscillator corresponds to the equation obtained by Agarwalcontaining non-rotating terms as pointed out by Kohen and co-workers (compare equa-tion (27) in [63] with equation (2.49) in [122]). It is clear that the BRME (2.33) and (2.35)do not preserve the positivity of the density matrix in general 17 although we neglectedthe time-dependency of the spectral function (2.22). Despite this drawback, it has beenused to describe, among others, light-harvesting complexes [59].

15We use c(j)k, l =∑i(W†)ki h

(j)i Wil , where we omitted the ω dependency of the friction matrix, such that for

first term in (2.33) we get ∑k,l c(j)k, l Fk,ω1

F†l,ω2

ρ = ∑i h(j)i (∑k WikFk,ω1

)(∑l Wil Fl,ω2 )†ρ = ∑i h(j)

i Ai,ω1 A†i,ω2

ρ.16To simplify the notations, we omitted the j index of W(j)

ik specifying that the matrix W(j) is associated tothe j reservoir since it is obtained from c(j)(ω). For the same reason, we used Ai,ω instead of A(j)

i, ω exploiting

the fact that the index j is provided for the spectral function h(j)i (ω) of the BRME equation (2.35).

17Various ad-hoc strategies exists to stay positive. For example, the BRME remains positive for slippagein the initial conditions [82, 123]. However, this leads to physical inconsistencies in the presence ofentanglement [123, 124]. Alternatively, one can use a time-dependent friction matrix to stay positive, atleast for two-level systems [125]. Finally, a partial SA can also ensure positive eigenvalues [56, 59]. For adiscussion of the “conventional Markov limit” (i.e. prior to the SA) see e.g. [126, 127] or Sec. 3.1. of [128] .

15


Third step: the secular approximation By applying the secular or random phaseapproximation, one ensures that the density matrix remains positive at all times andobtains the famous LDME [32, 33] sometimes denominated as the secular Redfield masterequation. To do so, we have now to go back to the last line of (2.15) and consider atime-rescaled interaction picture with t= τ/g2 so that τ = tg2 is finite for t∼∞ (longtime-limit) and g∼0 (weak-coupling limit). Under this time coarse-graining, only theslow dynamics survives implying that the fast oscillating phases or off-resonant termse−i(ω1−ω2)τ/g2→ δKr(ω1 − ω2) are eliminated in the last line of (2.15) allowing to writeω1=ω2=ω (i.e. only one of the two sums remains). To perform this approximation, itis crucial that [129]:

(i) there is no energy level degeneracy in the system Hamiltonian HS , i.e. looking atits eigenvalues or eigenenergies Em 6=En for m=n.

(ii) the energy gaps or frequencies computed from the set of eigenenergies differencesωnm =En − Em are not allowed to be degenerated, that is ωmn 6=ωrs for mn 6= rs 18.

More precisely, the problem arises from near-degeneracies when taking the long timeand weak-coupling limit as described beforehand leading from the BRME to the LDME.Note that applying a proper time-averaging procedure such as the one proposed byDavies [32], see the formulation (2.41) provided later on, permits to avoid these diffi-culties. The condition (ii) is sometimes designed as the Liouville or Liouvillian degenera-cies [130, 131, 132] since the energy gaps characterize the nature of the Liouvillian L ofa master equation, such as the BRME (2.33) or (2.35), written in the form ρ=Lρ. Suchdegeneracies most frequently arise for systems with densely packed energy levels andmainly matter at high temperature [131].

To go then back to the Schrödinger picture, we have to compute

e−iHS t (Fk,ωFl,−ωρ(I)t

)eiHS t

= (e−iHS tFk,ωeiHS t)(e−iHS tFl,−ωeiHS t)(e−iHS tρ(I)t eiHS t)

= (e−iωtFk,ω)(eiωtFl,−ω) ρt = Fk,ωF†l,ω ρt

(2.36)

and doing the same for the other terms. We thereby obtain ρ=−i[HS , ρ] + ∑Mj ∑N

k,l D(j)k, l ρ

depending on the following irreversible contributions

D(j)k, lρ =∑

ω

c(j)k, l(ω)

(F†

l,ωρFk,ω −12{Fk,ωF†

l,ω, ρ})

, (2.37)

whose friction matrix c(j)(ω)=(c(j)k, l(ω)), later designed as the Kossakowski matrix, is posi-

tive definite by construction similarly to the BRME. Thus, we are allowed to diagonalize

18These frequencies are the ones used in the two sums over ω1 and ω2 of the irreversible contributions ofthe BRME (2.33) or (2.35).

16


it to obtain the Lindblad-Davies master equation in its traditional form, later designed asthe LDME, being ρ=−i[HS , ρ] + ∑M

j ∑Ni D

(j)i ρ whose irreversible contributions read

D(j)i ρ = ∑

ω

h(j)i (ω)

(A†

i,ωρAi,ω −12{Ai,ω A†

i,ω, ρ})

(2.38)

expressed in terms of the Lindblad eigenoperators Ai,ω. The term A†i,ωρAi,ω is responsible

for the gain whereas {Ai,ω A†i,ω, ρ} is responsible for the loss. Contrary to the BRME, the

population and the coherence are decoupled from each other taking into account thatthe Hamiltonian is non-degenerate 19. Since the spectral function

h(j)i (ω) =

{e(j)

i (ω) , ω ≥ 0

a(j)i (−ω) , ω < 0,

(2.39)

can be expressed in terms of an emission e(j)i (ω) and absorption a(j)

i (ω) rates, satisfying themicroscopic detailed balance e(j)

i (ω)= eβ jω a(j)i (ω), we can rewrite (2.38) as

D(j)i ρ= ∑

ω≥0e(j)

i (ω)

(A†

i,ωρAi,ω−12{Ai,ω A†

i,ω, ρ})+ ∑

ω>0a(j)

i (ω)

(Ai,ωρA†

i,ω−12{A†

i,ω Ai,ω, ρ})

(2.40)

We have obtained the LDME by neglecting the fast oscillating contributions; by doing sowe have implicitly performed Davies’ time-averaging [32] on the BRME. In brief, thetime-averaging procedure can be summarized as follows

D(j)i = lim

T→∞

1T

∫ T/2

−T/2dq U−q(R(j)

i (Uq(ρq))), (2.41)

where Uq(·)= eiHS q(·) e−iHS q is the reversible evolution superoperator 20 while D(j)i and

R(j)i are, respectively, the LDME (2.38) and BRME (2.35) irreversible contributions. This

procedure is discussed in Sec. 5.2.3 in [23] or Sec. I-II of [35]. For most applications asingle coupling operator per reservoir is sufficient, so let us rewrite the LDME (2.38) as

ρ=−i[HS , ρ] +M

∑j

∑ω

hj(ω)

(A†

j,ωρAj,ω −12{Aj,ω A†

j,ω, ρ})

(2.42)

with a diagonal friction matrix characterized by hj(ω) =∫ ∞−∞ dt eiωt g2

j trRj(Φ(j)

j, t Φ(j)j, 0σβ j

)

and the eigenoperators Aj,ω =∑mn(Qj)mn δKr(En − Em, ω) |Em〉〈En| for the jth reservoir.

19A degenerate Hamiltonian can still couple them. On the other hand, note that near-degeneracies areproblematic to perform the secular approximation as previously mentioned.

20The super-prefix is used to distinguish a superoperator from the operators upon which it acts andnothing more (it is a typical nomenclature in the physics literature).

17


2.1.4 Alternative microscopic derivations

Singular coupling limit Various microscopic derivations lead to Markovian masterequations. The most known is the Lindblad equation obtained in the singular couplinglimit (SCL) 21. It relies on factorized states as the WCL but requires another rescaling,namely of the reservoir Hamiltonian HR→ς−2HR to accelerate the decay of the reservoircorrelations (i.e. tR→0), so that the two-time correlation functions (2.16) tend towardsa Dirac-delta C(j)

k,l (t, t − s) → ( 12 c(j)

k,l + is(j)k,l )δ(s) associated to a flat spectral function 22,

whereas the decay of the time scale of the system tS remains unaffected. To avoid anon-zero effect when taking the limit ς→0 one has to rescale the interaction HamiltonianHSR→ ς−1HSR too. A detailed calculation can be found in Sec. 3.3.3 of [87] as well asSec. 5.3 of [23] leading to the Lindblad singular master equation (LSME) 23

ρ = −i[HS , ρ] +M

∑j

N

∑k,l

c(j)k, l

(Qlρ Qk −

12{QkQl , ρ}

)(2.43)

whose irreversible contribution is independent of the Hamiltonian. To obtain a doublecommutator form, one has to diagonalize the positive definite friction matrix c(j)=(c(j)

k, l)

with the unitary W yielding the matrix h(j)=Wc(j)W† =diag(h(j)1 , ..., h(j)

N ) and rescale thecoupling operators Ci =∑k WikQk. Taken altogether, we finally obtain

ρ = −i[HS , ρ]− 12

M

∑j

N

∑i

h(j)i [Ci, [Ci, ρ]]. (2.44)

The two LSME (2.43) and (2.44) drive the system towards the fully mixed (or micro-canonical) state contrary to the BRME (2.33)–(2.35) or the LDME (2.37)–(2.38) leadingto a Gibbs (or canonical) state. This is due to the fact that we imposed a Dirac-deltatwo-time correlation (i.e. white noise) associated to an environment which is essentiallyclassical. Thus, a bosonic (or fermionic) reservoir only makes sense in the SCL for infinitetemperatures (assuming an appropriate spectral density is taken) in order to respectthe KMS condition. The SCL is therefore much more limited than the WCL, acceptingnon-flat spectral functions, and cannot lead to a NESS forbidding to study transport phe-nomena. On the other hand, the LSME is valid for strong couplings 24 since we considera ”slow motion“ with the SCL, namely the free motion (i.e. the reversible contribution)

21See either Sec. 3.3.3 in [20], Sec. 1.3.5 in [21], Sec. 5.3 in [23] or Sec. 3.3.3 of [87].22See the equation (5.44) and following in Sec. 5.3 [23] to obtain it more formally; the main point is

that in the two-time correlation function the scaling factors originating from the reservoir and interactionHamiltonians compensate each other in the limit ς→0 to lead to the Dirac-delta. Presently, it is sufficient tonotice that the element c(j)

k,l =∫ ∞

0 du c(j)k,l (u) is equal to c(j)

k,l (ω=0) earlier obtained in (2.26).23Compare the equation (141) of [87] with the second line of our calculation (2.15).24But only for a white noise/classical type of environment; for a colored-noise/quantum kind of envi-

ronment (yielding notably towards non-flat time-dependent spectral functions) one would end-up with anon-Markovian equation.

18


changes on a similar time scale as the interaction with the reservoir (i.e. the irreversiblecontribution), that is tH ∼ tS� tR∼ 0 [87]. Moreover, because the irreversible part ofthe LSME is independent of the system Hamiltonian HS , it can handle external time-dependent perturbations which are slow (i.e. adiabatic) as well as fast (i.e. non-adiabatic)contrary to the LDME which is limited to adiabatic perturbations 25. Consequently, itcan be employed to study the famous quantum Zeno effect generated by a continuousmeasurement process (see Sec. 3.5.2 in the textbook [20]). Finally, it is also well definedfor extended systems contrary to the BRME or the LDME since the decomposition intoeigenoperators (relying on a discrete energy spectrum) is not used [105]. Note thatone can also obtain a low-density limit Lindblad equation for a N-level quantum systemsurrounded by an ideal gas of bosons or fermions (see e.g. Sec. 1.3.3 of [21]).

Non-rotating-wave formulation To discuss systems which are strongly damped aswell as having closely spaced energy level, one can use the factorization assumption inconjunction with van Kampen’s cumulant expansion. See the equation (2.17) in [135]expressed in terms of eigenoperators (2.14)–(2.15) therein being equivalent to ours (2.13)–(2.14). Taking the same singular limit as for the LSME (2.43)–(2.44), i.e. a flat spectralfunction, one obtains from it the non-rotating-wave singular master equation (NRSME) 26

ρ = i[HS , X]− 12

(γ

β[Q, [Q, ρ]] +

γ

2[Q, {[Q, HS ], ρ}]

)(2.45)

which is not of Lindblad type and therefore doesn’t preserve the positivity of ρ in general.At high temperature (i.e. β�1) this equation is equal to the LSME (2.44) for Qi→Q andh(j)

i =γi/β j→γ/β with γ a friction coefficient embedding the system-reservoir couplingstrength. The NRSME has been used to treat a two-level system with narrowly spacedenergy levels where the ”optical Lindblad equation“, i.e. the LDME, does not hold (seeSec. III of [135]). On the other hand, it was applied to describe a damped free Brownianparticle (see Sec. IV of [135]) whose continuous energy spectrum (since the particle isnot confined) forbids to model it with the LDME as discussed in the Appendix 9.1.

High temperature limit To describe the motion of a free Brownian particle or a har-monic oscillator at high temperature, the Caldeira-Leggett master equation (CLME)is well suited. It is usually derived using the Feynman-Vernon theory 27. Only little

25Since the Hamiltonian enters in the irreversible part through the eigenoperators (2.13) under theMarkovian and secular approximations in the long time limit as pointed out in e.g. [133, 120, 134].

26We adapted the master equation (1.4)–(1.5) obtained in [135] according to our own interaction Hamilto-nian (2.3) but only for one reservoir and a single coupling operator for a better match.

27Originally, it was obtained in [136]. See the master equation formulation (5.12) therein; it is linked toDekker’s equation (55) obtained in [137] carrying two additional terms. Note that the ”official“ form ofCLME, which most people refer to and that we provide hereafter, can be found in equation (3.410) of the

19


benefit can be gained from repeating here this tedious calculation. In fact, it is by farmore instructive to reach it from the NRSME (2.45) by an appropriate choice of themodeling parameters. To do so, let us take a unique reservoir so that β j→β and a singlecoupling operator taken as the position operator Qj→ X. On top of that, we chooseHS=P2/(2m) + V(X) as system Hamiltonian modeling a particle of mass m confinedin a quadratic potential V(X)= (mω2/2)X2, where P is the momentum operator andω the angular frequency of the free harmonic oscillator. Consequently, the identity[X, HS ]= i(P/m) immediately leads to the CLME

ρ = −i[HS , ρ]− 12

(γ

β[X, [X, ρ]] + i

γ

2m[X, {P, ρ}]

). (2.46)

It can equally be obtained from the Redfield theory for a white noise bath 28 and leads inthe phase space representation to the Klein-Kramer Fokker-Planck equation [139, 66]

∂t f = {Hc, f }xp + Dp ∂p(p f ) + Dpp ∂2p f , (2.47)

where f = f (x, p, t) is the distribution function on the canonical position-momentumphase space (x, p), Hc is the classical Hamiltonian of the system and {Hc, f }xp standsfor the canonical Poisson bracket obtained in the classical limit of the Wigner-Moyalevolution operator 29. The coefficients Dp and Dpp are, respectively, the classical drift anddiffusion coefficients. The correspondence between the NRSME (see the equation (4.2) in[135]) and the CLME exists because they are both valid at high temperatures. Note thatthe CLME, like the NRSME, is not of Lindblad’s form and does not conserve positivity 30

and only leads to the proper Gibbs state at high temperature 31. However, for a freeparticle the Caldeira-Leggett model leads to the correct thermal equilibrium in the longtime limit as shown in the Sec. 3 of [143]. In addition, such a formulation is compatiblewith continuous energy spectrum characterizing a free particle. One can also recover anequation having the same double commutator structure as the LSME (2.44) for a “very”high temperature (i.e. β�1) or a heavy particle (i.e. m�1). For lower temperatures, itis necessary to resort to a generalized Feynman-Vernon influence functional as proposedby Grabert leading to non-Lindblad type of equations [147, 148, 149].

textbook [20] ( Sec. 3.6.1 therein provides a simple derivation of the CLME).28This is shown in Sec. III of [138]. Alternatively, one can take a look at equation (23) of [63] using the

identities [Xρ, X] + [X, ρX]=−[X, [X, ρ]] and [Pρ, X]− [X, ρP]= [X, {P, ρ}].29To switch from the density matrix to the Wigner representation one could use the set of rules presented

in the two tables on p. 4 of [140]; commutators and anticommutators are replaced by partial derivatives.30See the comment about positivity violation on short time-scales prior to equation (6.7) in [141] or in

the letter [142]. To recover Lindblad’s form, one has to add an additional term (γβ/16m)[P, [P, ρ]] havinga small impact on the dynamics at high temperature since β� 1, see equations (3)–(6) in [143]. For ageneralized Lindblad formulation in terms of diffusion coefficients, valid at intermediate temperatures, seethe equations (7.29) in [144] or (29) in [145]. The positivity of this master equation depends on the relationbetween the different diffusion coefficients (forming a Kossakowski matrix) as shown in Sec. 5 of [145].

31See the Sec. 2 of [143] or Fig. 3 of [146].

20


Stochastic limit The STL have been used by Accardi and co-workers [150, 151] toextend the van Hove limit in order to consider systems which are rapidly decaying (takea look at e.g. the Sec. II of [152]). Contrary to the WCL, applying the STL does not requirethe time-averaging or SA but can nevertheless lead to a Lindblad equation [153, 154]which we denominate as the LSTME. Note that the factorization assumption is necessaryto perform the STL. Such a procedure generates much richer decay processes than theWCL, namely non-exponential decays [155] and a temperature-dependent bifurcationof the relaxation constants [69]. Indeed, under the STL the coupling can be consideredas strong since the interaction is much larger than the energy scale of the system [69].The Lindblad equation obtained by Kimura and co-workers, see (26) in [153] or (5) in[69], has a similar form as the LSME (2.44) or the NRSME (2.45) in the sense that it isprovided in terms of the coupling operator appearing in the interaction Hamiltonian(2.3). Moreover, it can lead to the high-temperature version of CLME (2.46) containingonly the double commutator (see equation (28a)–(28b) in [153]).

In brief Clearly, all these microscopic derivations lead to master equations havingsimilar structures. We can mainly distinguish two classes: (i) the LDME obtained using theSA or time-averaging procedure in the WCL decoupling the population and coherenceand (ii) the non-SA equations. Among the second class, we can distinguish equationswhich are in the Lindblad form (LSME and LSTME) from the others (BRME, CLME andNRSME) which do not maintain the positivity in general. Looking at the non-SA and non-Lindblad equations we can differentiate two equations: the BRME expressed in termsof eigenoperators, which can neither be used to handle continuous energy spectrumnor non-adiabatic perturbations, and the CLME and NRSME depending directly on thesystem coupling operators which are free of these two deficiencies. Finally, examiningthe master equations expressed in terms of the coupling operators (LSME, LSTME,CLME and NRSME) one immediately notices that they are characterized by a flatspectral function and rely on high temperature and/or large couplings. The secondclass of master equations can handle numerous phenomena which cannot be capturedin the WCL as already pointed out by Accardi [152]. However, they present differentdrawbacks and it would be practical to have a set of master equations to treat thesevarious limits without spoiling the positivity, over-restricting the nature of the reservoirand allowing to treat continuous energy spectrum as well as non-adiabatic perturbationsat least on a phenomenological level.

21


2.1.5 Ergodicity property

We have seen that the KMS conditions (2.27)–(2.28) ensure the convergence to the Gibbsstate of the LDME and BRME presented in the previous section. Note that for certainparameters, it may happen that the BRME does not reach the Gibbs state due to positivityissues [128]. The convergence to the Gibbs state happens if and only if this state exists, isunique and displays global attractiveness 32. In practice, one can ensure that the systemreturns to the correct equilibrium for any initial state with the condition 33

[Aω,j, X] = [A−ω,j, X] = 0 ∀j and ω≥0 for X ∈ B(HS) and X = c1, (2.48)

where c is a scalar. If one finds a X being not proportional to an unitary 1 for which thecommutation relations [Aω,j, X] and [A−ω,j, X] vanish the final steady state is no longerunique (i.e. it will depend on the initial condition and is not necessarily diagonal, evenin the basis of the Hamiltonian). The criterion (2.48) is sometimes denominated as theergodicity property (EP) 34. It is often enough to have a correct ”mixing“ of the elementsof the density matrix during the time evolution by ensuring that every energy levelis connected to the others, at least indirectly. An EP formulated directly in terms ofthe coupling operators Qj and the system Hamiltonian HS is provided, for example, inequation (34) of [86]. In brief, one has to check the condition

[Qj, X] = 0 ∀j

[HS , X] = 0

}for X ∈ B(HS) and X = c1 (2.49)

with the ad hoc condition [HS , Qj] ∈ B(HS). This imply that Qj and HS do not sharea common eigenspace ensuring that all scattering channels are coupled [77, 78]. Thiscan be checked with [Qj, HS] 6= 0 for all j assuming a discrete energy spectrum [159].However, this does not fully discharge from checking the EP. Note that the secondformulation of the EP (2.49) is more general than the first one (2.48) in the sense that itdoes not rely on the eigenoperators (2.13) and (2.34). Interestingly, for some physicalrealization one would like that the steady-state is not unique, namely to study largedeviations based on rare current fluctuations [160]. On the other hand, breaking the EPis used to produce a quantum non-demolition interaction [161], leading to a dissipationlessdecoherence process [162] with [HS , Qj]=0 for all j, or to produce the so-called asymptoticentanglement for multipartite systems [163, 164, 165].

32See the algebraic Davies-Spohn-Frigerio convergence criterion [156, 157, 158]. The same is true for theNESS in the WCL [37, 86] as earlier mentioned in Sec. 2.1.3.

33See the equations (92)–(94) in Sec. 1.3.4 of the textbook discussing the case for the LDME [21]. Note thatit also holds for the BRME since both equations are expressed in terms of the eigenoperators and rely onthe aforementioned KMS conditions. This criteria holds for the Aω,i’s or Aω,j’s of the LDME (2.38) or (2.42)but not for the Fω,k’s of the LDME (2.37) directly, i.e. one has first to diagonalize the friction matrix or use asingle coupling operator per reservoir. For a sufficient but not necessary criterion directly associated to thefriction matrix, see the condition (76) at p. 58 of [21].

34The EP is also discussed in Sec. VI of [37] in an equivalent form, see Theorem 3 and following remarks.

22

2.2. Axiomatic formulation

2.2 Axiomatic formulation

2.2.1 Lindblad master equation

Instead of considering a microscopic derivation, one could postulate with the help of aset of axioms, namely 35

• convex linearity,

• preservation of the trace and hermiticity,

• complete positivity,

the following Lindblad-Kossakowski master equation (LKME) 36

ρ = Lρ = −i[H, ρ]−N

∑k,l

ηk,l

(L†

l ρLk −12{LkL†

l , ρ})

(2.50)

describing the dynamics of the system S interacting with an arbitrary environment butwithout specifying an underlying Hamiltonian picture as for the LDME (2.37)–(2.38). Itis sometimes also written in terms of commutators

ρ = −i[H, ρ]− 12

N

∑k,l

ηk,l

([L†

l , ρLk] + [L†l ρ, Lk]

). (2.51)

The operators Lk are the Lindblad operators while η is the Kossakowski matrix which has tobe positive-definite (see e.g. equations (4)–(6) of [124] or Sec. 4.3 of [23]). As previously,one can diagonalize the matrix η with an unitary W such that WηW† =diag(γ1, . . . , γN)

leading to the diagonal matrix γ=(γi) and Lindblad operators Ci =∑k WikLk yielding

ρ = −i[H, ρ]− 12

N

∑i

γi

([C†

i , ρCi] + [C†i ρ, Ci]

). (2.52)

Note that the Lk’s (or Ci’s) are not necessarily linked to the system Hamiltonian Hlike the Lindblad eigenoperators (2.13) and η (or γ) has not to fulfill a KMS condition(2.27)–(2.28) such as the friction matrix (2.26). Nevertheless, to ensure that the time

35The convex linearity implies that L(λρ1 + (1− λ)ρ2)=λL(ρ1) + (1− λ)L(ρ2) fulfilled by the Lindbladequation (2.50) because it is linear. The preservation of the trace requires that tr(Lρ)= tr(ρ) whereas thepreservation of the hermiticity reads (Lρ)† =Lρ. Both conditions can easily be checked for the Lindbladequation (2.50). Finally, complete positivity is a stronger requirement than positivity which is tightlyassociated to the notion of linearity; we postpone its discussion for later in Sec. 2.2.5.

36In fact, it is the Gorini-Kossakowski-Sudarshan-Lindblad master equation [33, 34]. Since we are hereinterested in the generator of the dynamics and no clear consensus exists in the literature to abbreviate it,we decided to use the Lindblad-Kossakowski nomenclature as in [166].

23


evolution leads to a unique steady-state (typically the fully mixed state 1/dS), theLindblad operators must respect an ergodic condition of the type [Ci, X]= [C†

i , X]=0 ∀ifor X= c1 corresponding to the first EP (2.48) earlier presented. In case one would liketo enforce the convergence to a given steady state ρ other than the fully mixed state, onecould perform the replacement ρ→ρ− ρ in the previous equations (2.50)–(2.51) at theexpense of adding a non-homogenous contribution so that the equation is no longer ofLindblad type. Note that the irreversible generator can be made dependent on H if ρ istaken as the Gibbs state ρβ =exp(−βH)/tr(exp(−βH)) to “mimic” an interaction with athermal environment (see the equation (21.42) in [167]). Finally, if the Lindblad operatorsCi of (2.52) are self-adjoint, one obtains a double commutator structure [Ci, [Ci, ρ]] havingthe same form as the LSME (2.44) obtained in the SCL.

2.2.2 The Schrödinger and Heisenberg picture: a duality relation

The LKME (2.50) formulated in the Schrödinger picture becomes in the Heisenbergpicture 37

A = L∗A = i[H, A]−N

∑k,l

ηk,l

(Lk AL†

l −12{LkL†

l , A})

(2.53)

which is provided in terms of the Hilbert-Schmidt dual L∗ of L defined with the help ofthe identity [90]

tr(L(ρ)A) = tr(ρL∗(A)) (2.54)

for all ρ and A. Clearly, the duality between L and L∗ corresponds to the duality be-tween the Schrödinger and the Heisenberg picture and is a consequence of the linearityof the LKME. Contrary to the LDME (2.37), the LKME (2.50) is not restricted to the WCL.For an appropriate set of Lindblad operators, one can consider strongly coupled popula-tions and coherences in photosynthetic complexes under a Markovian dynamics (seein [70] the equations (2.4)–(2.5) and (2.10) containing nonsecular terms responsible forthis coupling). Moreover, although it is often believed that entanglement 38 can only bedestroyed by the interaction with a noisy environment [21] one can construct a Lind-blad equation generating entanglement between two subsystems through a Markoviandynamics. In this respect, the off-diagonal elements of the Kossakowski friction matrixη in (2.53) must be non-zero so that the entanglement arises from the action of theenvironment onto the composite system [169]; the is equally true for the LDME [170] butthe choice of the elements of friction matrix is constrained by the nature of the reservoir,i.e. the spectral function (2.26). The same can be achieved for a many-body systemleading to “mesoscopic” entanglement under a Lindblad dynamics [171].

37Obtained by computing tr(AL(ρ)) = tr(ρL∗(A)) using (2.50), the cyclicity of the trace and isolatingL∗(A). The master equations (2.50)–(2.53) were originally provided in the seminal papers [33, 34, 168].

38Entanglement is a form of quantum correlations between multipartite quantum systems [88].

24


2.2.3 Linear dynamical maps

A Lindblad master equation [33], such as the LKME (2.50), is characterized by a generatorL leading to a uniformly continuous one-parameter family of linear dynamical maps {Λt}.Since Λt is differentiable and dΛt/dt=LΛt with Λ0=1, the only solution is 39

Λt = eLt, t ≥ 0 (2.55)

so that dΛt/dt|t=0 = L. Note that the generator L is also a map but not a dynamicalone. Usually, {Λt} is denominated as a quantum dynamical semigroup (QDS) [21]. It ischaracterized by the condition

ΛtΛs = Λt+s , t, s ≥ 0, (2.56)

which is equivalent to the additive composition rule and the Markov property 40, withthe additional requirement that the time-evolved average in the Schrödinger picture

〈A〉t = tr(ρt A) for ρt = Λtρ (2.57)

has to be a continuous function. The dual semigroup {Λ∗t } of (2.55) associated to thedual generator L∗ is given by

Λ∗t = eL∗t, t ≥ 0 (2.58)

implying Λ∗t A=At so that the average given in (2.57) reads in the Heisenberg picture〈A〉t = tr(ρAt). Note that a semigroup is not a group because it is missing the invertibilityproperty, i.e. the map Λt is not invertible (see Sec. 3.1.2 of [88]) which is a feature of thetime-irreversibility of open quantum systems.

2.2.4 Quantum detailed balance

When we computed the LDME in the Sec. 2.1.3, we encountered the microscopic detailedbalance (2.27) or KMS condition (2.28) which we will designate from now on as theKMS-DB. The notation of detailed balance was already studied by Agarwal in the70s [175] and Majewski in the 80’s [176] following the idea of time reversal invarianceand microreversibility. Note that contrary to the classical case, several formulations ofdetailed balance exist for quantum systems because the underlying algebra is complex

39See e.g. Sec. 1.3 of [23] for a formal introduction to semigroups (including proofs) or 3.2.1 of [88]; weprovide here only the key definitions. We may note in passing that a one-parameter family means that eachmember of the semigroup is given by a single period of time.

40So that (2.55) is sometimes referred to as a quantum Markov semigroup [172]. For open quantum systems,it is the only situation where one can establish with certainty that the time-evolution is Markovian. Ifa master equation doesn’t generate a QDS one can only check ”heuristically“ if it is Markovian usingmeasures such as the one proposed by Breuer and co-workers (based on the trace distance, see [173, 174],used to detect non-Markovian effects).

25


and the multiplication is no longer commutative. The time reversal arises from anantiunitary operator Θ, such that ΘΘ† =Θ†Θ=1, defining a reversing superoperator 41

of the form R(A)=ΘAΘ−1 producing a bounded operator if A is itself bounded. TheAgarwal-Majewski quantum detailed balance condition (AM-QDB) is defined with respect toa faithful invariant state ρ (i.e. positive definite steady-state) as

〈A;L(B)〉ρ,0 = 〈R(B);L(R(A†))〉ρ,0 (2.59)

relying on the so-called pre-scalar product 42

〈A; B〉ρ,λ = tr(

A†ρλBρ1−λ)

, λ ∈ [0, 1] (2.60)

evaluated at λ = 0 as well as on the property R(B)† = R(B†) given in the Proposition4 of [178]. On the other hand, Alicki [118] and Kossakowski [119] discussed it usingthe LKME (2.53) in the Heisenberg picture L∗A= i[H, A] + L∗d A generating a QDS, butwhose dissipator Ld (being the dual of L∗d) is not necessarily attached to an underlyingHamiltonian model. The resulting Alicki-Kossakowski quantum detailed balance (AK-QDB)condition for the positive-definite steady-state ρ reads

[H, ρ] = 0

〈A;L∗d(B)〉ρ,0 = 〈L∗d(A); B〉ρ,0 (2.61)

taken from the Sec. 1.3.4 of the book [21]. The two previous detailed balances (2.59) and(2.61) are sometimes also expressed in terms of the full generator L or the semigroup{Λt} [178]. The KMS-DB come along with the LDME (2.37)–(2.38) ensuring the conver-gence to the Gibbs state ρβ for a single reservoir when the EP is fulfilled; thus, it leadsto a ”microscopic“ dissipator Ld which can satisfy the AK-QDB (2.61) with ρ=ρβ. Onthe contrary, not all dissipators Ld satisfy the KMS-DB (e.g. considering an athermalreservoir). For further explorations of the notions of detailed balance conditions (2.59)–(2.61) see the papers of Fagnola and co-workers proposing a first draft of an unifiedframework [178, 177, 179] which can be used, among others, to discuss the link betweendetailed balance and entropy production 43.

41The standard notion of time reversal affects the parity of an observable: the position and energy of aparticle are both unaffected and are qualified as even whereas its momentum undergoes a sign flip and isconsequently said to be odd. For an OQS, we have to define a reversing superoperator R so that an evenobservable R(A)=A remains unaffected whereas an odd observable R(A)=−A changes its sign.

42The notion of pre-scalar product was introduced in [177]. A more ”symmetric“ version of the quantumdetailed balance (2.59) with λ=1/2 is examined therein (see the Definition 2) as well as in [178, 179] and isassociated to the notion of entropy functional (see e.g. Chap. 7 of the book [180]).

43See the formula (15) in [181]; the Lindblad equation used therein relies on the same kind of eigenopera-tors (2.13) as the LDME but without specifying the microscopic origin of the coupling operators.

26


2.2.5 Complete positivity

General considerations Despite their different origin, the various Lindblad masterequations (such as the LDME, the LSME or the LKME) produce a completely positive (CP)time-evolution [33]. A CP linear dynamical map ΛCP,t = eLCPt is associated to a generatorLCP giving rise to a completely positive time evolution ρ=LCP ρ assuming a Markoviandynamics 44. Note that ΛCP,t is implicitly assumed to be also trace preserving (it is aCPTP map). The notion of CP is closely linked to the concept of ancilla, a distant systemA not changing in time and whose Hilbert spaceHA possesses an arbitrary dimension 45,which is added to the system S but does not interact with it. When we apply thecomposite map (ΛCP,t ⊗ idA), where idA is the identity map characterizing the ancilla, onan arbitrary positive-definite state belonging to D(HS ⊗HA) the resulting state is stillpositive-definite [87]. The same remains true if one acts on a separable mixed state

(ΛCP,t ⊗ idA)

(∑

ipi(ρS ,i ⊗ ρA,i)

)= ∑

ipi ΛCP,t(ρS ,i)⊗ ρA,i ≥ 0, (2.62)

where the weights pi respect pi ≥ 0 and ∑i pi = 1. Moreover, for an entangled stateρS+A∈D(HS ⊗HA), i.e. a state which cannot be written in the separable form appearingin (2.62) according to Werner’s statement [185], we equally obtain 46

(ΛCP,t ⊗ idA)ρS+A ≥ 0. (2.63)

Note that the entangled (or correlated) state ρS+A was generated prior to the applicationof the combined operation by an interaction which is no longer present. The CP state-ment is reasonable: the combined map acting locally on the system S , which is assumedto be largely separate from the ancilla A so that it cannot influence it 47, produces aphysical state characterized by a positive-definite density matrix. On the other hand, alinear map which is only positive ΛP,t would produce 48, in general, negative eigenvaluesin this enlarged space under the action of (ΛP,t ⊗ idA). More precisely, it will alwaysoccur with at least one initial state where the system is entangled with the ancilla [87].For this reason CP linear maps have been privileged over just positive ones [21] toconsider entanglement in multipartite reduced systems [87]. Note that this definitionof CP is only meaningful for linear dynamics [187, 188, 189] and, as mentioned byPechukas [190] as well as by Shaji and Sudarshan [191], it does not need to be a generalphysical requirement in spite of its mathematical attractiveness.

44For a non-Markovian time-evolution, one has to proceed differently, see the review [182]. An exampleof CP non-Markovian dynamics can be found in [183, 184].

45The ancilla can be seen as a trivial heat reservoir not interacting with the system. Note that the HilbsertspaceHA is a vectorial space CN of arbitrary dimension N (usually finite dimensional for sake of simplicity).

46It can be used to check if a map Λt is completely positive, typically by using a state ρS+A which ismaximally entangled. This procedure is stated by the Choi-Jamiolkowski isomorphism [186].

47No correlation is generated between the system and the ancilla by the composite map (ΛCP,t ⊗ idA).48A positive map transforms a matrix with positive eigenvalues into a matrix preserving this property.

27


Correlations with the environment CP linear maps make sense, in general, when thesystem and the environment are uncorrelated (see e.g. Sec. 6.2 of [106] for a proof). Ifthe correlations are of classical nature [192, 193] or if the revival of the correlations isforbidden, for example, by excluding all anomalous backward flows of informationor heat [194], the linear map is CP. For further explanations, see Dominy et al. [189]proposing a framework for the discussion and analysis of CP for correlated initial statesand nonlinear time-evolutions, as well as the recent experiments [195, 196].

CP and non-secular terms The non-secular terms of the BRME (2.35) are eliminatedthrough the application of the SA; the resulting microscopic master equation, knownas the LDME (2.38), is CP. Unfortunately, the SA suppresses the coupling between thepopulations and the coherences as well as the coupling between the different coherences.To obtain a microscopic master equation in the WCR which is CP and contains non-secular terms, one could resort to the coarse-grained master equation (CGME) [197, 198,199, 104]. We privilege here Schaller and Brandes’ version [199] which is relativelystraightforward to use and leads to a dissipator of Lindblad’s form

Dj ρ = ∑ω1

∑ω2

hτj (ω1, ω2)

(Aj,ω1 ρA†

j,ω2− 1

2

{A†

j,ω2Aj,ω1 , ρ

}), (2.64)

where the coarse-grained spectral matrix elements

hτj (ω1, ω2) =

τ

2πei(ω2−ω1)τ/2

∫ ∞

−∞dν hj(ν) sinc

((ν + ω1)

τ

2

)sinc

((ν + ω2)

τ

2

)(2.65)

depend on the coarse-graining parameter τ and hj(ν)=∫ ∞−∞ dt eiνt g2

j trRj(Φ(j)

j, t Φ(j)j, 0σβ j

) theusual spectral function. The coarse-grained spectral matrix defined by (2.65) is positivesemi-definite which is crucial to ensure that the CGME is of Lindblad form 49. Note thatin the limit of infinite τ the CGME (2.64) converges towards the LDME (2.42) which isparticularly appealing since it permits to stay in contact with the WCL. The reason issimple: the integrand of the previous matrix elements (2.65) is asymptotic to the productof a Kronecker-delta δKr(ω1 + ω2) and a Dirac-delta δD(ν + ω1) in the limit

limτ→∞

τ sinc((ν + ω1)

τ

2

)sinc

((ν + ω2)

τ

2

)� 2πδKr(ω1 + ω2)δD(ν + ω1). (2.66)

The main difficulty with the CGME is to find how to fix τ; that is all the more true sincefor all τ’s different from infinity the equilibrium state is not of Gibbs’ type (see Sec. C.2of [199]). Besides, one can use the microscopic LSME (2.44) for strong couplings (or theLKME) which is also free of the SA but leads to the fully mixed state. To address theseissues, we introduce in the next chapter a set of thermodynamical master equationswhich contain at the same time non-secular terms and relax to the Gibbs state.

49For an alternative dynamical coarse-graining, see the time-averaging procedure proposed in [89].

28

3 Thermodynamic reformulation:nonlinear open quantum systems

In the previous chapter, we saw plenty of strategies to model open quantum systemsunder various regimes either by deriving master equations or postulating them withthe help of a restricted set of mathematical axioms. We have also seen that it is difficultto obtain a physical master equation which simultaneously includes non-secular termsand leads in the long time limit to the Gibbs state. The necessity to include non-secular contributions have been recognized over the past few years for a wide rangeof situations: they are essential for the creation of coherence [200, 201] and quantumcorrelations [202, 203], the generation of a coherent NESS [204, 60] and for a properdescription of local currents and conservation laws [49, 51, 54]. In this chapter weexplain how to include them at the expense of giving up linearity and, consequently,complete positivity. Therefore, we take a different route and favor thermodynamicalconsistency to guarantee a proper connection with the macroscopic world [37] in analogyto classical nonequilibrium systems [75, 76]. At the same time, we show that contrary towhat is claimed in [83, 84] the second law of thermodynamics can also hold for non-CPtime-evolutions. De facto, a positive time evolution suffices.

Hereafter, we start by introducing the modular dynamical semigroup (MDS) formal-ism [77, 78] and show that the LDME (2.38) generates a MDS. Thereafter, we introducethree nonlinear thermodynamics master equations (NTMEs) equally generating a MDS.The two first, namely the dynamically time-averaged NTME and pre-averaged NTME ob-tained in [77, 78], hold in the WCR and provide a firm statistical stem since they areconnected to the microscopic Hamiltonian description in the WCL 1. The third NTME,obtained in [205] and designed hereinafter as the modular-free NTME, is of a differentnature. Indeed, it provides a thermodynamic extension to the LSME valid for stronglydamped systems through their coupling to a classical environment. For a summary ofthe different master equations, see the Table 3.1.

1A time-averaging procedure [89] has to apply to the pre-averaged NTME to recover the LDME contraryto the dynamically time-averaged NTME which implicitly includes such a procedure (it depends on acoarse-graining parameter in a similar way as the CGME (2.64) provided in the previous chapter).

29

Chapter 3. Thermodynamic reformulation: nonlinear open quantum systems

3.1 Modular dynamical semigroup (MDS) in a nutshell

3.1.1 Time-evolution of the system of interest

The MDS formalism was originally designed to describe the time-evolution of the systemin contact with an arbitrary nonequilibrium environment [77, 78]. To keep it simple andallow comparisons with the microscopic master equations of the previous chapter, let usstart with an environment composed of M inexhaustible and independent reservoirsR = R1 ⊗ ...⊗RM at the inverse temperature βk = 1/Tk and k = 1, ..., M. Instead ofperforming a derivation from first principle, we postulate for the system observableA∈B(HS) a phenomenological time-evolution

∂t〈A〉= Ht(A) + St(A) (3.1)

with 〈A〉= tr(Aρ) its expectation and ρ∈D(HS) the density matrix of the system. Thefirst term Ht(A) in (3.1) is responsible for the reversible time-evolution dictated by thesystem Hamiltonian HS and reads

Ht(A)=−i〈[A, HS ]〉. (3.2)

It is expressed in terms of a quantum commutator between two observables

[A, B] = AB− BA (3.3)

which is a bilinear bracket satisfying Jacobi’s identity. The second term St(A) in (3.1) isresponsible for the irreversible time evolution

St(A)=M

∑k

∑αk

〈[[A, ∆Sk]]βkρ,αk〉 (3.4)

driving the system to a stationary state through the action of the relative modular operator∆Sk = SS − SS ,k. This quantity is given by the difference of the the system entropyoperator SS and the target entropy operator SS ,k associated to the kth reservoir. Since wewould like to end up in the equilibrium Gibbs state when connecting a single reservoirto the system, we choose for the system entropy SS the von Neumann entropy operatorS(ρ) = − ln ρ while for the target entropy SS ,k we take its steady counterpart S(πk)

which has been evaluated at the βk-KMS state πk associated to the kth reservoir. Theresulting self-adjoint operator ∆Sk = S(ρ) − S(πk), commonly used in the modulartheory 2, is nonlinear in ρ. The entropic contribution (3.4) is provided in terms of a set of

2The operator ∆Sk =S(ρ)− S(πk) is equal to the relative Hamiltonian lρ|πkused in modular theory [206].

Moreover, its average S(ρ||πk) = 〈∆Sk〉=−tr(ρ(log ρ− log πk)) is equal to Umegaki’s relative entropybeing the quantum analogue of the classical Kullback-Leibler divergence.

30

3.1. Modular dynamical semigroup (MDS) in a nutshell

(ρ, αk, βk)-modular dissipative bilinear brackets between two observables A and B [77, 78]

[[A, B]]βkρ,αk

=∫ 1

0dλ

(ζ

iλβk2

ρ (δXk,λαk(A†)

)†(ζ

iλβk2

ρ (δXk,λαk(B))

), (3.5)

where δXk,λαk(·) = i[Xk,λ

αk, · ] can be seen as a quantum differentiation 3. It depends on the

bounded scattering operator Xk,λαk

which encodes the interaction with the kth reservoir anddepends on the modular parameter λ ∈ [0, 1]. Besides,

ζtρ(A) = eiHρt A e−iHρt (3.6)

is a time-evolution superoperator with t= iλβk/2 an imaginary microscopic time andHρ =β−1

k Sρ a modular Hamiltonian. The bracket (3.5) is clearly positive semidefinite 4

[[A†, A]]βk

ρ,αk≥ 0 (3.7)

as well as symmetric ([[A, B]]βkρ,αk)† = [[B†, A†]]

βk

ρ,αkand [[1, · ]]βk

ρ,αk= 0 with 1 the identity

operator. The abstract formulation (3.1)–(3.7) has the advantage to remain valid evenin the thermodynamic limit for extended systems assuming ∆Sk is bounded. It can,therefore, provide a natural thermodynamical extension to the quantum dynamicalsemigroups [19, 20, 21] through the modular formalism 5. It should be noted that thesecond sum in the entropic superoperator (3.4) over the set index αk ∈ I is used tospecify the nature of interaction between the system and the kth reservoir 6 by means ofthe scattering operator Xk,λ

αkappearing in the dissipative bracket (3.5). Finally, although

the relative modular operator ∆Sk enforces the convergence to the proper thermal staterelative to HS it is important to underline that it happens if and only if it exists, is uniqueand displays global attractiveness. To this end, an EP as introduced in Sec. 2.1.5 has tobe fulfilled by the dissipative bracket. Consequently, for a single reservoir, or multiplereservoirs with equal βk = β, the system ends up in the β-KMS state ρβ whereas formultiple reservoirs at different inverse temperatures it converges to a NESS ρ+. Note thatthe choice of the von Neumann entropy as system entropy SS is not the only possiblechoice (although it seems to be the most natural one) and we could instead have used,among others, Tsallis [207, 208, 209, 210] or Rényi [211, 212, 213, 214] entropies.

3The quantum differentiation δO(·)= i[O, · ] relative to the operator O is, in general, used to express thereversible time evolution of this operator ∂tOt =−δOt (HS ). To get convinced, one can simply compute thisquantity ∂tOt =∂t(eiHS tOe−iHS t)= iHSOt − iOt HS=−i[Ot, HS ]=−δOt (HS ).

4In classical nonequilibrium thermodynamics, if we can decompose the friction matrix M as M = BT B,where B is a so-called noise matrix, we know it is positive-definite (this decomposition is not necessarilyunique). This strategy is used, for example, to write down the stochastic process associated to a given Fokker-Planck equation, see Eq. (1.57)–(1.58) of Öttinger’s book about GENERIC [76]. The present dissipativebracket (3.5) is written using such kind of decomposition to enforce its positive definite nature.

5See Chap. 4.3.11-12 of [206] for details about modular structures.6We could have absorbed the sum over k into a generalized index α to build a single sum but we

wanted to emphasis the fact that we consider here multiple reservoirs. Clearly, the index αk can be amultidimensional parameter αk =(αk,1, αk,2, . . .) and each of its component can be discrete or continuous (thesum corresponding to a continuous index has to be replaced by an integral).

31


3.1.2 MDS master equation for the system of interest

General setting To streamline the previous abstract formulation, we assume that wework with trace class operators and rewrite the time-evolution (3.1)–(3.7) as a masterequation. First we notice that the superoperator (3.6) can be recasted in the somehowsimpler form ζ

iλβk/2ρ (A) = ρλ/2 A ρ−λ/2. Consequently, the expectation of the bracket

(3.5) becomes after few rearrangements

〈[[A, B]]βkρ,αk〉=

∫ 1

0dλ tr

((i[Xk,λ

αk, A†])

†ρλ (i[Xk,λ

αk, B])ρ1−λ

)(3.8)

The entropic superoperator (3.4) can be rewritten with the help of its expectation (3.8)as St(A) = tr(J (ρ)A) so that the time-evolution (3.1) becomes ∂t〈A〉= tr(−i[HS , ρ]A)+

tr(J (ρ)A) yielding the following non-necessarily linear MDS master equation in theSchrödinger picture 7

∂tρ = −i[HS , ρ] + J (ρ), (3.9)

where the irreversible contribution J (ρ) = ∑Mk Jk(ρ) represents the action of the M

reservoirs on the system taking the following double commutator form

Jk(ρ) = ∑αk

∫ 1

0dλ [(Xk,λ

αk)†, ρλ[Xk,λ

αk, Ψk]ρ

1−λ] (3.10)

depending on the bounded and self-adjoint kth Massieu operator

Ψk = S(ρ)− βk HS (3.11)

introduced thanks to the Gibbs state πk = Z−1βk

e−βk HS with Zβk= tr(e−βk HS ) the partition

function associated to the reservoir k. The double commutator structure of the masterequation (3.10) insures the preservation of the normalization. The Massieu operator Ψk

is a more natural (but less general) choice than the relative modular operator ∆Sk. Theydiffer from each other by the identity multiplied by a constant c = ln Zβk which can beneglected inside of the commutator 8. For later convenience, it is particularly useful to

7Passing from the Schrödinger to the Heisenberg picture, and conversely, is generally not achievable fora nonlinear master equation (for the Lindblad master equation [33], being of linear nature, this can easilybe done as seen in Sec. 2.2.2). Since one can extract nearly all necessary information to describe the systemfrom the perspective of the statistical operator, we decided to privilege the Schrödinger picture. The priceto pay for this choice is that one cannot easily define multitime correlation function which requires theHeisenberg picture (one way out is to linearize the equation near equilibrium; however, this reduces therange of validity of the resulting correlation functions, see e.g. Sec. II.E in [215]).

8The Massieu ”potential“ 〈Ψk〉 = 〈S〉 − βk〈HS 〉 appears as a Legendre transform of the entropy 〈S〉being then treated as a state variable (changing according to the state of the system); it reaches an extremumat the steady-state and is clearly a better choice for irreversible thermodynamics than the Helmoltz free

32


recast the MDS master equation (3.9) as follows

∂tρ = K(ρ) (3.12)

and notice that in case it is explicitly linear in the density matrix we can rewrite it∂tρ = Lρ in reference to the linear dynamical maps of Sec. 2.2.3. The system convergestowards a unique steady-state ρ implying for the irreversible time-evolution

J (ρ) = 0 (3.13)

assuming that the aforementioned EP hold. For a single reservoir ρ is a Gibbs state π (orρβ being the other notation used throughout this work) while for multiple reservoirs at adifferent temperature ρ is a NESS ρ+ as previously stated. However, if one would like toend in a so-called generalized Gibbs state, for example, associated to earlier mentionedTsallis entropy [207, 209] one has to modify the Massieu operator (3.11) accordingly (seeAppendix 9.2 for further details). Finally, to make sure that J is self-adjoint, we have toimpose the following constraint on the individual scattering operators

(Xk,λαk)† = Xk,1−λ

f (αk)(3.14)

for an invertible function f : I → I such that f ( f (αk)) = αk. Moreover, for the well-posedness of (3.10) we have to enforce the EP following the condition (2.49) but directlyfor the scattering operator Xk,λ

αk. Briefly summarized, this means that the commutators

[Xk,λαk

, HS ] must be bounded as well as HS and Xk,λαk

must not share common eigenspaces.

Modular friction matrix The constraint (3.14) supplemented by the EP allows to fur-ther specify the MDS formulation by decomposing the scattering operators into twocontributions

Xk,λαk

= ∑α′k

Ck,λαkα′k

Yα′k, (Ck,λ

αkα′k)∗ = Ck,1−λ

f (αk) f (α′k)(3.15)

allowing to construct a positive definite matrix whose (α′k, α′′k ) elements read

Dk,λα′kα′′k

=∑αk

(Ck,λαkα′k

)∗Ck,λ

αkα′′k, (Dk,λ

α′kα′′k)∗=Dk,1−λ

f (α′k) f (α′′k ). (3.16)

Now, taking (3.10)–(3.16) altogether we finally obtain for the kth reservoir

Jk(ρ) = ∑α′k ,α′′k

∫ 1

0dλ Dk,λ

α′kα′′k[(Yα′k

)†, ρλ[Yα′′k, Ψk]ρ

1−λ]. (3.17)

energy ”potential“ 〈Fk〉 = 〈HS 〉 − Tk〈S〉 being a Legendre transform of the energy (see the discussion atthe end of Appendix A.4 in [76] itself referring to Callen’s book [216]).

33


Looking back at the LDME (2.37) obtained in the microscopic context or the LKME(2.50) deduced from axiomatic arguments suggests that we have obtained with (3.16) amodular friction matrix: it models the degree of freedom of the environment affecting thesystem (such as a correlation function) and fully carries the dependency on the modularparameter λ. Following this analogy, the interaction operators Yαk

play a similar role asthe eigenoperators in the LDME (2.37) or the Lindblad operators in the LKME (2.50),i.e. they mediate the interaction with the environment. The time-evolution (3.1)–(3.7)as well as the master equations (3.9)–(3.10) or (3.17) have to be regarded as forming aframework since most of its components remain to be defined to reach a physical model.

Generalized environments Finally, to further generalize the MDS master equation wecan get rid of the dependence of the temperature [77, 78]. For example, assuming thereservoir is in a non-thermal stationary state (e.g. invariant under the application of itsreversible dynamics) we can use the following ansatz for the friction matrix

βkDk,λα′kα′′k

:= −∂λ(Dk,λ

α′kα′′k)

g(α′k, α′′k ), (3.18)

where ∂λ is a partial derivative relatively to the modular parameter λ and g(α′k, α′′k )

is a regular function which has to be determined. The ansatz (3.18) allows to link theprevious expression to a non-equilibrium environment where a temperature cannot bedefined in general but one can define an entropy SE . Now, following [78] we get withthe Hamiltonian of the environment HE the environmental dissipative brackets

[[HE , HE ]]k, λ

E ,α′kα′′k= Dk,λ

α′kα′′k, [[HE , SE ]]

k, λ

E ,α′kα′′k= ∂λ(Dk,λ

α′kα′′k)/g(α′k, α′′k ), (3.19)

The main assumptions are that theses brackets do not change appreciably during thetime-evolution of the reduced system and that the energy fluctuations are small so thatthe total system can be regarded as closed in analogy to the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) for classical system [76]. Thus,the environment performs an adiabatic action on the system ensuring that the state ofthe system and environment remain factorized at all times and the dynamics of the fullsystem is Markovian. When combined with the irreversible contribution (3.17) and thedefinition of the Massieu operator (3.11), we finally obtain

Jk(ρ) = ∑α′k ,α′′k

∫ 1

0dλ[[HE , HE ]]

k, λ

E ,α′kα′′k[(Yα′k

)†, ρλ[Yα′′k, S(ρ)]ρ1−λ]

+[[HE , SE ]]k, λ

E ,α′kα′′k[(Yα′k

)†, ρλ[Yα′′k, HS ]ρ1−λ].

(3.20)

This idea was originally proposed in [205] and later generalized in [77] by allowing theλ-dependence of the friction matrix (3.16). The MDS master equation (3.20) correspondsto the equation (2) of [78] but with a more explicit expression for the irreversible brackets.

34


3.1.3 Entropy production

As the size of a physical system comes closer to the nanoscale, quantum effects start tomatter and it is crucial to reconsider the laws of thermodynamics. To verify that we stillhave a positive entropy production σS(ρ), we can use the space-integrated local form ofthe entropy balance [217, 218, 37]

∂t〈S(ρ)〉 = JS(ρ) + σS(ρ). (3.21)

This balance equation depends on the time-evolution of the entropy operator ∂t〈S(ρ)〉determined by (3.1)–(3.8) with again ∆Sk ∼ Ψk and where the reversible term drops outfor A=S(ρ) yielding 9

∂t〈S(ρ)〉 = tr(J (ρ)S(ρ)) = ∑k

∑αk

〈[[S(ρ), Ψk]]βkρ,αk〉 (3.22)

and on the entropy density flux

JS(ρ) = ∑k

βk J kQ(ρ) (3.23)

associated to the heat fluxes of the inidividual reservoirs

J kQ(ρ) = tr(Jk(ρ)HS) = ∑

αk

〈[[HS , Ψk]]βkρ,αk〉. (3.24)

To get the entropy flux (3.23) we used the master equation (3.9)–(3.10) and the fact thatthe reversible term for the observable βk HS is zero. The equation (3.23)–(3.24) models theentropy exchange between the system and the M reservoirs composing the environment.Thus, combining previous relations (3.21)–(3.24) and reforming the Massieu operator(3.11) we obtain for the entropy production

σS(ρ) = ∑k

∑αk

〈[[Ψk, Ψk]]βkρ,αk〉 ≥ 0 (3.25)

fulfilling the second law of thermodynamics according to the inequality (3.7) as expected.

3.1.4 Positivity argument for the master equation

To see that the density matrix ρ remains positive, we can compare the change of entropy(3.22) with the entropy production (3.25) as proposed in [77, 78]. To this end, weassume that we have a discrete energy spectrum and look at the evolution of the vonNeumann entropy 〈S(ρ)〉=−tr(ρ ln ρ)=−∑n pn ln pn with pn ∈ ]0, 1[ the eigenvaluesof the faithful state ρ. Since this quantity is a sum of positive terms it must remainpositive. The only “danger” is when one (or more) eigenvalue(s) converge to zero; if

9The reversible contribution 〈ρt〉=−i tr(ρ[H, ln ρ])= i tr(H[ρ, ln ρ]) vanishes since ρ and ln ρ commute.

35


the time-evolution did not preserve the positivity negative eigenvalues would arise.This could typically happen at low temperature when the system evolves towards asteady state carrying one eigenvalue close to one whereas all others are exponentiallysuppressed. In this case, the von Neumann entropy decreases and would even vanish ifa pure state is reached (in the zero temperature limit we would end up with a pure state).Accordingly, showing that 〈Sρ〉>0 would prove that ρ>0 10. One needs to realize thatthe aforementioned EP insures that for a set αk∈I ⊂ I there exist [Xk,λ

αk, Ψk]∼ [Xk,λ

αk, ln ρ]

since [Xk,λαk

, HS ] is assumed to be bounded (again ∆Sk ∼ Ψk). More precisely, from (3.15)we have that Ck,λ

αkα′kis usually proportional to a correlation function of the environment

which rapidly decays and Yα′kis bounded. Therefore, we can safely assert that for the

eigenvalues pn going to zero || ln ρ ||� ||HS || so that ∂t〈S(ρ)〉→σS(ρ)≥0 implying thatρ≥ 0. Depending on the nature of the scattering operator Xk,λ

αk(or equivalently of the

time-evolution) we either have ρ > 0 (pure states inaccessible) or ρ ≥ 0 (pure statesaccessible). The zero temperature limit can only be touched in the second case since thesteady-state is then a pure state, see the Appendix 9.3 for further explanations.

3.2 From the MDS to the thermodynamic master equations

In the previous section, we have defined the MDS master equation (3.9)–(3.11) underfairly general settings. To connect the master equation generated by (3.10) to the Hamil-tonian description (2.1)–(2.3), the following functional dependence of the kth scatteringoperator could be chosen

Xk,λω = Xλ(Ak,ω, hk(ω)) (3.26)

depending on the Lindblad eigenoperators Ak,ω and spectral functions hk(ω) whichwe encountered earlier in Sec. 2.1.3. In this case, the sum over αk is the sum over theenergy gap ω=Em − En expressed in terms of the system eigenenergies extracted fromthe system Hamiltonian HS (see e.g. (2.12)–(2.14) but for a single coupling operator). Toprevent the explicit use of the eigenoperators, one could also write the scattering operatoras Xk,λ

t = Xλ(HS , Qk, Ck(t)) with Ck(t) the homogeneous two-time correlation function(2.16) of the kth reservoir; the sum over αk is then an integral over time as appearingin the first line of the calculation (2.15). More generally, to avoid the introduction ofa spectral function/two-time correlation one could write Xk,λ

αk= Xλ

αk(HS , Qk, Φk) at the

expense of not yet specifying the nature of αk. In this section, we provide differentscattering operators yielding the LDME as well as three NTMEs. The first NTMEdepends on a dynamical cut-off controlling the time-averaging procedure (i.e. generatinga dynamical SA) while the second one is free of such a procedure and therefore easierto handle (analytically and numerically). Finally, the third NTME relies on scatteringoperators free of the modular parameter λ.

10The shorthand notation ρ>0 indicates that the zero eigenvalue is never attained by the spectrum of thedensity matrix being strictly positive.

36

3.2. From the MDS to the thermodynamic master equations

3.2.1 The Lindblad-Davies master equation

Choice of the scattering operators Our aim is to obtain the LDME (2.42) from theMDS master equation (3.10)–(3.11). Therefore, let us go back to the scattering operator(3.26) earlier proposed and write it explicitly as 11

Xk,λω =

1√2

e−λβkω

2

√hk(ω) Ak,ω (3.27)

satisfying the self-adjointness property (3.14) since

(Xk,λω )† =

1√2

e−(1−λ)βk(−ω)

2

√hk(−ω) Ak,−ω = Xk,1− λ

f (ω) , (3.28)

where we used the property A†k,ω = Ak,−ω, the invertible function f (ω) = −ω and the

detailed balance hk(ω) = eβkω hk(−ω) associated to the KMS condition (2.27)–(2.28). Wealso notice that the commutators [Xk,λ

ω , HS ] for the scattering operator (3.27) are bounded

||[Xk,λω , HS ]|| ∝ |ω|

∣∣∣e−λβkω

2

∣∣∣√hk(ω) ||Ak,ω|| (3.29)

since ||Ak,ω||< ∞ because ||HS || and ||Qk||< ∞ as well as |hk(ω)|< ∞. Moreover, since||HS || < ∞ the frequencies |ω| < ∞ and the large values generated by |e−λβkω/2| aresuppressed by the cut-off contained in the spectral function hk(ω).

We previously provided a different form (3.17) of the MDS master equation in terms of afriction matrix and interaction operators instead of the scattering operators. To obtainthe LDME, we have that to set α′k, α′′k → ω in the modular friction matrix (3.16) yielding

Dk,λω = e−λβkω hk(ω) (3.30)

and the interaction operators

Yk,ω = Ak,ω, Y†k,ω = Ak,−ω. (3.31)

The friction matrix (3.30) depends on the modular spectral function

hk,λ(ω) := e−λβkω hk(ω). (3.32)

Note that (3.30) really becomes a matrix when considering multiple coupling operatorsper reservoir (i.e. the indices α′k and α′′k play a trivial role here). Note that for (3.32)the self-adjointness property (3.14) is translated into the identity hk,λ(ω)= hk,1−λ( f (ω))

with f (ω)=−ω. Clearly, the second version of the MDS irreversible contribution (3.17)is somehow easier to use than the first one (3.10).

11The αk index in the sum of the equation (3.10) is replaced by the discrete set of Bohr frequencies ω.

37


Calculation of the master equation

Inserting the scattering operators (3.27) into the irreversible part of the master equation(3.10)–(3.11) yields the seemingly nonlinear irreversible contribution [77, 78]

Jk(ρ) = −12 ∑

ω

hk(ω) [A†k,ω, Kk,−ω

ρ [Ak,ω, ln ρ + βk HS ]], (3.33)

where we defined a frequency-dependent Kubo-Mori superoperator

Kk,ωρ B =

∫ 1

0dλ eλβkωρλBρ1−λ. (3.34)

Using the fact that the eigenoperators Ak,ω fulfill the standard identity

[HS , Ak,ω] = −ωAk,ω (3.35)

we prove in Appendix 9.4 the property [77, 78]

Kk,−ωρ [Ak,ω, ln ρ + βk HS ] = Ak,ωρ− e−βkωρAk,ω (3.36)

which we exploit to rewrite the irreversible term (3.33) as

Jkρ = −12∑

ω

[A†k,ω, hk(ω)Ak,ωρ− e−βkω hk(ω)ρAk,ω]

= −12∑

ω

hk(ω)([A†

k,ω, Ak,ωρ]− [Ak,ω, ρA†k,ω]) (3.37)

being now clearly linear in ρ. To go from the first to the second line we used e−βkω hk(ω) =

hk(−ω), the change of variable −ω → ω and Ak,−ω =A†k,ω. Finally, we expand the two

commutators to produce an anticommutator so that

Jkρ = ∑ω

hk(ω)

(A†

k,ωρAk,ω −12{Ak,ω A†

k,ω, ρ})

(3.38)

leading to the LDME (2.42) obtained from a microscopic derivation.

As we saw above, obtaining the LDME is purely fortuitous and is possible thanks to thecommutation relation (3.36) valid for thermal reservoirs and for a system Hamiltonianhaving a discrete energy spectrum. Breaking one of these conditions and the equationremains nonlinear. Therefore, let us look at the case of a non-thermal environment [77,78]. We first write the LDME in the athermal form with (3.19)–(3.20) noticing that theansatz (3.18) is fulfilled for the friction matrix (3.30), i.e. −(1/ω) ∂λDk, λ

ω =βDk, λω , so that

38


the interaction operators (3.31) and the modular spectral function (3.32) give [77, 78]

Jk(ρ) = ∑ω

∫ 1

0dλ hk,λ(ω)[Ak,−ω, ρλ[Ak,ω, S]ρ1−λ]

+1ω

∂λ(hk,λ(ω))[Ak,−ω, ρλ[Ak,ω, HS ]ρ1−λ]

(3.39)

which is equal to the LDME despite its unusual form. In a second step, we rewritethe modular spectral function (3.32) in terms of Fourier transform of the microscopictwo-time correlation function and compute

hk,λ(ω) =∫ ∞

0dt eiω(t+iλβk) g2

k tr(

Φ(k)t Φ(k)

0 πRk

)=∫ ∞

0dt eiωt g2

k tr(

Φ(k)t− iλβk

Φ(k)0 πRk

)=∫ ∞

0dt eiωt g2

k tr(

π−λRk

Φ(k)t πλ

RkΦ(k)

0 πRk

)=∫ ∞

0dt eiωt hk,λ(t),

(3.40)

where Φ(k)t = eiHRktΦke−iHRkt, yielding the modular two-time correlation function

hk,λ(t) = g2k tr(

Φ(k)t πλ

RkΦ(k)

0 π1− λRk

)(3.41)

obtained by performing a change of variable t→ t + iλβ in the second line of (3.40) andthen rewriting the reversible evolution operator in terms of the Gibbs state of the kthreservoir πRk

= e−βk HRk /tr(e−βk HRk ) as follows

Φ(k)t− iλβk

= eiHRk (t−iβkλ)Φke−iHRk (t−iβkλ) = e−βk(−λ)HRk Φ(k)t e−βkλHRk = π−λ

RkΦ(k)

t πλRk

. (3.42)

Now, for a non-thermal reservoir the modular two-time correlation functions become

hk,λ(t) = g2k tr(

Φ(k)t σλ

RkΦ(k)

0 σ1− λRk

), (3.43)

where σRkis the non-thermal state associated to the kth reservoir which is invariant under

the application of the environment’s reversible dynamics. In practice, one only needsto know this state to obtain the modular spectral function hk,λ(ω) =

∫ ∞0 dt eiωthk,λ(t)

which in turn when inserted into the master equation (3.39) make it truly nonlinearsince the calculations (3.40)–(3.42) can no longer be performed to recover the LDME.The resulting master equation is not equal to the NTMEs discussed later and is thereforea MDS nonlinear master equation in its own right.

We provide with the irreversible contribution (3.33) the first proper double commutatorformulation of the LDME. It is well known that the LDME asymptotically matches theHamiltonian dynamics for vanishing coupling strength [35, 36]. However, even for smallbut finite couplings, this link is broken so that the equation becomes one among otherphenomenological equations although it remains thermodynamical robust [37, 219, 220].Moreover, it neither contains non-secular terms nor can be used for systems possessinga continuous energy spectra.

39


3.2.2 The dynamically time-averaged NTME

Choice of the scattering operators To leave the strict WCL as well as being able toafford continuous energy spectrum, we have to specify a different scattering operatorthan the one associated to the LDME (3.27). This has to be done in order to stay (withinsome limit) in contact with the underlying microscopic picture as presented in Sec. 2.1.3.To this end, we introduce a dynamical coarse-graining time Tg, being a function of thesystem-reservoir coupling strength g, whose role is to ensure that in limit Tg → ∞ werecover the LDME. Keeping these elements in mind, let us choose for the scatteringoperators 12

Xk,λν =

1√2

e−λβkν

2

√hk(ν)Ak,ν, (3.44)

where we deliberately select for the frequencies the index ν instead of ω to avoid anyconfusion with the LDME eigenoperators (this choice of notation will become obviouswhen later specifying the equation for a discrete energy spectrum). They depend on thefollowing modified eigenoperators

Ak,ν =∫

Reiνt√

δTg(t) Qk,t dt, (3.45)

inspired from the Proposition 7.3 of [89], giving rise to the eigenvalue equation [77, 78]

[Ak,ν, HS ] = νAk,ν + 2ωg2 ∂ν Ak,ν (3.46)

cautiously proven in the Appendix 9.5 and which is associated to the coarse-grainingparameter ωg = 1/(2Tg) for all reservoirs 13. The smoothed eigenoperators (3.45) dependon the coupling operator Qk, which is attached to the kth reservoir, evolved according to

Qk,t = eiHS tQke−iHS t (3.47)

as well as on a normalized Gaussian distribution with zero mean

δTg(t) =1√

2πTgexp

(− t2

2T2g

)(3.48)

whose deviation is given by Tg. Hence, the Ak,ν’s are bounded operators compatiblewith discrete and continuous energy spectrum assuming that the coupling operator Qk

12The sum over αk in (3.10) is replaced by an integral over the continuous set of frequencies ν.13Ideally, one should use one dynamical coarse-graining time parameter per reservoir, i.e. Tk,g for the

kth scattering operator producing the contribution Jk(ρ) to the irreversible time-evolution J (ρ). For themoment, we stay with a single one for sake of simplicity.

40


is itself bounded. Indeed, they are bounded by a Tg-dependent expression

||Ak,ν|| ≤(∫

Rdt√

δTg(t))||Qk|| ≤ (8π)1/4

√Tg ||Qk||, (3.49)

where the term in parenthesis associated to (3.48) is a Gaussian integral always boundedfor Tg ∈ (0, ∞). We later discuss the transition from a NTME with continuous to adiscrete energy spectrum for Tg → ∞; see mainly the equations (4.4)–(4.9). Note alsothat the additional term in the eigenvalue equation (3.46) proportional to ωg =1/(2Tg)

vanishes in the same limit so that we recover the eigenvalue equation (3.35). Besides,the scattering operators (3.44) satisfy the self-adjointness property (3.14) since

(Xk,λν )† = (1/

√2) e−(1−λ)

βk(−ν)2

√hk(−ν) Ak,−ν,= Xk,1− λ

f (ν) , (3.50)

where we used the definition of the eigenoperators (3.45) giving the property

A†k,ν =

∫R

dt ei(−ν)t√

δTg(t) Q†k,t = Ak,−ν (3.51)

since Q†k,t = (Qke−iHS t)†(eiHS t)† = Qk,t according to (3.47) as well as the invertible func-

tion f (ν) = −ν and the microscopic detailed balance hk(ν) = eβνhk(−ν). In addition,

||[Xk,λν , HS ]|| = (1/

√2)∣∣∣e−λ

βkν

2

∣∣∣√hk(ν) ||[Ak,ν, HS ]|| < ∞ (3.52)

since ||[Ak,ν, HS ]|| is bounded by a Tg-dependent expression 14.

Calculation of the master equation Inserting the scattering operators (3.44) into theirreversible part of MDS master equation (3.10)–(3.11), where the set index αk runs nowover a continuous set of frequencies ν, yields the truly nonlinear contributions [77, 78]

Jk(ρ) =12

∫R

dν hk(ν) [A†k,ν, Kk,−ν

ρ [Ak,ν, Ψk]], (3.53)

expressed in terms of the frequency-dependent Kubo superoperator (3.34). It is no longerlinear since the operators Ak,ν appearing in the equation are associated to a modifiedeigenvalue equation (3.46) so that the analogue of the property (3.36) does not apply. Theirreversible contribution (3.53) implicitly embeds a dynamical time-averaging procedureas provided in the Definition 4.1 of [89] extending Davies’ technique (2.41), namely 15

Jk(ρ) =∫

Rdq δTg(q)U−q(Jk(Uq(ρ))), (3.54)

14See the calculation (9.15) at the end of Appendix 9.5 as well as |hk(ν)|<∞.15We explain in the Appendix 9.6 how to uncover the time-averaging procedure (3.54) from (3.53).

41


where Uq = eiHS q(·) e−iHS q and Jk is the time-evolution prior to the time-averagingwhich does not necessarily produce a positive time-evolution 16. This procedure alsoensures that one recovers the LDME in the WCL as later shown. Note that the NTME(3.53) includes a very similar dynamical time-averaging procedure as the CGME (2.64)but thanks to its nonlinear it leads the system towards the Gibbs state.

Non-thermal reservoir Similarly to what we did for the LDME (3.39) by rewriting theirreversible contribution (3.53) to make it temperature-independent, we obtain [77, 78]

Jk(ρ) =∫

ν

∫ 1

0dλ hk,λ(ν)[A†

k,ν, ρλ[Ak,ν, S]ρ1−λ]

+1ν

∂λ

(hk,λ(ν)

)[A†

k,ν, ρλ[Ak,ν, HS ]ρ1−λ]

(3.55)

whose modular spectral function hk,λ(ν) is given by (3.43). It corresponds to the masterequation (4) obtained in [78] and could be compared to the ”nonlinear LDME“ (3.39) byassuming a discrete energy spectrum.

3.2.3 The pre-averaged NTME

Choice of the scattering operators The irreversible contribution of the first NTME(3.53) has been “smoothly” attached to the LDME at the expense of introducing an ad hocparameter, the dynamical coarse-graining time Tg. However, it is not necessarily obvioushow to choose it according to a given physical setup. Looking back at the traditionalmaster equation derivations [20], one ends up at “halfway” of the LDME (2.38) with theBRME (2.35), namely after application of the Born-Markov approximation but beforeresorting to SA. The BRME, being a pre-averaged master equation, is widely used tomodel experiments despite the fact that it neither conserves the positivity of the densitymatrix nor of the entropy production [83, 84]. Can we postulate an equation which issimilar to the BRME but without its drawbacks? Let us introduce

Xk,λ =1√2

∑ω

e−λβkω

2

√hk(ω)Ak,ω, (3.56)

valid only for discrete energy spectrum, satisfying as expected the self-adjointnessproperty (3.14) since

(Xk,λ)† =1√2

∑ω

e−(1−λ)βk(−ω)

2

√hk(−ω) Ak,−ω =Xk,1− λ (3.57)

obtained in a similar way as for the LDME calculation (3.51). Again as for the LDME,the commutators [Xk,λ, HS ] for the scattering operator (3.56) are bounded which can beshown following the arguments which have been used to prove (3.29) or (3.52).

16Similarly to the case of BRME which is a “pre-averaged version” of the LDME.

42


Calculation of the master equation Inserting the scattering operators (3.56) into theirreversible part of master equation (3.10)–(3.11), where the set index αk contains onlyone scattering operator, yields the nonlinear irreversible contribution [77, 78]

Jk(ρ) =12 ∑

ω1,ω2

√hk(ω1)hk(ω2)

∫ 1

0dλ e−λβk

ω1+ω22 [A†

k,ω1, ρλ[Ak,ω2 , Ψk]ρ

1−λ], (3.58)

which could be rewritten in a more compact form with the frequency-dependent Kubosuperoperator (3.34). This equation is not equal to the LDME mainly because we havenow two sets of frequencies, i.e. ω1 and ω2. This equation is the corrected pre-averagedcounterpart of the dynamically time-averaged equation (3.53) as can be seen in theAppendix 9.6. In this respect, it should be compared to the BRME since it carries thesame type of non-secular terms playing a key role in physical applications as mentionedearlier. One immediately notices that the BRME (2.35) is not symmetric at the level ofthe spectral function compared to the pre-averaged NTME (3.58). In fact, one recognizesin the pre-averaged NTME (3.58) a kind of modular spectral matrix

hk,λ(ω1, ω2) =√

hk,λ(ω1)hk,λ(ω2) (3.59)

composed of two modular spectral functions (3.32) evaluated at a different frequencywhich reminds us of the coarse-grained spectral matrix (2.65) carried by the CGME (2.64)but without the coarse-graining parameter. For a closer analogue, see the time-averagedversion (4.12) provided later on which equally depends on a coarse-graining parameter.Let us also point out that the nonlinearity of the pre-averaged NTME (3.58) ensures theconvergence towards the correct Gibbs state contrary to the CGME 17.

Non-thermal reservoir Finally, to consider a non-thermal reservoir we again proceedas for the LDME (3.39) and recast the irreversible contribution (3.58) as [77, 78]

Jk(ρ) = ∑ω1,ω2

∫ 1

0dλ hk,λ(ω1, ω2)[A†

k,ω1, ρλ[Ak,ω2 , S]ρ1−λ]

+∂λ(hk,λ(ω1, ω2))(ω1+ω2

2

) [A†k,ω1

, ρλ[Ak,ω2 , HS ]ρ1−λ]

(3.60)

which is independent from the temperature thanks to the introduction of the modularspectral matrix hk,λ(ω1, ω2) provided in (3.59).

17The stationary state of the CGME depends on the select coarse-graining parameter.

43


3.2.4 The modular-free NTME

Choice of the scattering operators Up to now, we have exclusively used scatteringoperators (or friction matrices — see the MDS master equation (3.17) and associatedexplanations) depending on λ for the LDME (3.27) as well as the two NTMEs (3.44)and (3.56). At first glance, this choice seemed innocent but it is not. De facto, usingλ-independent scattering operators leading to a modular-free NTME implies that:

(i) the property (3.14), insuring the self-adjointness of the time-evolution, can nolonger be fulfilled with the help of the microscopic detailed balance,

(ii) the contact with the underlying Hamiltonian in the WCL is lost, i.e. we cannotrecover the LDME for vanishing coupling strengths.

To illustrate this, let us take the following scattering operators

Xkω =

1√2

√hk(ω) Ak,ω. (3.61)

Note that it is equal to the choice for the LDME (3.27) but with λ = 0 so that it does notsatisfy the self-adjointness property (3.14), namely

(Xkω)

† =1√2

e−βk(−ω)

2

√hk(−ω) Ak,−ω 6= Xk

f (ω), (3.62)

where we used, as previously, the property A†k,ω = Ak,−ω, the microscopic detailed

balance hk(ω) = eβkω hk(−ω) and the invertible function f (ω) = −ω. To avoid thisproblem, one could select a flat spectral function hk(ω)→ hk(0) for all ω’s to recover(Xk

ω)† =Xk

f (ω). Note also that the commutators [Xkω, HS ] for the scattering operators (3.61)

can be shown to be bounded following the calculations performed for the LDME (3.29)or the first NTME (3.52). In addition, the time-evolution still converges to the propersteady-state as long as the EP mentioned in Sec. 2.1.5 is respected. Since the role of themicroscopic detailed balance is lost for the scattering (3.61), one can fully give up thespectral decomposition of the Qk and use them per se

Xk =1√2

√hk(0) Qk, (3.63)

where one could take hk(0)=γk/βk with γk the friction associated to the kth reservoir.Since we can always choose coupling operators Qk being self-adjoint without lossof generality, we have that (Xk)† = Xk implying the time-evolution is self-adjoint.Moreover, the commutator [Xk, HS ] is bounded for a bounded Qk. It is noteworthythat this choice is compatible with unconfined systems (i.e. with a continuous energyspectrum) since it does not rely on the Lindblad eigenoperators.

44


Calculation of the master equation Let us start with the first scattering operator (3.61)and insert it into the irreversible part of the master equation (3.10)–(3.11) for a singlecoupling operator per reservoir, noticing the set index αk runs once again over a discreteset of frequencies ω, yielding the modular free NTME irreversible contribution 18

Jk(ρ) = −hk(0)

2 ∑ω

[A†k,ω, Kρ[Ak,ω, ln ρ + βk HS ]] (3.64)

which depends on the frequency-independent Kubo-Mori superoperator obtained from(3.34) for ω=0 or later given in (4.18). Now for the second choice (3.63) we obtain

Jk(ρ) = −γk

2βk[Qk, Kρ[Qk, ln ρ + βk HS ]]. (3.65)

Alternatively, one could write it as [221, 205, 77, 78]

Jk(ρ) = −12

(γk

βk[Qk, [Qk, ρ]]− γk [Qk, Kρ[Qk, HS ]]

)(3.66)

using the property Kρ[A, ln ρ] = [A, ρ] demonstrated in the Appendix 9.7 to compute[Qk, Kρ[Qk, ln ρ]] = [Qk, [Qk, ρ]]. The equation (3.66) structurally matches Grabert’sequation (5.22) derived by means of projection operators [221] since he uses the sametype of interaction Hamiltonian 19. On the other hand, Öttinger equally obtained theequation (3.66) in [205] using thermodynamical considerations, see equation (7) in [146].The factor 1

2 appearing in the three equations (3.64)–(3.66), absent from Grabert’s andÖttinger’s formulation, is important hereafter to compare it to the LDME (3.38), theLSME (2.44) and the other NTMEs (3.53) and (3.58). Note that one can obtain thefirst irreversible contribution (3.64) from the second one (3.65) by applying a time-averaging procedure as shown in the Appendix 9.8. Later, when we will look at thelinear response of a single qubit connected to a heat reservoir we will see clearly seethat the modular-free NTMEs (3.64)–(3.66) does not describe the same physics as theLDME (2.38); these nonlinear equation are valid in a different regime, namely the strong-damping or Brownian motion limit (BML) as point out by Grabert 20 when he considereda heavy particle in contact with a reservoir of lighter particles.

18By modular free we mean that the factor λ, originating from the dissipative bracket (3.5), is missing inthe scattering operator (3.63) leading to the irreversible contribution.

19Compare Grabert’s interaction Hamiltonian (5.8) in [221] with our choice (2.3).20This was Grabert’s intention in [221] but he made a mistake from equation (5.11)–(5.12) and therefore

only accidentally ended up with the nonlinear equation (5.22) therein matching (3.66). Nevertheless, as wewill see his original idea of BML was right. More generally, we will see that it can describe systems stronglycoupled to a hot classical Markovian environment such as the LSME (2.44) and NRSME (2.45).

45


Nam

eIrreversible

contributionρ(t)

>0

ρ(∞

)cont.spectum

?regim

eorigin

no

BRM

E−

12∑

ω1 ,ω

2 h(ω

2 ) (Aω

1 A†ω

2 ρ−A

†ω2 ρA

ω1+

h.c. )no

πno

WC

Rm

icro(2.35)

LDM

E∑

ωh(ω

) (A†ω

ρAω−

12 {Aω

A†ω ,ρ} )

yesπ

noW

CL

micro

(2.38)

LSME

γβ (QρQ−

12 {QQ

,ρ} )=−

12 (γβ[Q

,[Q,ρ]] )

yes1

yesH

TR,SC

Rm

icro(2.44)

NR

SME

−12 (

γβ[Q

,[Q,ρ]]+

γ2[Q

,{[Q

,HS ],ρ}

] )no

∼π

yesH

TR,SC

Rm

icro(2.45)

CLM

E−

12 (γβ[X

,[X,ρ]]+

iγ2m[X

,{P,ρ}

] )no

∼π

yesH

TR,BM

Lm

icro(2.46)

LKM

Eη (L

†ρL−

12 {LL†,ρ} )

=−

12 ([L†,L

ρ]−

[L,ρL†] )

yes1

yesunspecified

pheno(2.50)

CG

ME

∑ω

1 ,ω2 h

τ(ω1 ,ω

2 ) (Aω

1 ρA†ω

2 −12 {

A†ω

2 Aω

1 ,ρ })yes

∼π

noW

CR

hybrid(2.64)

1stNTM

E12 ∫

Rd

νh(ν)[A

†ν ,K−

νρ

[Aν ,Ψ

]]yes

πyes

WC

Rpheno

(3.53)

2ndN

TME

12∑

ω1 ,ω

2 √h(ω

1 )h(ω

2 )[A†ω

1 ,K−

ω1+

ω2

2ρ

[Aω

2 ,Ψ]]

yesπ

noW

CR

pheno(3.58)

3rdN

TME

−12 (

γβ[Q

,[Q,ρ]]+

γ[Q

,Kρ [Q

,HS ]] )

yesπ

yesH

TR,SC

Rpheno

(3.66)

3rdN

TME*

−12 (

γβ[X

,[X,ρ]]+

iγm[X

,Kρ P

] )yes

πyes

HTR

,BML

pheno(5.102)

Table3.1

–Su

mm

aryof

them

asterequ

ations;we

use

au

nique

reservoirand

coup

lingop

erator.T

heequ

ationscontain

eithera

spectralfunctionh(ω

)as

givenin

(2.25),acoarse-graining

matrix

hτ(ω

1 ,ω2 )

asprovided

in(2.65),a

frictionγ

fora

reservoiror

ηfor

anarbitrary

environment.Besides,they

dependeither

oneigenoperators

Aω

orA

νas

providedin

(2.12)–(2.13)or(3.45),a

Lindbladop

eratorL

ora

coup

lingop

eratorQ

(canbe

takenas

position

operator

Xfor

aharm

onicoscillator

ora

Brow

nianparticle).

Forthe

steady-state

ρ(∞

)the

equationsconverge

eitherto

theG

ibbsstate

π,only

partiallyto

theG

ibbsstate

∼π

orto

thefully

mixed

state1.W

eind

icateifa

master

equation

canafford

acontinu

ous

energysp

ectrum

(cont.spectru

m).A

bbreviations:Bloch-R

edfi

eldm

asterequation

(BRM

E),Lindblad-Davies

master

equation(LD

ME),Lindblad

singularm

asterequation

(LSME),non-rotating-w

avesingu

larm

asterequ

ation(N

RSM

E),C

aldeira-L

eggettmaster

equation

(CL

ME

),Lind

blad-K

ossakowskim

asterequ

ation(L

KM

E),

coarse-grainedm

asterequation

(CG

ME),1stor

dynamically

time-averaged

nonlineartherm

odynamic

master

equation(1stN

TME),

2ndor

pre-averagednonlinear

thermod

ynamic

master

equation

(2ndN

TM

E)and

3rdor

mod

ular

freenonlinear

thermod

ynamic

master

equation

(3rdN

TM

E).T

heM

assieuop

eratorΨ

isgiven

by(3.11)w

hilethe

frequency-d

epend

entKu

bosu

perop

eratorK

ωρ

comes

from(3.34)for

the1st/2nd

NTM

E.The3rd

NTM

Edepends

onthe

standardK

ubosuperoperator

Kρ

givenin

(4.18);thesecond

version(3rd

NT

ME*)is

fora

harmonic

oscillator/Brownian

particle.D

ifferentregimes:w

eak-couplinglim

it(WC

L),weak-coupling

regime

(WC

R),Brow

nianm

otionlim

it(BML),strong-coupling

regime

(SCR

)fora

white

noise(classical)reservoir,high

temperature

regime

(HT

R)

oru

nspecifi

ed.

Finally,w

eind

icateif

theequ

ationw

asobtained

through

am

icroscopic

derivation

(micro),

byphenom

enologicalmeans

(pheno)ora

mixture

ofthem(hybrid).

46

4 Practical implementation of thenonlinear master equations

The master equations presented in the Chapter 3, apart from the LDME (3.38), cannoteasily be studied due to their nonlinearity. The first idea which comes to mind is tolinearize these equations, for example, around the stationary state. The resulting linearequations can be studied analytically for simple systems such a single qubit exposedto a heat reservoir. However, to examine more complicated systems one has to resortto numerical implementations requiring a more explicit formulation for the differentNTMEs (3.53), (3.58) or (3.66) as well as for their linearized counterparts.

We begin with the discrete representation of the dynamically time-averaged NTME(3.53). Note that the dynamical coarse-graining time Tg appearing in this equation has tobe treated with care. We explain how to deal with it and how it can be fitted accordingto a reference time-evolution. Note that such a procedure is not required for the pre-averaged NTME (3.58) since it is free of the Tg parameter but in return is only validfor systems characterized by a discrete energy spectrum. Nevertheless, we provide atime-averaged version of this equation too for the sake of completeness. We then discussthe respective role of these two NTMEs. The modular free NTME (3.66) is simpler tohandle since it does not depend on the Lindblad eigenoperators Ak,ω but directly onthe coupling operator Qk. Thereafter, we furnish a general linearization procedure andapply it to the NTMEs. Finally, explicit formulations of the NTMEs in the density matrixbasis are computed which can be directly used to solve these equations numericallyusing appropriate integration schemes (e.g. Euler, Runge-Kutta or Adams-Bashforthnumerical integrators).

47

Chapter 4. Practical implementation of the nonlinear master equations

4.1 Discret formulation of the dynamically averaged NTME

When considering a system with discrete energy levels, we can simplify the Ak,ν’s byspecifying the time evolved coupling operator Qk,t in terms of the usual eigenoperators

Qk,t = ∑ω

e−iωt Ak,ω, (4.1)

where the sum runs over the spectrum of the reversible contribution ω ∈ sp(i[HS , ρ]).Thus, the modified eigenoperators (3.45) with (4.1) becomes

Ak,ν = ∑ω′

Ak,ω

∫R

dt ei(ν−ω)t√

δTg(t) =√

2π ∑ω

Ak,ω′√

δωg(ν−ω) (4.2)

depending on the frequency-dependent Gaussian

δωg(ν) =1√

2πωgexp

(− ν2

2ω2g

)(4.3)

obtained by computing the Fourier transform of√

δTg(t) in the first line of equation (4.2)

and using ωg = 1/(2Tg). We can now insert this result into the irreversible contribution(3.53) and get for the kth reservoir

Jk(ρ) =π

2 ∑ω1,ω2

∫R

dν hk(ν)√

2δωg(ν−ω1)√

2δωg(ν−ω2) [Ak,−ω1 , Kk,−νρ [Ak,ω2 , Ψk]].

(4.4)

Now using the equation (21) in [89] to rewrite the Dirac-delta functions

√2δωg(ν−ω1)

√2δωg(ν−ω2) =

√√√√δωg

(√2(ω1 −ω2)

2

)δωg

(√2(

ν− (ω1 + ω2)

2

))(4.5)

we can simplify the irreversible contributions (4.4) as

Jk(ρ) =12 ∑

ω1,ω2

∫ 1

0dλ hTg

k,λ(ω1, ω2)[Ak,−ω1 , ρλ[Ak,ω2 , Ψk]ρ1−λ], (4.6)

where we defined a Tg-dependent modular spectral matrix elements

hTgk,λ(ω1, ω2) = exp

(− (ω1 −ω2)2

8ω2g

) ∫R

dν hk,λ(ν)1√

2πωgexp

(−(ν− ω1+ω2

2 )2

2ω2g

)(4.7)

48

4.2. How to use the dynamical coarse-graining time?

expressed in terms of the modular spectral function hk,λ(ν) earlier defined in (3.32).Clearly, in the weak-coupling limit

limTg→∞

hTgk,λ(ω1, ω2) � δKr(ω1 −ω2) δD(ν− (ω1 + ω2)/2) hk,λ(ν), (4.8)

namely the first exponential in the Tg-dependent modular spectral matrix elements(4.7) produces a Kronecker-delta function δKr(ω1 − ω2) erasing all unequal discretefrequencies leaving us with a single set of frequencies ω1=ω2=ω simplifying ∑ω1,ω2

→∑ω. Besides, the second exponential in (4.7) produces in the same limit a Dirac-deltafunction δD(ν − (ω1 + ω2)/2) → δD(ν − ω) eliminating the integral over the set ofcontinuous frequencies ν. This fact is highly remarkable because it leads to

Jk ρ =12 ∑

ω

∫ 1

0dλ hk,λ(ω)[Ak,−ω, ρλ[Ak,ω, Ψk]ρ

1−λ] (4.9)

equal to the seemingly nonlinear irreversible contribution (3.33) according to the modu-lar spectral function (3.32) and frequency-dependent Kubo superoperator (3.34). More-over, from Sec. 3.2.1 and the property (3.36), one rapidely notices that we recovered theLDME. This dynamical time-averaging procedure is very similar to the one used in theCGME (2.64). Note that one could replace the double sinc function of the coarse-grainedspectral matrix (2.65) by the double Gaussians appearing in the Tg-dependent modularspectral matrix (4.7) which are numerically more robust (i.e. no oscillations).

4.2 How to use the dynamical coarse-graining time?

4.2.1 Role of the dynamical coarse-graining time

Mathematically, the Tg parameter plays the role of a “damping factor” for the integralof the modified eigenoperator Ak,ν as defined in (3.45) allowing to extend it to theentire real line R [77, 78]. Moreover, when considering continuous energy spectrum theGaussian nature of these eigenoperators insure that they are bounded and do not givea zero contribution contrary to the usual Lindblad eigenoperators Ak,ω; see the bound(3.49) and the Appendix 9.1 for further explanations. From a physical point of view, themodified Lindblad eigenoperators Ak,ν’s are designed to

(i) to ensure that we recover the LDME in the limit Tg → ∞,

(ii) be compatible with system displaying small energy gaps ∆min ∼ g.

(iii) accept slow processes in the system associated to a continuous energy spectrum.

To better understand the two last points, we have come back to the time scales separationpresented earlier in Sec. 2.1.2. To get ride of the second condition in (2.7) it is crucial

49


to not break the first one when selecting Tg. To this end, we choose it according tothe spectral function hk(ν) containing all the relevant information about the reservoir.More precisely, the inverse Fourier transform hk(t) of hk(ν) possesses a given relaxationprofile linked to the time scale tR for which one has to design a function f (t) ∝ e−t/Tg

being its envelope. The time Tg is then the smallest acceptable overestimate of tR suchthat one retains the full contribution of the correlation function. Thus, dependingon the examined physical system, it may (or may not) exist such an equation beingsimultaneously physically admissible and different from the usual LDME (i.e. if Tg isvery large, one would barely notice a difference). Thus, Tg can be understood in thiscase as an overestimation of the reservoir correlation time tR. It is useful to think how tointerpret the Tg parameter from the point of view of different physical setups. First of all,it can be understood as the observation time required to distinguish the characteristicfrequencies of a system possessing discrete energy levels or equivalently understandωg =1/(2Tg) as the energy resolution of the measurement apparatus [35]. Hence, fora Tg → ∞ we perfectly resolve the energy levels whereas for Tg → 0 we look at thespectrum as if it were composed of pure white noise because the ”measurement error“ωg is infinite implying that we are totally unable to perceive the individual energylevels. Alternatively, Tg can be understood as the time required to experience the energyconservation during the scattering process taking place between the system and thereservoir. For example, considering a heavy particle interacting with the lighter particlesof the reservoir the parameter Tg represents the time required to capture the energyconservation up to an accuracy ωg. Accordingly, for intermediate values we measurea “gaussified” energy conservation illustrated in the Fig. 4.1. It becomes ”sharp“ forωg → 0 whereas for ωg → ∞ the conservation process is totally blurred although themomentum conservation is maintained.

4.2.2 Redefinition of the standard timescale separations

We can now rewrite the Markovian time scale separation (2.7) including Tg, that is

tR � Tg � tS ∼ g−2 (4.10)

with Tg = g−ζ T with T > 0 and 0 < ζ < 2 with ζ a free parameter 1 typically taken asζ=1. The parameter T can be used for fine-tuning when considering a precise physicalsetup. Let us look at two situations where the extended version of the Markovian timescale separation (4.10) is helpful. Firstly, we know that when the energy splitting islarger than the coupling strengths the RWA breaks down (i.e. ∆max� g). This problemis similar to the limit of a continuous energy spectrum 2. This issue does not arise

1In [89] the parameter ζ = 1 is selected to converge for Tg→∞ to Davies’ generator. The time scaleseparation τ appearing in the CGME (2.64) play a similar role as Tg (see also the relation (14) in [198]).

2The limit of continuous energy spectrum is a much more stringent constraint than the break downlimit of the RWA ∆max & g or tH . g−1. Note also that we “relax” the SA when introducing the dynamicaltime-averaging since we no longer eliminate the fast oscillating phases.

50

4.2. How to use the dynamical coarse-graining time?

Figure 4.1 – Feynman diagram of the collision process between the particle of interestand the reservoir modeled according to the dynamically time-averaged NTME (3.53):the heavy system’s particle, which is carrying a momentum p′ and energy ε(p′), collideswith one of the multiple lighter reservoir’s particles having a momentum q and absorbsduring the process an energy ν (note that we use h= 1) so that it back-scatters with amomentum p and energy ε(p). The disk with a radius Eg =ωg represents the Gaussian(3.48) of the modified eigenoperators (3.45). For an infinite Tg we have ωg = 0 suchthat the Gaussian δTg(ε) is replaced by a Dirac-delta and we recover a “sharp” collisionprocess with exact energy conservation (i.e. the sphere shrinks to a single point).

with the dynamically time-averaged NTME (3.53) or (4.6)–(4.7) for which the boundtS� tH is replaced by tS� Tg being independent of the energy splitting for a smallbut non-vanishing coupling g. Thus, it will be possible to study setups where the RWAbreaks down. Secondly, it is well-known that the realization of quantum logic gates isgenerally not possible in the WCL since the short-time dynamics is washed out [222].Alicki pointed out in [133] that the time required to apply a quantum gate is of theorder tgate ∼ 1/∆gate with ∆gate the difference between the typical Bohr frequencies ofthe Hamiltonian Hgate generating the gate 3. This becomes obvious when noting that aquantum gate is nothing else than an unitary transformation 4 reading Ugate = e−itgate Hgate .However, the Markovian limit imposes that the observation time tS is much larger thantgate which is problematic for small gaps ∆gate required for fast pulses (the effective Bohrspectrum is then quasi-continuous). With the dynamically time-averaged NTME thisis not a problem; Tg is the free parameter missing in the standard Davies theory whichsolves this issue 5 through the modified eigenoperators Ak,ν as provided in (3.45).

3The Hamiltonian Hgate more or less corresponds to the system Hamiltonian HS of the microscopicdescription of Sec. 2.1. Note also that for a two-state system ∆gate =ω2 −ω1

4To be more precise, this bound is estimated from the time required to transform an initial state to itsorthogonal counterpart which needs at least tmin =π/2ω according to Margolus-Levitin theorem. Thus, iftgate is scaled up it implies that the eigenvalues of Hgate must scale down and vice-versa [133].

5Nonetheless, one has still to provide a proper framework for time-dependent Hamiltonians. In theadiabatic limit for slow pulses, we do it later in the linear response regime in Sec. 5.2. For fast periodicpulses, one could simply adapt Alicki’s scheme used to model the quantum logic gate in [133] startingfrom the tutorial [223] or looking at [224]. In this case, the dynamical coarse-graining time Tg permits todisconnect the perturbation scheme from the condition (46) given in [133].

51


4.2.3 Matching a given time-evolution

The coarse-graining time Tg can be exploited to ensure that the time-evolution generatedby the Tg-dependent master equation (3.53) gets closer to a target time-evolution (e.g. anexact result or experimental data). To this end, we can resort to the integrated trace-normdistance between the Tg and target time-evolutions 6

Dtmax(ρTg , ρtarget)=1

2tmax

∫ tmax

0dt ||ρTg(t)− ρtarget(t)||1, (4.11)

where ||A||1=√

A† A is the Schatten 1-norm 7. If we assume that ζ=1 for Tg = g−ζ T, weonly have to find the value of T which minimizes the quantity (4.11) under the time-scalerestriction (4.10). To determine a reasonable functional dependence for T, one repeatsthe analysis for a different set of parameters characterizing the system of interest.

4.2.4 Time-averaging the pre-averaged NTME

It is also important to point out that even if (3.58) gives similar results as the LDME for(very) small coupling constants one has to apply the time-averaging procedure in thespirit of what is done in [89] (see Definition 4.1 therein) to exactly recover the LDME.Luckily, since the nonlinear equation (3.58) is expressed in terms of the eigenoperatorsAk,ω and we are already in the Schrödinger picture it only remains to integrate thetime-evolution over a pair of Gaussian ”smoothing“ functions as defined in (4.3) and totake the limit of vanishing coupling. Exploiting the fact we are working with a discreteenergy spectrum, we can ”recycle“ our calculations performed for the first NTME (4.4)and modify the irreversible contribution (3.58) as follows

J (Tg)

k (ρ) = 2π∫

Rdν√

δωg(ν−ω1)√

δωg(ν−ω2)×(12 ∑

ω1,ω2

√hk(ω1)hk(ω2)

∫ 1

0dλe−λβk

ω1+ω22 [A†

k,ω1, [ρλ[Ak,ω2 , Ψk]ρ

1−λ]

)

=12 ∑

ω1,ω2

exp

(− (ω1 −ω2)2

8ω2g

)√hk(ω1)hk(ω2)×

∫R

dν δωg

(ν− ω1 + ω2

2

) ∫ 1

0dλe−λβkν [A†


1−λ]

=12 ∑

ω1,ω2

∫ 1

0dλ h

Tgk,λ(ω1, ω2) [A†


1−λ]

(4.12)

6This strategy has been exploited to study a master equation depending on a similar kind of scalingparameter, see Sec. VI and VII of [104].

7A summary of useful norms and relation among them can e.g. be found in the Appendix A of [120].

52

4.3. Why two NTMEs to explore the WCR?

depending on new Tg-dependent modular spectral matrix which reads

hTgk,λ(ω1, ω2) = exp

(− (ω1 −ω2)2

8ω2g

)√hk(ω1)hk(ω2)

∫R

dν δωg

(ν− ω1 + ω2

2

)e−λβkν,

(4.13)

where the underlining indicates that it is not equal to the Tg-dependent the modularspectral matrix obtained in (4.7) for the dynamically time-averaged NTME (4.4). In thefirst case (4.7) the spectral function is inside of the integral over ν whereas in the secondcase (4.13) the product of two spectral functions is outside of the integral. Thus, the twotime-averaged NTMEs (4.4) and (4.12) are not equal. Obviously, in the weak-couplinglimit Tg→∞ we obtain the expected Kronecker-delta and Dirac-delta function

limTg→∞

hTgk,λ(ω1, ω2) �

√hk(ω1)hk(ω2)δKr(ω1−ω2) δD(ν− (ω1 + ω2)/2) e−λβkν (4.14)

so that NTME (4.12)–(4.13) leads to the LDME in the form (4.9).

4.3 Why two NTMEs to explore the WCR?

One could think that having two NTMEs (in fact three if the NTME (4.13) is considered),to model a system in the WCR is somehow redundant 8. We have seen in Appendix 9.6that one cannot obtain a pre-averaged equation (i.e. independent of the dynamicalcoarse-graining) directly from the NTME given by (3.53). On the other hand, the coarse-graining time Tg can be optimized to approach a given time-evolution more closely asseen in Sec. 4.2.3. The BRME have been obtained from first principles whereas the CGMEcontains an ad-hoc coarse-graining parameter while the NTMEs have been postulated;this is in fact not really a problem because the contact to the underlying Hamiltoniandynamics can only be established in the WCL so that a priori none of these equationsare better than any others for non-vanishing couplings under this sole criteria. Thus,it is only once they are confronted with precise experimental outcomes that one canreally judge their respective range of validity. Nevertheless, the pre-average NTME(3.58) is very similar to the BRME (2.35) but without its drawbacks. Moreover, contraryto the CGME (2.64) it relaxes to the correct Gibbs state and is free of the coarse-grainingparameter making it, in practice, particularly appealing.

8The third NTME (3.66) should be used in the BML as briefly pointed out in Sec. 3.2.4. Note also that thetime-averaged version (4.12) of the pre-averaged NTME (3.58) is restricted to discrete energy spectrumsand is therefore less general than the dynamically time-averaged NTME (3.53).

53


4.4 Linearization around equilibrium

4.4.1 General procedure: small perturbation around the Gibbs state

We follow Grabert’s thermodynamic method to linearize master equations [225, 221]. Itrelies on a density matrix brought out of the equilibrium by the application of a smallperturbation to the Gibbs state using an arbitrary force operator δA with ||δA|| � 1,namely

ρδA =e−βk(HS−δA)

ZδA, (4.15)

where ZδA = tr(e−βk(HS−δA)), so that in the limit of vanishing δA we recover Zβk = ZδA→0

and πk = ρδA→0. Therewith, we have to show that the central commutator in the masterequation (3.9) becomes near equilibrium

ρλ[Xk,λαk

, Ψk]ρ1−λ → −πk

λ[Xk,λαk

, K−1πk

ρ]πk1−λ + o(||δA||). (4.16)

To this end, we linearize the density matrix in δA ignoring all higher order terms.Accordingly, up to a c-number depending on Zβk and ZδA, one computes the deviationfrom the kth Gibbs state

ρδA − πk = βkKπk(δA) + o(||δA||). (4.17)

thoughtfully derived in the Appendix 9.9. The previous formula is expressed in termsof the famous Kubo-Mori superoperator

Kρ(B) =∫ 1

0dλ ρλBρ1−λ (4.18)

applied to an arbitrary operator B. From the inversion of the expression (4.17) for ρδA = ρ

we get

K−1πk

(ρ− πk) = βkδA + o(||δA||) = ln ρ− ln πk + o(||δA||), (4.19)

using the natural logarithm of the perturbed density matrix (4.15) giving βkδA = ln ρ−ln πk − ln (Zβk /ZδA) and ignoring the c-number ln (Zβk /ZδA); note that we used theinverse of the Kubo superoperator (4.18) which can be written

K−1ρ (B)=

∫ ∞

0ds (ρ + s1)−1B (ρ + s1)−1. (4.20)

Be aware that we do not explicitly use this integral representation but only the fact thatthe inverse of the Kubo superoperator exists and is well defined (since it is a Krausmap [226] mapping positive operators to positive operators); we indicated it here forthe sake of completeness. We are then ready to rewrite with (4.19) the Massieu operator

54

4.4. Linearization around equilibrium

(3.11) near equilibrium as

−Ψk = ln ρ+ βk HS = ln ρ− ln (πk) = K−1πk

(ρ−πk) + o(||δA||) = K−1πk

ρ−1+ o(||δA||),(4.21)

where once again the c-number produced by Zβk was neglected. Notice that we com-puted K−1

πkπk = 1 from Kπk1 = πk using the Kubo superoperator (4.18). Now we can

evaluate the central commutator present in Eq. (3.10) containing the Massieu operatorwith (4.21) yielding

[Xk,λαk

, Ψk] = −[Xk,λαk

, K−1πk

ρ] + o(||δA||) (4.22)

so that all terms proportional to the identity vanish (justifying why we neglected thec-numbers in the previous calculations). Our last approximation consists in evaluatingthe density matrices which surround the previous commutator (4.22) at equilibriumfinally giving the sought relation (4.16) because ρλ

δAδAρ1−λδA = πk

λδAπk1−λ + o(||δA||).

One should never forget that such type of linearization has a tendency to create a non-positive time evolution for states being too far away from equilibrium as pointed out byGrabert 9. As we will later see, this is not a problem since such an equation is exclusivelyintended for the response theory close to equilibrium. If one still would like to useit away from this regime, one should always check that the density matrix remainspositive (similarly as one would do when working with the BRME).

Thus, from (4.16) we obtain the linear counterpart of the nonlinear term (3.10) associatedto the kth reservoir for a vanishing perturbation ||δA|| → 0, namely

Jkρ = −∑αk

∫ 1

0dλ [(Xk,λ

αk)†, πλ

k [Xk,λαk

, K−1πk

ρ]π1−λk ] (4.23)

giving rise to the linearized master equation

∂tρ = −i[HS , ρ] + J ρ, J ρ =M

∑kJkρ. (4.24)

For later convenience, we denote L the self-adjoint generator of the linearized reduceddynamics so that (4.24) reads

∂tρ = Lρ (4.25)

9In fact, the positivity of the density matrix during the time-evolution depends on the choice of thescattering operator defining the nature of the master equation. For example, the linearization of the LDMEis the LDME itself so that it preserves the positive of ρ.

55


permitting to define next to equilibrium the propagator or evolution operator 10

Wt = eLt, ρt = Wtρ0 (4.26)

as well as a Heisenberg evolution of an observable

A(t) = eL∗t A, (4.27)

where the dual L∗ is defined with the help of the identity tr(L(A)B) = tr(AL∗(B)) forall A and B.

4.4.2 Linearization of the dynamically time-averaged NTME

We can now linearize the obtained master equation (3.53) next to equilibrium followingthe method exposed in Sec. 4.4.1. One can directly use the result (4.23) with the scatteringoperator (3.44) yielding

Jkρ = −12

∫R


πk[Ak,ν, K−1

πkρ]]. (4.28)

This equation has a tendency to create a non-positive time evolution for states beingtoo far away from equilibrium. Neverthless, it is perfect to study the FDT (see Sec. 5.2)and the GKF (see Sec. 5.3), valid next to equilibrium presented, beyond the strict WCL.This linearization has the additional benefit to agree with the LDME in the limit Tg → ∞since the double commutator becomes

[Ak,−ν, Kk,−νπk

[Ak,ν, K−1πk

ρ]]Tg→∞, ν→ω−→ [Ak,−ω, Ak,ωρ− e−βkωρAk,ω] (4.29)

as explained in details in Appendix 9.10. Notice that we went from a set of continuousfrequencies to a discrete one; this is clearly illustrated in the next section where weprovide a formulation exclusively valid for discrete energy spectrum. We are then onestep away from the LDME, i.e. take a look at the equations (3.36)–(3.37) of Sec. 3.2.1.

4.4.3 Linearization of the pre-averaged NTME

Following the same strategy as for the first NTME, we can linearize (3.58) near equilib-rium following Sec. 4.4.1 or directly take the linearized equation (4.23) with the scatteringoperator (3.56) yielding

Jkρ = −12 ∑

ω1,ω2

√hk(ω1)hk(ω2) [A†

k,ω1, Kk,− ω1+ω2

2πk [Ak,ω2 , K−1

πkρ]]. (4.30)

10Up to now, since the master equation generating the MDS was not necessarily linear, it was impossibleto define the propagator Wt, i.e. the exponential generator of the dynamics. Notice that one has still tospecify the scattering operators to fully define Wt.

56

4.5. Explicit formulation of the NTMEs

The pre-averaged NTME (3.58) and its linearization (4.30) are much simpler to handleanalytically as well as numerically than its dynamically time-averaged counterpart (3.53)and (4.28). Applying a time-averging as we did in Sec. 4.2.4 to this equation leads to theLDME.

4.4.4 Linearization of the modular-free NTME

As previously, we perform the linearization of (3.65) expressed in terms of the Massieuoperator (3.11) following Sec. 4.4.1 or directly take the linearized equation (4.23) withthe scattering operator (3.63) yielding

Jk(ρ) = −γk

2βk[Qk, Kπk [Qk, K−1

πkρ]] (4.31)

which is much simpler in comparison to the two other NTMEs (4.28) and (4.30). If weperform the same treatment on the time-averaged version (3.64) following the Appendix9.8 one immediately notices that it is not equivalent to the LDME. This point will becomeobvious when looking at the linear response of a two-level system.

4.5 Explicit formulation of the NTMEs

4.5.1 The dynamically time-averaged NTME

Now coming back to the master equation (4.4), we rewrite it using: (i) the explicit defini-tion of the frequency-dependent superoperator (3.34), (ii) the density matrix decomposi-tion ρ=∑n pn |pn〉〈pn|, where pn and |pn〉 are its eigenvalues and orthonormal eigen-vectors, and the identity matrix as 1=∑n |pn〉〈pn|, (iii) the property ρλ |pn〉= pλ

n |pn〉.Thus, we get

Jk(ρ) =−12 ∑

ω1,ω2

εω1ω2ωg ∑

n,m

∫R

dν hk(ν) ξω1ω2ωg

(ν)×

∫ 1

0dλ e−βkνλ pλ

n p1−λm [Ak,−ω1 , |pn〉〈pn| [Ak,ω2 , ln ρ + βk HS ] |pm〉〈pm|]

=− 12 ∑

ω1,ω2

εω1ω2ωg ∑

n,m

∫R


(ν)×

βkω2 − ln (pn/pm)

βkν − ln (pn/pm)

(e−βkν pn − pm

)[Ak,−ω1 , (Ak,ω2)nm |pn〉〈pm|],

(4.32)

where we used the following shorthand notations for the Gaussian functions

εω1ω2ωg

= exp

(− (ω1 −ω2)2

8ω2g

), ξω1ω2

ωg(ν) = δωg

(ν− (ω1 + ω2)

2

). (4.33)

57


We applied the definition of the Massieu operator (3.11) and the eigenvalue equation[Aω2 , HS ]=ω2Aω2 so that

([Ak,ω2 , ln ρ + βk HS ])nm = (Ak,ω2)nm(βkω2 − ln (pn/pm)), (4.34)

where (·)nm = 〈pn| · |pm〉, and solved the integral

∫ 1

0dλ e−λ(βkν−ln (pn/pm)) =

e−βkν pn − pm

(βkν− ln (pn/pm)). (4.35)

Thereby, we can recast the previous master equation into a compact form by definingthe function

f k,ω1ω2nm =

∫R

dν hk(ν) ξω1,ω2ωg

(ν) gk,ω2nm (ν) (4.36)

depending on the Gaussian (4.33) and on the eigenvalues of the density matrix through

gk,ω2nm (ν) =


) βkω2 − ln (pn/pm)

βkν − ln (pn/pm). (4.37)

Finally, the nonlinear equation (4.32) becomes

Jk(ρ) = −12 ∑

ω1,ω2

[Ak,−ω1 ,Ak,ω1ω2ωg

(ρ)] (4.38)

with the superoperator

Ak,ω1ω2ωg

(ρ) = εω1ω2ωg ∑

n,mf k,ω1ω2nm (Ak,ω2)nm |pn〉〈pm| , (4.39)

where the kth reservoir acts on the system through the coefficients f k,ω1ω2nm given in (4.36)

depending on the spectral function hk(ν). Now, if we take the Gaussian functions (4.33)in the limit Tg → ∞ in conjunction with the master equation (4.38) we obtain

Jk(ρ) = −12 ∑

ω

[Ak,−ω,Ak,ω0 (ρ)], (4.40)

where the superoperator (4.39) has been evaluated at ωg = 0 yielding

Ak,ω0 (ρ) = hk(ω)∑

nm


)(Ak,ω)nm |pn〉〈pm| (4.41)

being linear in ρ and equal to the LDME provided in Sec. 3.2.1 but written in the densitymatrix basis. In practice, one can directly implement the equations (4.38)–(4.39) and(4.40)–(4.41) and solve them numerically, for example, to compare them.

58


The same procedure can be applied to the linearization (4.29) yielding11

Jkρ = −12 ∑

ω1,ω2

εω1ω2ωg

∫R


(ν)[Ak,−ω1 , Kk,−νπk

[Ak,ω2 , K−1πk

ρ]]

= −12 ∑

ω1,ω2

εω1ω2ωg

∫R


(ν)×

∑n,m

f k,νnm ∑

r,sgk,0

rs ρrs[Ak,−ω1 , 〈 pk,n| [Ak,ω2 , | pk,r〉〈 pk,s|] | pk,m〉 | pk,n〉〈 pk,m|]

(4.42)

provided in terms of the eigenvalues pk,n of the equilibrium density matrix πk andits eigenvectors | pk,n〉 such that πk | pk,n〉 = pk,n | pk,n〉 as well as HS | pk,n〉 = Ek,n | pk,n〉because πk ∝ e−βk HS . Moreover, we have that ρrs = 〈 pk,r| ρ | pk,s〉. In the second line wehave introduced the following two coefficients

f k,νnm = − e−βkν pk,n − pk,m

βkν− ln ( pk,n/ pk,m)(4.43)

gk,0rs =

ln pk,r − ln pk,s

pk,r − pk,s(4.44)

again expressed in terms of the eigenvalues pk,n of πk; notice that f k,0nm = (gk,0

rs )−1.

It is worth to notice that this basis is proportional to the Hamiltonian basis of thesystem because the equilibrium state attached to kth reservoir is πk ∝ exp (−βk HS). SeeAppendix 9.11 for a thorough derivation of equation (4.42). One can show that thisequation is not of Lindblad type according to the criteria of complete positivity for adiscrete N-level system proposed by Gorini and co-workers [34, 168] for the case of atwo-level system. Now for convenience we can recast the equation (4.42) in a form

Jkρ = −12 ∑

ω1,ω2

εω1ω2ωg

[Ak,−ω1 , Ck,ω1ρ (Ak,ω2)] (4.45)

close to the original nonlinear master equation (4.38) with the superoperator

Ck,ω1ρ (Aω2)=

∫R

dν hk(ν)ξω1ω2ωg

(ν) ∑n,m

χk,ω2nm (ν) | pk,n〉〈 pk,m| (4.46)

depending on the coefficients

χk,ω2nm (ν) = f k,ν

nm ∑r,s

g0rs ρrs 〈 pk,n| [Ak,−ω2 , | pk,r〉〈 pk,s|] | pk,m〉

which are clearly linear in ρ and depend on (4.43) and (4.44).11The main advantage compared to the nonlinear form is that the diagonalization of ρ is replaced by the

diagonalization of πk which needs only to be carried out once.

59


4.5.2 The pre-averaged NTME

We proceed as previously and express (3.58) with the definition of the Massieu operator(3.11) in the density matrix basis {|pn〉} giving

Jk(ρ) =−12 ∑

ω1,ω2

√hk(ω1)hk(ω2) [A†

ω1, ∑

n,m|pn〉〈pn| ρλ[Aω2 , ln ρ + βk HS ]ρ1−λ |pm〉〈pm|]

=− 12 ∑

ω1,ω2

√hk(ω1)hk(ω2) pm

(∫ 1

0dλ e−λβk

ω1+ω22 (pn/pm)

λ)×

(ln (pn/pm)− βkω2) [A†ω1

, (Aω2)nm |pn〉〈pm|]

=− 12 ∑

ω1,ω2

√hk(ω1)hk(ω2)

(ln (pn/pm)− βkω2

ln (pn/pm)− βkω1+ω2

2

)×(

e−βkω1+ω2

2 pn − pm

)[A†

ω1, (Aω2)nm |pn〉〈pm|],

(4.47)

where we used [Aω2 , HS ]=ω2Aω2 and computed

∫ 1

0dλ e−λ

(ln(

pnpm

)−βk

ω1+ω22

)=

(e−βk

ω1+ω22

pnpm− 1)

ln (pn/pm)− βkω1+ω2

2

. (4.48)

Note that imposing ω1=ω2 exactly matches the LDME (4.40)–(4.41) 12. For the linerizedform (4.30), one uses the basis of the equilibrium density matrix {| pk,n〉} leading to

Jkρ = −12 ∑

ω1,ω2

√hk(ω1)hk(ω2) [Ak,−ω1 , Kk,−( ω1+ω2

2 )πk [Ak,ω2 , K−1

πkρ]]

= −12 ∑

ω1,ω2

√hk(ω1)hk(ω2)×

∑n,m

f k,( ω1+ω22 )

nm ∑r,s

gk,0rs ρrs[Ak,−ω1 , 〈 pk,n| [Ak,ω2 , | pk,r〉〈 pk,s|] | pk,m〉 | pk,n〉〈 pk,m|],

(4.49)

where to perform the calculations we apply the relations (9.44) and (9.47) of Ap-pendix 9.11. The functions f k,ω

nm and gk,0rs are respectively given by (4.43) and (4.44).

12In this way, we eliminate “by hand” all non-secular terms characterized by ω1 6= ω2.

60


4.5.3 The modular-free NTME

Because the λ-independent NTME (3.65) is not expressed in terms of the eigenoperatorsAk,ω, we cannot use an identity of the type (4.34). Instead, let us focus on the explicitcalculation of the Kubo superoperator (4.18) appearing in (3.65) adapting the calculation(9.46) of the Appendix 9.11, namely

Kρ[Qk, ln ρ + βk HS ] = ∑n,m

∫ 1

0dλ ρλ |pn〉〈pn| [Qk, ln ρ + βk HS ] |pm〉〈pm| ρλ

= ∑n,m

(pn − pm

ln pn − ln pm

)〈pn| [Qk, ln ρ + βk HS ] |pm〉 |pn〉〈pm|

(4.50)

so that the irreversible contribution reads

Jk(ρ) = −γk

2βk∑n,m

(pn − pm

ln pn − ln pm

)[Qk, 〈pn| [Qk, ln ρ + βk HS ] |pm〉 |pn〉〈pm|] (4.51)

On the other hand, for linearized form (4.31) we have to compute

Kπk [Qk, K−1πk

ρ] = ∑n,m

∫ 1

0dλ πλ

k | pn〉〈 pn| [Qk, K−1πk

ρ] | pm〉〈 pm|πλk

= ∑n,m

∑r,s

(gk,0

rs

gk,0nm

)ρrs | pn〉〈 pn| [Qk, | pr〉〈 ps|] | pm〉〈 pm|

(4.52)

since K−1πk

ρ=∑r,s gk,0rs ρrs | pr〉〈 ps| according to the calculation (9.47), where as previously

pk,r and | pk,r〉 are the eigenvalues and eigenvectors of πk while the coefficients gk,0nm and

gk,0rs are given by (4.44). Thus, we obtain the linearized irreversible contribution

Jk(ρ) = −γk

2βk∑n,m

∑r,s

(gk,0

rs

gk,0nm

)ρrs [Qk, | pn〉〈 pn| [Qk, | pr〉〈 ps|] | pm〉〈 pm|]. (4.53)

Clearly, the first NTME (4.32) is the most tedious nonlinear master equation to solveanalytically as well as numerically whereas the second (4.47) and the third (4.51) NTMEsare easier to use in practice. In fact, the third one is the simplest since it is free of the Qk

decomposition into the eigenoperators Ak,ω.

61

5 Response theory

The master equation (3.9)–(3.11) has been designed to treat situations where the re-duced system is far away from equilibrium. However, many physical properties canalready be extracted when the system of interest is set close to equilibrium, typicallyin the linear response regime. In this context, two universal statements providing struc-tural information about the system can be obtained, namely the Callen-Welton-Kubofluctuation-dissipation theorem (FDT) [216, 227] which gives susceptibilities whereasthe Green-Kubo formula (GKF) [228, 229] yields transport coefficients. The idea behindthe FDT and the GKF is that the average linear response of the system exposed to a weakperturbation is equal to its internal equilibrium fluctuations in absence of disturbancecharacterized by a canonical correlation function. Briefly said, these schemes allow toperform a two-point measurement on an inspected setup by relating in time a pair ofphysical quantities. The two approaches rely on different sources to produce the distur-bance. The FDT requires to apply an external adiabatically time-varying driving on thereduced system connected to a single reservoir. On the other hand, the GKF demands toconnect two or more reservoirs, having different temperatures or chemical potentials,to the reduced system so that the fluxes crossing it produce a weak thermodynamicaldriving. Note that one could develop other kind of perturbation schemes as pointed outin the Appendix 9.12 and study nonequilibrium fluctuations by adapting the strategyused in [86] for the NTMEs as we propose in [230]. We have discussed the FDT forOQS in [231] following the original idea of [232] whereas the GKF, and its associatedOnsager-reciprocity relations (ORR) [233], have been examined in [37, 234, 219, 220]. Wefollow here the formulation proposed in [77, 78].

Although the MDS master equation (3.9)–(3.11) is in general nonlinear, we use thelinearization obtain in Sec. 4.4 of the previous chapter to compute the FDT and GKFunder fairly general settings. To do so, we first linearize the entropy production, showthat the linearized equation (4.23)–(4.24) is purely dissipative and define a Kubo-kind ofquantum detailed balance. Finally, the FDT and GKF are applied to a two level systemexposed to a thermal environment.

63

Chapter 5. Response theory

5.1 Essential tools to work in the linear response regime

In this section, we develop the tools necessary to obtain a proper FDT and GKF. Webegin by showing how to linearize the entropy production (3.25) and that it remainspositive. We then proof that that the linearized time-evolution obtained in Sec. (4.4.1) ispurely dissipative 1 and provide a thermodynamical reformulation of the usual quantumdetailed balance presented in Sec. 2.2.4.

5.1.1 Linearized entropy production

The entropic contribution St(A) = tr(J (ρ)A) introduced at the beginning of Sec. 3.1.1with (3.4) becomes linear near equilibrium since the time-evolution is there given by(4.23)–(4.24) yielding after few rearrangements

St(A)=∑k

tr((Jkρ)A

)=−

M

∑k

∑αk

〈[[A, K−1πk

ρ]]βk

πk ,αk〉 (5.1)

whose individual averaged dissipative brackets are given by the previous expression(3.8) but with the replacement ρ→ πk. One can also recompute the entropy productionas previously or simply take (3.25) and evaluate the dissipative bracket with (4.22) giving

σS(ρ) = ∑k

∑αk

〈[[K−1πk

ρ, K−1πk

ρ]]βk

πk ,αk〉 ≥ 0 (5.2)

again fulfilling the second law of thermodynamics but this time close to equilibrium.

5.1.2 Purely dissipative nature of the linearized evolution superoperator

Because the generator L in (4.23)–(4.25) is linear, we have access to the Liouville spacerepresentation (e.g. see Chap. 3 of [235]; highlighted hereafter with a bold notation)contrary to the nonlinear form. There, one can show that the obtained generator is purelydissipative 2. More precisely, the eigenvalue equation for the irreversible contributiontakes the form

J ρ = ηρ, Re(η) ≤ 0, (5.3)

where J is a N2×N2 Liouville matrix, ρ is a N2× 1 state vector and N is the dimensionof the discrete Hilbert spaceH. The real part of the eigenvalues η are all negative apart

1This notion is in the same vein as a relaxing semigroup, see Sec. 4.4 and Theorem 4.4.2 of [23]. If L isthe generator of the dynamics (obtained for the NTMEs by performing a linearization around equilibrium,see Sec. 4.4), it means that real part of the eigenvalues of L is strictly negative aprat from one eigenvaluebeing equal to zero and which is associated to the stationary (or steady) state. See Sec. 5.1.2 below.

2This does not mean that the Hamiltonian part of the generator is null but that (5.3) holds.

64

5.1. Essential tools to work in the linear response regime

from one which is equal to zero and is attached to the steady-state (i.e. η = 0 for Lπ = 0).Since J ρ = ∑k J kρ we only have to show that each individual contribution is purelydissipative, i.e. J kρ = ηkρ with Re(ηk) ≤ 0 for k = 1, . . . , M. This can be achieved bycomputing the identity

tr((J kρ)(K−1

πkρ)†)= tr

((ηkρ)(K−1

πkρ)†)

(5.4)

with the help of the definition of the kth dissipative bracket given by (3.8) evaluated atπk and (4.23) for the left-hand side of the equation whereas on the right-hand side weuse the expression for the inverse Kubo superoperator (4.20) obtained in the basis ofthe density matrix {|pn〉} furnished in the Appendix 9.13 and decompose the trace astr(·) = ∑n 〈pn| · |pn〉 yielding altogether

−〈[[K−1πk

ρ, K−1πk

ρ]]βk

πk ,αk〉 = ηk ∑

n,m

(ln pn − ln pm

pn − pm

)| 〈pn| A |pm〉 |2 (5.5)

implying that Re(ηk)≤0 since 〈[[K−1πk

ρ, K−1πk

ρ]]βk

πk ,αk〉 ≥ 0 because the dissipative bracket is

positive definite according to (3.7) and (ln pn− ln pm)(pn− pm) ≥ 0 for a faithful state asshown in the second part of Appendix 9.13. Finally, for the zero eigenvalue associated toρ=πk the left-hand side of (5.5) vanishes since K−1

πkπk =1 implying that the dissipative

bracket (3.8) for the kth reservoir is null because [Xk,λαk

,1]=0 which concludes the proof.

5.1.3 Quantum detailed balance revisited

To elaborate the linear response theory in the MDS context, it is particularly convenientto introduce the following detailed balance [231]

L∗ = K−1π LKπ (5.6)

relating, for example, the generator L of the linearized reduced dynamics (4.24)–(4.25)to its dual L∗ through the Kubo-Mori superoperator (4.18). This formulation appearedfirst in Grabert’s book 3 and will be of particular importance to obtain hereafter a properGKF. It closely resembles the quantum detailed balances earlier encountered in Sec. 2.2.4suggesting to denominate it as the Grabert quantum detailed balance (G-QDB). To gain aclearer understanding, let us rewrite the Kubo-Mori superoperator (4.18) by resorting tothe notations used by Fagnola and Umanitá in [177] yielding here

Kπ(B) =∫ 1

0dλ L(πλ)R(π1−λ)B, (5.7)

3See Eq. (5.5.12) in [225]. To be more precise, the property (5.6) depends on the antiunitary time reversaloperator Θ, i.e. L∗ = ΘK−1

π LKπΘ, but it is possible to link Θ to the identity for its individual componentsand simplify its expression.

65


where L(ρ)B=ρB and R(ρ)B=Bρ are respectively a multiplication from the left and theright. In comparison with G-QDB (5.6)–(5.7), the standard quantum detailed balancetakes the form [118]

L∗=L(π−1)LL(π), (5.8)

associated to the Hilbert-Schmidt scalar product given by the pre-scalar product (2.60)evaluated at λ=0, whereas its more symmetrical version reads [177]

L∗=(L(π1/2)R(π1/2))L(L(π1/2)R(π1/2)) (5.9)

leading to a symmetric scalar product obtained by inserting λ=1/2 in (2.60). The G-QDB(5.6) is associated to its own scalar product, namely the Kubo-Mori-Bogoliubov (KMB)scalar product. It depends on the superoperator (4.18) and reads

〈A; B〉ρ = tr(

AKρ(B))

. (5.10)

To obtain it, one has to act with (5.6) on a self-adjoint observable A followed by theapplication of the superoperator Kπ on the left

Kπ(L∗(A)) = L(Kπ(A)) (5.11)

and multiply from the right by a second self-adjoint observable C before taking the trace

tr(Kπ(L∗(A))C) = tr(L(Kπ(A))C) = tr(Kπ(A)L∗(C))= tr(AKπ(L∗(C))) = 〈A;L∗(C)〉π ,

(5.12)

where we used the dual property of L followed by application of the definition of theKubo superoperator (4.18) and the cyclicity of the trace. One recognizes in the previousexpression the KMB scalar product (5.10) with ρ=π and B=L∗(C). We have now alltools in hand to describe the linear response theory from the point of view of a dynamicsgenerating a MDS.

5.2 Fluctuation-dissipation theorem (FDT)

5.2.1 Definition

The Callen-Welton-Kubo FDT [216, 227] connects a correlation function capturing theintensity of the spontaneous fluctuations of an equilibrium state to the response functionmeasuring the amount of the dissipation produced by the relaxation of a state putout-of-equilibrium. The response function or linear susceptibility of a reduced system

66

5.2. Fluctuation-dissipation theorem (FDT)

connected to a single reservoir reads [231]

χ(1)AB(t) = β ∂t 〈A(t); B〉π (5.13)

expressed in terms of the Kubo scalar product (5.10) evaluated at the Gibbs state π andA(t) = eL

∗t A the Heisenberg time evolution. The generator L and its dual L∗, whichis defined by tr(CLρ) = tr((L∗C)ρ), are both associated to the linearized equation(4.24)–(4.25) for a unique reservoir (M = 1). This Kubo canonical correlation structure isoften absent in the susceptibilities computed for open quantum system [236, 237] which isnot thermodynamically consistent when considering a finite system-reservoir couplingstrength in conjunction with an adiabatic driving [231].

5.2.2 Standard mistake for open quantum systems

To understand the problem, it is instructive to start with the susceptibility of the totalclosed system. Therefore, we consider the total system described by the density matrix$ whose dynamics is dictated by the reversible von Neumann equation $ = LTot$ =

−i[HTot, $] and LTot$β = 0, with $β the thermal equilibrium state of the entire system 4,so that the evolution of the total density matrix is given by $(t) = eL

∗Tott$. Thus, the

imaginary part of the two-time correlation function 5 between the observables of interestATot =(A⊗ 1R) and BTot =(B⊗ 1R) yields 6

i 〈[ATot(t), BTot]〉$β= i tr

((eL

∗Tott ATot)[BTot, $β]

)= β tr

(AToteL

tTot(LTot(K$β

BTot)))

= β ∂t tr((eL∗Tott ATot)(K$β

BTot)) = β ∂t 〈ATot(t); BTot〉$β.

(5.14)

This is nothing else than the susceptibility cast in the form (5.13), but for the entiresystem, being a real quantity as it should be. It was obtained by using in the first linethe cyclicity of the trace and the Heisenberg evolution ATot(t) = eL

∗Tott ATot. To reach

the second line we used the definition of the dual L∗ and the equilibrium identity[BTot, $β]=−iβLTot(K$β

BTot) computed in Appendix 9.14. Clearly, for the closed systemthe situation is fine.

Now, to emphasize the mistake generally committed when considering an open system,we use the fact that the two observables ATot and BTot are exclusively attached to the

4We postulate that the closed quantum system can follow a thermalization process. To see under whichconditions this can happen one could, for example, take a look at [238, 239, 240].

5To put it more precisely, the real quantity i 〈[ATot(t), BTot]〉$β =(i/2)(cAB(t)− c∗AB(t))= c′′AB(t) is attachedto the correlation function cAB(t) = c′AB(t) + ic′′AB(t) = tr(ATot(t)BTot$β) and c′′AB(t) = β ∂t 〈ATot(t); BTot〉$β

is the imaginary part of the correlation function. We used the notation cAB(t) to distinguish it from thecorrelation function of the reduced system h(t). Moreover, to clearly display the causality one in generalmultiplies the previous expression by the unit step function Θ(t).

6The two observables are exclusively attached to the system of interest; this tensorial notation has beenpreviously introduced with the total Hamiltonian (2.1)–(2.3) for the underlying microscopic dynamics.

67


system of interest to “naively” restrict the first line of (5.14) to the reduced system (i.e. ac-cording to the notations used in the previous sections for the reduced system we haveATot→A, BTot→B, $→ρ, $β→π and LTot → L). The susceptibility is then associated to

i 〈[A(t), B]〉π = i tr((eL

∗t A)[B, π])= tr (AWt(V×0 (π))) , (5.15)

where Wt = eLt is the unperturbed propagator of the reduced time evolution (4.24)–(4.25)whereas the quantity

V×0 (·) = i[B, ·] (5.16)

is a reversible interaction superoperator 7. Such an expression is typically obtained throughMukamel’s perturbative expansion technique [235] (e.g. see equation (12) of [237]; theLindblad/Redfield master equation are considered in Sec. 4 and 8 therein). However, oneimmediately notices that it is impossible to recover the Kubo structure from it! Indeed,the time-derivative in (5.14) arises only because the generator LTot is simultaneouslyresponsible for the evolution of ATot(t) and for the commutator [BTot, $β]. On the contrary,for the restricted version (5.15) we have, in general, that the equilibrium contributionV×0 (π)= i[B, π] = β(−i[HS , KπB]) 6= βL(KπB), i.e. the irreversible part is missing (seeAppendix 9.14 for calculation details). Indeed, this problem disappears in the limit ofvanishing coupling strengths (the ratio of the reversible over irreversible part tends tozero as it should in the WCL) but remains for any finite couplings which are requiredfor most sensible applications.

5.2.3 Thermodynamical reformulation

This issue has already been pointed out by Grabert 8 for classical and quantum systemsin the early 80s. The absence of the irreversible term originates from an inappropriateperturbation scheme to derive the FDT when considering the reduced system. The“secret” to obtain a proper FDT is to consistently perturb the reversible and irreversiblecontributions of the reduced master equation [231]. To find out how to do it, one shouldgo back to Davies’ calculations [32] but now in presence of a total Hamiltonian HTot(t)whose time-dependency is entirely carried by the system Hamiltonian (HS)

ε

t ⊗ 1B with

(HS)ε

t = HS + εE(t)B, (5.17)

7This designation becomes obvious when performing the perturbative expansion to compute the linear(and nonlinear) response properly, i.e. it will then depend on the irreversible contribution too.

8Grabert revisited the WCL in his book [225] providing his own master equation and perturbationscheme to derive a proper FDT; see Eq. (5.4.48) for the master equation and Eq. (6.5.23) for the FDT. However,it is possible to show that this master equation is nothing else than the LDME (see Sec. VI.A and AppendixB of [231] for a proof). In addition, Grabert proposed in [221] a similar strategy for the so-called Brownianmotion regime: see therein Sec. 2 with Eq. (2.19)–(2.22) displaying the problem and Sec. 6 with Eq. (6.15)showing how to solve it. In both situations, the perturbation scheme was designed so that the irreversiblecontribution of the master equation depends on the external perturbation to recover Kubo’s structure.

68


where ε is a small real parameter, E(t) is a real slowly-varying function modelingthe external perturbation in time and B is a self-adjoint operator. The perturbationgiving rise to (5.17) is applied at t > 0 while for t ≤ 0 the function E(t) is equal tozero. For an adiabatic driving, i.e. whose time modulation takes place over the energy-relaxation time scale, the resulting time evolution depends simultaneously on ε and tin a regular manner [232, 241, 242, 120]. Indeed, the irreversible contribution inheritsthe perturbation so that the relaxation of the system occurs in the instantaneous energyeigenbasis which varies slowly on the time scale of the reservoir in the spirit of thestandard adiabatic theorem [243]. Such time-dependent Lindblad generators have beenrecently subject of renewed interest in the context of Cooper-pair pumping [53] andadiabatic quantum computing [244, 245, 134, 246]. Following this line of thought, thenon-necessarily linear master equation (3.12) under the adiabatic perturbation (5.17)becomes

∂tρεt = Kε

t (ρεt ), (5.18)

where ρεt is the perturbed density matrix, assuming that

(i) Kεt is bounded, self-adjoint and trace preserving,

(ii) Kεt is constant and equal to a time-independent version K for t ≤ 0;

in addition, we have that K0t = K at all times,

(iii) Kεt is purely dissipative according to definition of Sec. 5.1.2,

but does not necessarily produce a positive evolution far away from equilibrium simi-larly to the unperturbed linearized generator L obtained in Sec. (4.4). Moreover, it hasalso to fulfill the condition

Kεt (π

εt ) = 0 (5.19)

with respect to the Gibbs time-local equilibrium state

πεt =

e−β(HS )εt

tr(e−β(HS )εt )

(5.20)

sometimes designed as the accompanying density matrix since it represents at the instant tthe equilibrium state towards which the system converges. Note that πε

t is constant andequal to π for t ≤ 0. In the linear response regime we equally have for the linearizedmaster equation (4.25) near equilibrium 9

∂tρεt = Lε

t ρεt (5.21)

9To better see that reversible and irreversible contributions experience the driving one can write withoutloss of generality ∂tρ

εt = −i[(HS )

εt , ρε

t ] + J εt ρε

t in accordance with unperturbed version (4.24).

69


and

Lεt πε

t = 0 (5.22)

implying that the quasi-steady-state properties of the linearized dynamics are compatiblewith the adiabatic perturbation. Indeed, Kε

t (ρ)=Lεt ρ + o(||ρ− πε

t ||) and the expansion∂tρ

εt =πε

t + εδρ +O(ε2) yields

Kεt (ρ

εt ) = Lε

t ρεt + o(ε). (5.23)

We are now ready to define the perturbed evolution operator Wεt of Lε

t given in (5.21)from the initial to the current time t as

Wεt = T exp

(∫ t

0Lε

t′ dt′)

, (5.24)

where T indicates the time ordering. Note that for ε=0 we recover Wt = eLt being theevolution operator associated to the unperturbed dynamics L given in (4.26). Combining(5.23)–(5.24) yields an expression for the perturbed density matrix near equilibrium interms of the evolution operator

ρεt = Wε

t π + o(ε), (5.25)

where π is the Gibbs state. On the other hand, the time evolution of the propagator canbe written

∂tWεt = Lε

t ρεt . (5.26)

We show in the Appendix 9.15 that the perturbed propagator (5.24) can be rewritten as

Wεt = Wt +

∫ t

0ds Wt−s (Lε

s −L)Wεs . (5.27)

Moreover, since we are only interested in the linear response regime we can expand theperturbed generator as follows

Lεt = L+ εE(t)V×1 + o(ε) (5.28)

with the extended interaction superoperator 10

V×1 =1

E(t)L′t with L′t := (∂ε Lε

t )∣∣∣ε=0

. (5.29)

10This makes sense regarding the aforementioned reversible interaction superoperator (5.16) previouslydefined, i.e. V×0 (·)= i[B, ·]=(1/E(t)) (∂ε i[(HS )

εt , ·])|ε=0 where the index 0 indicates that only the reversible

contribution of the generator Lεt is considered.

70


Now, to obtain the susceptibility (5.13) we have to compute the average value of anobservable A evaluated at the perturbed state. To that end, we use (5.25)–(5.29) givingaltogether

〈A〉ρεt= tr(Aρε

t ) = tr(AWεt π) = 〈A〉π + ε

∫ t

0ds tr(AWt−sL

′sπ) + o(ε), (5.30)

where we also used tr(AWtπ) = tr(Aπ) = 〈A〉π and L′s = (∂ε Lεs )|ε=0. To further

compute the previous expression, we use the perturbed linearized evolution whichvanishes at the instantaneous Gibbs state (5.22) yielding

0 = ∂ε(Lεt πε

t )|ε=0 = L′tπ + Lπ′

t, π′

t = ∂ε(πεt )|ε=0. (5.31)

In the Appendix 9.16 we evaluate

π′

t = −βE(t)KπB, (5.32)

giving with (5.31) the identity L′tπ=βE(t)LKπB so that the average (5.30) reads

〈A〉ρεt= 〈A〉π + ε

∫ t

0ds E(s) β tr(AeL(t−s)LKπB) + o(ε)

= 〈A〉π + ε∫ t

0ds E(t− s) χ(1)

AB(s) + o(ε),(5.33)

where we used the change of variable s → t − s as well as eLsL = ∂s(eLs) to finallyreconstruct with β ∂str(eL

∗s AKπB) the sought susceptibility (5.13) for the reduced system.In the frequency space, we obtain after a Fourier transform of (5.13) the followingsusceptibility profile

χ(1)AB(ω) = −β tr

(A

LL− iω1

Kπ B)

(5.34)

which is easier to access experimentally than its counterpart in the time-domain. We havechecked the validity of the presented scheme by numerically extracting the susceptibilityχ(1)

AB(ω) in the frequency space following the method presented in the Appendix 9.17 fora two-level system with the LDME and the different NTMEs (see the Fig. 5.1 where theLDME and modular-free NTME are compared).

5.2.4 Perturbation of the MDS scattering operators

To obtain a meaningful perturbation scheme, we still have to provide the scatteringoperators for the perturbed master equation. One possible strategy is to rely on theunderlying microscopic description presented in Sec. 2.1.3 and to affect the driving ofthe system Hamiltonian (5.17) to the scattering operator (3.26) for a single coupling

71


operator yielding

(Xλ)ε

t = Xλ(Aεω,t, h(ω)), (5.35)

where the spectral function h(ω) remains unchanged since the reservoir dynamics isunaffected by the perturbation of the system, or the more general version

(Xλ)ε

t = Xλ((HS)ε

t , Q). (5.36)

Without entering into details, the Lindblad eigenoperators Aεω,t appearing in (5.35) carry

the time-dependency and fulfill the instantaneous eigenvalue equation [Aεω,t, (HS)

ε

t ] =

ωAεω,t leading to the generator Lε

t . Thus, one can designate the Aεω,t as instantaneous

eigenoperators. For a more cautious definition, see e.g. equations (54)–(57) of [120]or [246] as well as the justification in terms of relative time scales therein. Underan adiabatic driving, this criteria is enough to ensure that πε

t is the accompanyingGibbs state and that the instantaneous steady-state condition (5.22) holds. Note that theperturbation scheme needs not to be unique since one could choose the second scatteringoperator (5.36) or use Grabert’s perturbation technique summarized in Sec. VI.C of [231].

To conclude, the present approach does not fully apply when connecting multiplereservoirs to the system. The main reason is that we computed L′sπ in (5.30)–(5.32) forthe Gibbs state π which no longer holds for the NESS, i.e. one has to compute L′sρ+

which will inevitably spoil the Kubo structure (see the remarks in the introduction of[247] and references therein). Even if an analytic expression cannot be obtained for thesusceptibility one could still extract it numerically using the scheme presented in theAppendix 9.17 . Moreover, in the non-adiabatic regime (i.e. for a fast and/or strongdriving) our method also does not apply; this can be seen by looking at the correctionterms obtained for the time-dependent LDME in Sec. 4.4 of [120]. In the non-adiabaticcase, where Kubo’s FDT does not hold anymore, a Floquet type of approaches can beused when the applied driving is periodic [223, 248, 224, 249].

5.3 Green-Kubo formula (GKF)

5.3.1 Definition

Contrary to the FDT, one has to connect multiple reservoirs to the system to obtain aGKF for Onsager’s transport or kinetic coefficients [77, 78]

Lkl(β) =∫ ∞

0dt 〈 J

k,βQ (t) ; J

l,βQ 〉π (5.37)

72

5.3. Green-Kubo formula (GKF)

expressed in terms of the Kubo scalar product (5.10) evaluated at the stationary Gibbsstate π with a single coupling operator per reservoir to simplify the notation 11. Itcaptures the equilibrium fluctuations as does the FDT. The operator

Jk,βQ (t) = eL

∗βt

Jk,βQ is the flux operator J

k,βQ = J

k,βkQ

∣∣∣βk =β

(5.38)

which has been evolved in the Heisenberg picture; it is associated to the reservoir k buthas been evaluated at the inverse temperature βk = β. Note that the generator Lβ isgiven by the generator L of the linearized equation (4.24)–(4.25) but with all reservoirsset at βk =β. We show at the end of this section that the famous Onsager’s reciprocityrelations Lkl(β) = Llk(β) hold for k, l = 1, . . . , M. The GKF (5.37) can be regarded as“more thermodynamical” than the version obtained by Spohn and Lebowitz in [37]which is not expressed in terms of the Kubo scalar product. Moreover, it is compatiblewith their formulation in the WCL where the dynamics is dictated by the LDME. Tomodel such a thermodynamically driven OQS, one has to assume that:

(i) the different “ideal” independent reservoirs are defined in the thermodynamiclimit (volume V→∞ for a fixed particle density),

(ii) the thermodynamic forces are small, i.e. the intensive parameters of the differentreservoirs are close to each other,

(iii) the system has reached a non-equilibrium steady-state (NESS) for long times(t→∞) being independent of the initial state and characterized by a non-vanishingentropy production σS >0.

It is then possible to decouple the thermodynamic limit from the two other limits whichexist and are interchangeable [220].

5.3.2 Onsager coefficients

The transport coefficients are computed by expanding the entropy production up to thelinear order. Therefore, we return back to the entropy balance equation (3.21) but nowevaluated at the NESS. Thus, ∂t〈S(ρ+)〉= 0 according to (3.13) and (3.22) simplifyingthe entropy production to σS(ρ+) = −JS(ρ+) where the entropy flux is still given by(3.23)–(3.24) yielding

σS(ρ+) = ∑k

σS,k(ρ+) = −∑k

βk J kQ,+ = −∑

k∑αk

〈[[βk HS , Ψk,+]]βkρ+ ,αk〉, (5.39)

where the Massieu operator Ψk,+=S(ρ+)− βk H is evaluated at the NESS too. The totalheat flux is obtained by summing up the M individual reservoirs contributions (3.24)

11The index Q in Jl,βQ refer to the standard notation of a heat and not to the coupling operator.

73


and reads

JQ,+ =M

∑k

J kQ,+ =

M

∑k

tr (Jk(ρ+)HS) = tr (J (ρ+)HS) = 0. (5.40)

It vanishes becauseJ (ρ+)=0 implying that the inward and outward flows exactly canceleach other out. This means that for two reservoirs we have J 1

Q,+=−J 2Q,+. The steady-

state ρ+ = ρ+(β1, . . . , βM) of the previous expression is parametrized by the inversetemperature of the M reservoirs. Now, using the fact that the total flux (5.40) is equalto zero we can reexpress the entropy production (5.39) in terms of the thermodynamicaldriving forces xk = xk(β), often designed as macroscopic affinities [250], through

σS(ρ+) = −(

∑k

βk J kQ,+ − βJQ,+

)= ∑

kxk J k

Q,+, (5.41)

where we defined the kth affinity by

xk = β− βk. (5.42)

In the previous expressions we introduced the inverse temperature β of the system;when the reservoir k is at the same temperature as the system the affinity xk vanishesand the heat fluxes between the system and the kth reservoir dry up. The different fluxesJ kQ,+ can be chosen as depending on the affinity xk itself but also on all the other affinities

J kQ,+ ≡ J k

Q,+ (x1, x2, . . . , xM) = J kQ,+

({xj}

), (5.43)

where {xj} designs the set of affinities xj with j=1, 2, . . . , M, permitting lastly to expandthe heat fluxes in terms of the affinities. Hence, close to equilibrium one gets

J kQ,+ = J k,(0)

Q,+ + J k,(1)Q,+ + J k,(2)

Q,+ + . . . (5.44)

where the zero-order term J k,(0)Q,+ stands for the fluxes at equilibrium and automatically

vanishes while the first order contribution is

J k,(1)Q,+ =

M

∑l=1

Lkl xl (5.45)

with the Onsager transport coefficient

Lkl =∂

∂xlJ kQ,+({xj})

∣∣∣∣∣{xj}=0

=∂

∂βlJ kQ,+({β j})

∣∣∣∣∣{β j}=β

(5.46)

74

5.3. Green-Kubo formula (GKF)

whose kth heat flux is given by (3.24). The coefficients Lkl =Lkl(β) are the linear materialproperty of the system measured at the inverse temperature β (corresponding to theaffinity xj =β j − β=0) by changing the temperature of the connected reservoirs. It canbe understood as the linear response to the applied weak thermal drift. The first indexk of the transport coefficient Lkl refers to the flux J k

Q,+ whereas the index l stands forthe derivative against the affinity. Thus, the first order contribution to the steady-stateentropy production

σ(1)S (ρ+) =

M

∑k,l=1

xk Lkl xl = xt · L · x, (5.47)

where we introduced a scalar product notation with the affinity vector x=(x1, . . . , xM)T

and a M×M Onsager transport matrix L.

Notice that the transport coefficients (5.46) given as a derivative of the heat flux cor-respond to the expression (VII.2) obtained by Spohn and Lebowitz in [37] so that wecan hope to use their Lemma 1 (given on p. 135) to deduce the GKF. However, since ourdynamics dictated by (3.9)–(3.10) is not-necessarily linear two obstacles remain beforewe could do so. The first is that we cannot unambiguously define a heat flux operatorwhich requires the introduction of a Heisenberg evolution and the second that we need adetailed balance. Luckily, since the Onsager coefficients are defined next to equilibriumwe can use the linearization proposed in Sec. 4.4 providing the Heisenberg evolution(4.27) as well as the G-QDB (5.6). Moreover, close to equilibrium the entropy productionof the linearization (5.2), evaluated at ρ = ρ+, is positive. Thus, assuming the Gibbsstate π is a local minimum of σS(ρ+) ≥ 0 the Onsager matrix L appearing in the firstorder contribution (5.47) is positive definite [77, 78]. In addition, we can also take thelinearized version of the heat flux (3.24) for the same reason

J kQ,+ = tr

(Jk(ρ+)HS

)= tr

(ρ+J ∗k (HS)

)(5.48)

permitting to define a heat flux operator at the inverse temperature βk which reads

Jk,βkQ = J ∗k (HS). (5.49)

We are now ready to compute the Onsager coefficient (5.46) following the Lemma 1 of [37],where in (VII.3) therein we trade J

l,βQ π with Kπ(J

l,βQ ) using G-QDB (5.6), yielding [77, 78]

Lkl(β) = −tr(

Jk,βQ

1Lβ

Kπ(Jl,βQ )

), (5.50)

where the index β indicates that all objects have been evaluated at equilibrium (i.e. theflux operator as well as the linearized generator as earlier mentioned); the connectionwith the mentioned Lemma 1 is examined more in detail in the Appendix 9.18. The

75


spectrum of 1/Lβ contains a zero eigenvalue associated to the steady-state makingit singular. The issue is settled since one can show that the previous expression isequivalent to the integrated version (5.37); the integral exists and is finite because thegenerator Lβ, or equivalently L∗β, are both purely dissipative (their spectrum containsonly negative eigenvalues apart one zero associated to the steady state, see Sec. 5.1.2).This “Kubo-Mori formulation” is in the same vein as what was done for the FDT inSec. 5.2; it is thermodynamically more robust than the expression proposed by Spohnand Lebowitz in (VII.4) of [37] or by Jaksic an co-workers in Sec. 4.1 of [220] since itoffers a natural generalization for finite system-reservoir coupling strength.

5.3.3 Strategy to deal with the singular nature of the integrated GKF

It is useful to explicitly calculate how to go from the integrated (5.37) to the non-integrated version of the Onsager coefficient (5.50) because it provides a numericalstrategy to handle the singular nature of the integrated formula. To do so, we insert the

factor e−εt in the GKF (5.37) so that Jk,βQ (t)→J

k,βQ eL

εk,βt with Lε

k,β =(Lk,β − ε1) for ε→ 0

Lkl(β) =∫ ∞

0limε→0

tr(

Jk,βQ eL

εk,βtKπ(J

l,βQ ))

dt

=∫ ∞

0limε→0

tr(

Jk,βQ ∂t(e

Lεk,βt)(Lε

k,β)−1

Kπ(Jl,βQ ))

dt

=∫ ∞

0∂t

(limε→0

tr(

e−εteL∗k,βt

Jk,βQ (Lε

k,β)−1Kπ(J

l,βQ )))

dt

(5.51)

using tr(AeLεk,βtB)= tr(e−εteL

∗k,βt AB) to reach the last line and yielding after integration

Lkl(β) = − limε→0

tr(

Jk,βQ

1Lk,β − ε1

Kπ(Jl,βQ )

). (5.52)

This expression is identical in the limit ε→0 to (5.50) but is non-singular for ε 6=0, sinceit does not diverge for the zero eigenvalue, and is thus more appropriate for numericalcalculations. In practice, one performs the limit by evaluating the previous expression fora range of ε values and then carry out a linear regression. Finally, one can obtain a moreexplicit expression for the previous transport coefficient Lkl(β) with the trivial identityJ

k,βQ = J ∗k,β(HS)=K−1

π (Jk,β(Kπ(HS)))=K−1π (Jk,β(πHS)) as well as Kπ(J

l,βQ )= Jl(πHS)

obtained from the definition of the heat flux operator (5.49), the G-QDB (5.6) and theKubo superoperator (4.18) leading altogether to

Lkl(β) = − limε→0

tr(

K−1π (Jk(πHS))

1Lk,β − ε1

Jl(πHS))

. (5.53)

This form is perfect to obtain the different transport coefficients by numerical means.

76

5.4. Applications

5.3.4 The Onsager reciprocity relation

Using the GKF (5.37) naturally leads to the Onsager reciprocity relation [77, 78]

Llk(β) = Lkl(β). (5.54)

Therefore, we first notice that the G-QDB (5.6) implies eL∗βt =K−1

π eLβtKπ so that we canfully rewrite the time-evolved heat flux operator J

l,βQ (t)= eL

∗βt(J

l,βQ )=K−1

π (eLβt(Kπ(Jl,βQ ))).

Using this result we go back to the GKF (5.37), where the indices k↔ l were deliberatelyexchanged for the proof, and compute

Llk(β) =∫ ∞

0dt tr

(K−1

π (eLβt(Kπ(Jl,βQ )))Kπ(J

k,βQ )

)=∫ ∞

0dt tr

(eLβt(Kπ(J

l,βQ ))) J

k,βQ

)=∫ ∞

0dt tr

(Kπ(J

l,βQ ) eL

∗βt(J

k,βQ )

)=∫ ∞

0dt 〈 J

k,βQ (t) ; J

l,βQ 〉π = Lkl(β).

(5.55)

To finish, note that it is necessary to fully check the validity of the obtained GKF (5.37)which relies on some assumptions such as the Gibbs state being a local minimum ofthe steady-state entropy production as well as the choice of the G-QDB (5.6). To do so,we have numerically computed the Onsager coefficients directly from (5.46) carryingout simulations at different temperatures to numerically perform the derivative using astandard fitting strategy 12 and compare it with (5.52) arising from the GKF (5.37). Wehave obtained an excellent agreement between (5.46) and (5.52) for the LDME as well asfor the different NTMEs supporting the present version of the GKF.

5.4 Applications

We are ready to apply the FDT and GKF to a simple system, namely a single qubitconnected to a thermal environment for the LDME, the BRME and the different NTMEs.Since from an experimental point of view it is much easier to isolate the susceptibilityprofile of a quantum system than its transport coefficient 13, we focus most of ourattention on the FDT and link the obtained results to recent experimental findings. Bydoing so, we resume the work done in [251] and extend it further considering, amongothers, the effect of pure dephasing by numerical means. Besides, we will check thecorrectness of the thermal conductance extracted from the “Kubo-Mori formulation” ofthe GKF but in this case without linking the results to concrete data.

12In the same spirit as done for the FDT in the Appendix 9.17; further explanations are provided later onwhen computing the thermal conductance in Sec. 5.4.3.

13It is particularly challenging to construct a quantum setup with two reservoirs having a differenttemperature so that less data are available.

77


5.4.1 The FDT to study the decoherence process of a single qubit

Introduction The duration of the energy relaxation and decoherence is of significancefor a wide scope of quantum nanodevices: the archetypical example is a qubit whosecoherence time must be longer than the duration of a logic-gate operation to adequatelycarry out a quantum computation [7]. Indeed, preserving the coherence is a particularlychallenging task in the presence of noisy environments [9]. Theoretically, the energyrelaxation and decoherence lifetimes, respectively denoted by T1 and T2, are oftencomputed in the context of open quantum systems from the celebrated LDME obtainedby applying the SA in the WCL. However, due to its structural properties the LDMEexclusively predicts polarization decays of exponential kind for a two-level system,corresponding to Lorentzian profiles for the susceptibility. Precisely such features areviolated in a plethora of experimental findings ranging from electron/nuclear spins [40,252, 253, 254, 42] or nitrogen-vacancy (NV) centers [255, 256] to molecular qubits [257]and chromophoric molecules [258, 41] which display a biexponential decay of thepolarizations leading to two T2 times, a short and a long one. This decay process wasclearly associated to homogeneous non-Lorentzian susceptibility profiles in quantumdots (QDs) [259, 39, 260, 261, 262, 263, 264] and NV centers [265, 266]. In addition, usingmaterials doped with rare-earth ions, the decoherence can be slowed down by one orderof magnitude for an initial Bloch vector being properly sized and oriented [267, 268]. Onthe other hand, the SA leading to the LDME implies that T2 is at most twice as large as T1,while equality is reached only in the absence of pure dephasing [168, 38]. Although noexperimental evidence has yet broken the theoretical bound T2 ≤ 2T1 it has been highlydisputed, see [269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279]. Indeed, the LDME isjust one possible phenomenological Markovian master equation for finite couplings. Itis limited on one hand by the absence of non-secular terms [51, 52, 53, 54, 55, 56, 57, 58,59, 60, 61] and on the other by CP [187, 190, 280, 188, 102, 191, 106]. To abandon thisstrict paradigm, one can terminate the microscopic derivation before applying the SAgiving the famous BRME valid in the WCR. Unfortunately, it can violate the positivityof the density matrix and the second law of thermodynamics [83]. On the other hand,the NTMEs (3.53), (3.58) and (3.64) or (3.65) are free of these drawbacks.

Outline We start by computing the generators for the different master equations inthe Liouville representation. Note that since the decay times are customarily studied inthe linear response regime, it is legitimate to linearize the NTMEs around equilibriumfor this purpose. Thereafter, we obtain the different decay times from the real part of theLiouville generators’ eigenvalues. We find that before a friction-dependent threshold,depending on the nature of the master equation under study, we recover the usual T1

and T2 times. However, we observe with the BRME and the pre-averaged NTME in theWCR that the polarizations follow an oscillating exponential decay whose dynamicsdepends on the initial phase of the system contrary to the LDME. Beyond this threshold,we observe with the LSME and the modular-free NTME in the strong damping regime

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5.4. Applications

two T2 times attached to a biexponential decay of the polarizations. We then explain whydescribing the bifurcation event from one to two T2 times using the standard microscopicderivations is delicate and that it requires instead a phenomenological modeling in termsof generic Bloch equations which couple the coherences together. To better understandthe nature of the biexponential decay, we as well compute the susceptibility with theFDT showing that the profiles are neither Gaussian nor Lorentzian. We then solve thetime evolution for the coherences and explain how to generate a major prolongationof the coherence survival by selecting suitable initial states. Finally, we discuss therelevance of the T2 ≤ 2T1 bound, how to consider it in presence of two decoherencetimes and the impact of our findings on the T1T2T3 model arising when the populationsand coherences are coupled in presence of pure dephasing.

The single qubit model For the sake of simplicity, we parametrize the different oper-ators characterizing the qubit connected to a single reservoir with the Pauli matricesσx, σy, σz and σ±=σx ± iσy. We choose a diagonal system Hamiltonian HS =−(∆/2)σz

with the energy gap ∆=E2 − E1 > 0 whereas the coupling operator Q, dimensionlessso that the units of energy are fully assigned to spectral function h(ω), carries the realelements Q11 = q/2 and Q22 = −q/2 on the diagonal while the off-diagonal elementsread Q12 = eiθ = Q∗

21 with θ ∈ [0, 2π). The Hamiltonian HS gives then rise to the set offrequencies ω = {−∆, 0, ∆} respectively associated to the eigenoperators

A−∆ = eiθ σ+, A0 = qσz and A∆ = e−iθ σ− (5.56)

of the different master equations which we study (see the Table 3.1). The Pauli matrixσz, which characterizes the eigenoperator A0 and commutes with the Hamiltonian HS,is responsible for the pure dephasing process coupling the populations ρ11 and ρ22 tothe coherences ρ12 and ρ21. When possible, all relevant quantities are expressed with theabsorption rate a(ω)= h(−ω) for ω≥0 such that the quantity a(0) is finite 14. Finally,let us define for later convenience the dimensionless inverse temperature β=β∆.

Liouville space representation To perform the calculations, it is advantageous toswitch to the Liouville space (see e.g. Chap. 3 of [235] or the matrix-to-vector mapping(34) provided in [128]) highlighted hereafter with a bold notation. Choosing the vectorrepresentation ρ=(ρ11, ρ12, ρ21, ρ22)

T for the density matrix coefficients in the Hamiltonianbasis, we compute the 4×4 Liouville matrix L giving the equation of motion ρ=Lρ.Let us first consider the case of zero pure dephasing by selecting the coupling operator Qwith Q11 =Q22 =0 or Q11 −Q22 =0. This ensures that the populations decouple from thecoherences. In this regime, the LDME (3.38), the LSME (2.44) , the BRME (2.35) as well

14Typically a(0) ∝ γ/β, with γ the coupling strength, for the bosonic spectral function (2.25). Theabsorption rate a(0) is particularly relevant for the LSME (2.44) or the modular-free NTME (4.31) whichexclusively hold for white noise environments (i.e. flat spectral function).

79


as the linearized NTMEs (4.28), (4.30) and (4.31) all give rise to the Liouville matrix 15

L =

−u 0 0 v

0 −y + i∆ z e2iθ 0

0 z e2iθ −y− i∆ 0

u 0 0 −v

, (5.57)

where u, v, x, y and z are real. The parameters u and v are responsible for the relaxationof the populations and the ratio u/v determines the fraction of the population which arein the excited and ground state in the long time limit. In case the microscopic detailedbalance (2.27) holds, we have that u/v= e−β. On the other hand, the parameter y takescare for the decay of the coherences to zero while z is responsible for the couplingbetween the coherences. Now for the BRME (2.35), we obtain the parameters

u = a(∆),

v = u eβ,

y = (1 + eβ) a(∆)/2, (5.58)

z = y.

Besides, for the linearized dynamically time-averaged NTME (4.28) we get

u =∫

Rdν h(−∆ + ν) δωg(ν) ((1− e−βν)/βν),

v =∫

Rdν h(−∆ + ν) eβ δωg(ν) ((1− e−βν)/βν) = u eβ,

y =∫

Rdν h(−∆ + ν) δωg(ν) f (ν)/2, (5.59)

z =∫

Rdν h(−∆ + ν)

√δωg(ν)δωg(2∆− ν) f (ν)/2

depending on the frequency-dependent Gaussian δωg(ν) given in (4.3) and

f (ν) =(

1 + e β) 1− e(β−βν)

1− e β

β

(β− βν). (5.60)

15This structure holds for the great majority of Markovian master equations in absence of pure dephasingincluding the CGME (2.64); surprisingly, it is also valid for certain non-Markovian equations [269, 281]with an additional time dependency. Note that in this case the final state is not necessarily of Gibbs typeforbidding to apply the FDT formulated in terms of the susceptibility (5.13). Instead, one has to use thenumerical strategy described in the Appendix 9.17 valid for an arbitrary stationary state to extract thesusceptibility.

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5.4. Applications

For the linearized pre-averaged NTME (4.30) the parameters read

u = a(∆),

v = u eβ,

y = (1 + eβ) a(∆)/2, (5.61)

z = (β/2) eβ/2 coth(β/2) a(∆)

while for the linearized modular-free NTME (4.31) we obtain

u = a(0),

v = u eβ,

y = a(0) β coth(β/2), (5.62)

z = y.

The parameters of the modular-free NTME are closely associated to the LSME (2.44)characterized by u = v = y = a(0) and z = y which relaxes to the fully mixed state 16.For the LDME (3.38) we get the same expression as for the BRME (5.58) with z=0; theparameter z vanishes due to the time-averaging (2.41). This can be seen more clearlyfrom the 1st NTME (4.28) giving the LDME in the limit ωg → 0 due to the implicitdynamical time-averaging contained in this equation 17.

Spectral analysis The four eigenvalues of the generator (5.57) read

Λ0 = 0, Λ1 = −(1 + eβ) x, Λ± = −(y± κ), (5.63)

where κ=√

z2 − ∆2 is either real or imaginary. From the generator L as given in (5.57)and its eigenvalues it naturally follows that Γ1 =1/T1 =(1+ eβ) x is the energy relaxationrate. On the other hand, the absolute value of the real part of Λ± is associated to eitherone decoherence rate Γ2 =1/T2 =y, for z<∆, or two decoherence rates Γ2±=1/T2±=Λ±

for z>∆. It is noteworthy that all decay rates are relative either to the energy relaxationor decoherence, i.e. no Γ3 is involved in absence of populations and coherence coupling.The transition from one to two decoherence rates happens for z > ∆, or equivalently,

16Contrary to the modular-free NTME leading the system towards the Gibbs state due to the identityv = u eβ arising from the underlying MDS Kubo structure (and not from a microscopic detailed balance).

17This can be shown with: (i) x→ a(∆) since δωg (ν) gives a Dirac-delta δ0(ν), so that∫

Rdνg(ν)δ0(ν)=

g(0), implying (1− eβν)/(βν)→1 and h(−∆ + ν)→ a(∆), (ii) y→ (1 + eβ) a(∆)/2 since f (ν)→ (1 + eβ) aswell as h(−∆ + ν)→ a(∆) for the same reason and (iii) z=0 due to the Dirac-delta δ0(ν)δ0(2∆− ν).

81


past the absorption rate a(∆) thresholds

a2nd NTMEthr = ∆ e−β/2 tanh (β/2)/(β/2) (5.64)

a3rd NTMEthr = ∆ tanh (β/2)/(β/2) (5.65)

aBRMEthr = 2∆/(1 + eβ) (5.66)

aLSMEthr = ∆ (5.67)

obtained by computing z=∆ for the different master equations of interest. For the 1stNTME (4.28) associated to the parameters (5.59) one realizes that it is hard to get suchan analytical expression because the spectral function, or equivalently the absorption, isembedded in an integral. Thus, one has to compute this threshold for a given spectralfunction numerically to compare it with the other equations, see e.g. the Fig. 5.5 below.Let us further point out at that

• before the threshold, we observe that the 1st and 2nd NTMEs (4.28) and (4.30) aswell as the BRME (2.35), valid in the WCR, predict a single decoherence time asdoes the LDME (3.38). Yet, the dynamics in the WCR depends on the initial phasecontrary to the time evolution in the WCL as we are going to see further on.

• when looking at the 3rd NTME (4.31) we find before the threshold the decay rateratio Γ2/(2Γ1)=(2/β2) sinh(β) tanh(β/2)≥1, i.e. it diverges at low temperaturesfor all coupling strengths which is highly unphysical (see the Fig. 5.1 below). Thus,this equation can only be trusted at intermediate/high temperatures.

• the origin of the two decoherence rates Γ2± can be tracked back to the z parametercoupling the coherences together. From a microscopic point of view, the z param-eter arises in the WCR from the non-secular terms of the BRME (2.35). On theother hand, z arises in the strong damping regime directly through the couplingoperator of the LSME (2.43) which are not decomposed into eigenoperators.

• it would be very tempting to claim that the bifurcation Γ2→ Γ2± can always beobserved at low enough temperatures since athr→ 0 for β→∞ according to thethresholds (5.64)–(5.66). However, this strategy does not work out because a smallabsorption rate a(∆) is generally associated through the KMS condition to anexponentially large emission rate e(∆)= eβ∆a(∆) at low temperature producinga dissipative contribution which is generally much larger than the reversibleone 18. Describing the bifurcation microscopically would require an approachwhich permits to continuously go from the weak to the strong coupling regime or,equivalently, from the Markovian to the non-Markovian regime [222, 282].

18This comes from the fact that to compute the thresholds (5.64)–(5.67) we have implicitly fixed theabsorption rate a(∆). In the WCL one usually does the reverse for the low temperature limit, namely fixingthe emission rate e(∆) so that a(∆)= e−β∆e(∆)→0 for β→∞ as discussed in the Appendix 9.3.

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5.4. Applications

• we need a strong coupling to cross the threshold because the parameter z is, ingeneral, proportional to the system-reservoir coupling strength (or friction) γ.Thus, to go from one to two decoherence rates we need a friction γ > ∆. Sincewe assumed that the BRME, the pre-averaged NTME (4.30) and the dynamicallytime-averaged NTME (4.28) hold in the WCR (requiring a small γ � ∆), theycannot be used to describe the decoherence rate bifurcation by the sole increase ofγ. To sum up, the bifurcation takes place when the reversible contribution is of thesame order of magnitude as the irreversible one.

• looking back at the thresholds (5.64)–(5.67), one notices that they are all propor-tional to the energy gap ∆. One alternative strategy to get athr→0 at small couplingto describe the bifurcation Γ2→Γ2± could be to use ∆�1. This is not possible withthe BRME but it works with the dynamically time-averaged NTME (4.28) whichcan handle vanishingly small energy splitting. However, a small energy gap alsoimplies a small reversible contribution so that this strategy equally fails.

• to describe the regime beyond the threshold (displaying two distinct decoherencerates Γ2±) one could use the LSME (2.44) valid for a rapidly decaying systemstrongly coupled to a white noise reservoir 19. As we will see it later in Sec. 5.4.4,the 3rd NTME (3.66) produces essentially the same physics as the LSME (2.44)obtained in the singular limit 20 and could be used for this purpose.

Clearly, in order to model the bifurcation one has to use the Liouville matrix (5.57) forz 6= 0. However, the resulting Bloch equations coupling the coherences together haveto be considered per se, i.e. detached from any underlying Hamiltonian picture sincethe transition Γ2→ Γ2± takes place in the regime γ∼∆ which cannot be described bystandard microscopic Markovian master equation [20, 120]. Accepting to work with thisphenomenological model, one can then describe the temperature-dependent transitionfrom one to two decoherence times which was reported in low temperature experimentsof nuclear spins’ impurities in silicon crystals [254, 42] or InGaAs QDs [260]. The samekind of emergence of a biexponential decay for low temperatures, associated to a shortand a very long decay time, has been measured for molecular copper complex kind ofqubit [257]. On the other hand, the microscopic LSME predicts two decoherence ratesfor strongly damped systems coupled to a hot environment. This result is supported byrecent measurements performed on light-harvesting complex of green plants at roomtemperature which revealed a biexponential decay of the coherence [41]. To obtain arealistic model in the SCL for such a system, one can include the spatial structure of theenvironment through a spectral function containing spatial components (5.103)–(5.104)as discussed later on in Sec. 5.4.4. Therefore, one can use the modular free NTME (5.105)which is the thermodynamical consistent counterpart of the LSME (2.44).

19A similar kind of bifurcation phenomenon has been obtained using a modified version of Accardi’sstochastic limit [152, 153]; see the resulting Lindblad equation (5) in [69].

20In fact, it is more accurate since it relaxes towards the correct Gibbs state contrary to the LSME.

83


Figure 5.1 – Comparison of the imaginary part of the susceptibility χDD(ω) for the LDME(3.38) in black and the modular-free NTME (3.66) in red for a weak-coupling (γ=0.1)at low and high temperatures (β= 5 on the left, β= 1/6 on the right) using a bosonicspectral function (2.25) with J(ω)=γω (including an exponential cut-off). The full lineshave been computed with the FDT in the frequency domain while the black dots/redsquares have been obtained with the numerical scheme presented in the Appendix 9.17.Clearly, at low temperature the modular-free NTME is unphysical for small couplingssince its susceptibility diverges (implying that the ratio T2/(2T1)→∞).

!-2 -1 0 1 2

-1.5

-1

-0.5

0

0.5

1

1.5LDMENTMEBRME

!-200 -100 0 100 200

-0.3

-0.2

-0.1

0

0.1

0.2

0.3LDMENTMEBRME

Figure 5.2 – Comparison of the imaginary part of the susceptibility χDD(ω) for the LDME(3.38), BRME (2.35) and 2nd NTME (4.30) at low temperature (β=6) using two differentabsorptions: a(∆) = 0.001∆ on the left (before the threshold) and a(∆) = 0.1∆ on theright (after the threshold) while aBRME

thr =0.005∆ and a2nd NTMEthr =0.016∆.

84

5.4. Applications

Susceptibility analysis Let us examine the standard susceptibility χDD(t) associatedto the dipole operator D=µ(eiψσ+ + e−iψσ−) with µ>0. The frequency domain versionof the FDT (5.34) can be rewritten as χAB(ν)=−β A·L(L− iν1)−1 Kπ B, where the scalarproduct is defined as X ·Y = ∑4

1 x∗i yi. A and B are again two self-adjoint observablesnow expressed as 4×1 vectors whereas 1 is a 4× 4 identity matrix and Kπ is a 4×4diagonal Kubo matrix evaluated at equilibrium whose non-zero elements are

{ (1 + e−β)−1

; β−1 tanh (β/2); β−1 tanh (β/2); (1 + eβ)−1 }.

The dipolar susceptibility arising from the FDT then reads

χDD(ν) = (µ2/∆) tanh (β/2) ∑±

(1± z

κcos (2(θ − ψ))

) Λ±

Λ± + iν, (5.68)

where Λ± = y±√

z2 ± ∆2 are two eigenvalues (5.63) of the Liouville generator (5.57).Prior to the bifurcation, the absorption Im(χDD(ν)) can essentially be approximated by asuperposition of two Lorentzians, centered at Im(Λ±), with a unique linewidth equalto Γ2 for frequencies ν ' Im(Λ±), i.e. close to the resonances. Besides, one exactlyrecovers a superposition of two Lorentzians for z = 0 typically obtained in the WCL.Thus, the susceptibility lineshapes differ between the LDME and the the pre-averagedNMTE/BRME at small but finite couplings (see the left panel of Fig. 5.2). Moreover, thelineshapes in the WCR depend on the phases θ and ψ which are, respectively, associatedto the coupling operator Q and the dipole operator D. The difference in the susceptibilityprofiles between the WCL and the WCR is reflected in the time evolution of the systemas we will see later on. After the bifurcation, however, the two resonances are bothlocated at the frequency ν = 0, each of which is characterized by its own linewidthΓ2±, being neither of Lorentzian nor Gaussian type. Indeed, we have a superpositionof two non-Lorentzian functions Im(χDD(ν)) = ∑± c±ν/(Λ2

± ± ν2) with c± suitablereal coefficients 21. Notably, non-Lorentzian lineshapes were measured for nuclearspins [40], a wide range of QDs [259, 39, 260, 261, 262], NV centers [265, 266] and for aquantum well [283]. Moreover, if one of the two contributions is much narrower thanthe other, a peak which closely resembles a zero-phonon line (ZPL) arises [284, 285,286]. Non-Lorentzian profiles can be achieved for large frictions at high temperaturesmicroscopically with the LSME (2.44) and phenomenologically with the 3rd NTME(3.66). Our analysis shows that non-Markovian effects [276] are not necessarily requiredto produce non-Lorentzian profiles.

21The approximation ν=0 at the numerator cannot be used to restore a combination of Lorentzian incase c± and Λ±=0 so that the profile is truly non-Lorentzian.

85


Dynamical analysis To compute the evolution equations in absence of pure dephasing,we can consider separately the coherence (c) and population (p) generators by splittingthe Liouville matrix (5.57) into two sub-generators 22

Lc =

(−y + i∆ z e2iθ

z e−2iθ −y− i∆

)and Lp =

(−u v

u −v

). (5.69)

The solution to the differential equations ρc = Lc ρc and ρp = Lp ρp for the state vectorsof the coherences ρp=(ρ12, ρ21)

T and populations ρp=(ρ11, ρ22)T simply read

ρc(t)=Xc(t)ρ(0)c and ρp(t)=Xp(t)ρ(0)

p . (5.70)

The initial state vectors for the coherences ρ(0)p = (ρ(0)

12 , ρ(0)21 )

Tas well as for the populations

ρ(0)p = (ρ(0)

11 , ρ(0)22 )

Tare evolved with the solution matrices

Xc(t) =1

2κ

(κ−e−tΛ+ + κ+e−tΛ− ze2iθ(e−tΛ− − e−tΛ+)

ze−2iθ(e−tΛ− − e−tΛ+) κ−e−tΛ− + κ+e−tΛ+

), (5.71)

Xp(t) =1χ

(v + ue−tχ v(1− e−tχ)

u(1− e−tχ) u + ve−tχ

)(5.72)

with the parameters κ±= κ ± i∆, Λ± =−(y± κ), κ =√

z2 − ∆2 while χ = u + v is oneof the eigenvalues of Lp. To fully capture the implications of the eigenvalues Λ±, letus write down the solution for the coherence ρ12 with the initial condition ρ(0)

12 = r0 eiφ0 ,whose modulus must respect the positivity condition |ρ(0)

12 |2= r0

2 ≤ ρ(0)11 ρ(0)

22 , as

ρ12(t) =r0

2κ

{e−tΛ+

[κ−eiφ0 − ze−i(φ0−2θ)

]+ e−tΛ−

[κ+eiφ0 + ze−i(φ0−2θ)

]}. (5.73)

Before the threshold (i.e. z < ∆, Λ± is complex) one observes a single oscillating ex-ponential decay linked to a unique decoherence time T2 whereas beyond the thresh-old (i.e. z > ∆, Λ± is real) we have a non-oscillating biexponential decay typicallyassociated to a short T2+ and a long T2− decoherence time. Both cases are illustrated inFig. 5.3 with the pre-averaged NTME 23. The dependence on the orientation φ0 of theinitial Bloch vector, displayed by the NTMEs as well as the BRME (but not by the LDME),has been qualitatively measured in presence of a spin bath [287, 288]. To understand thetime evolution’s dependence on the initial phase, let us rewrite the solution (5.73) in the

22More generally, we could have re-absorbed the phase e2iθ originating from the element Q12 = eiθ ofthe coupling operator Q into the variable z to make it complex. This makes sense when considering theCGME [199] containing a friction matrix whose off-diagonal elements can be complex and which affect theelements Lc,12 and Lc,21 of the sub-generator Lc or L23 and L32 of the generator (5.57).

23For the sake of correctness, one should use the LSME or the modular free NTME (valid for strongdamping, i.e. large γ) beyond the threshold. Nevertheless, the pre-averaged NTME used in Fig. 5.3 sufficesto illustrate the phenomenon since it produces a qualitatively similar biexponential decay.

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5.4. Applications

Figure 5.3 – Time evolution before (left panel, absorption a(∆) = 0.0075∆) and af-ter (right panel, absorption a(∆) = 0.075∆) the threshold aNTME

thr = 0.065∆ in absenceof pure dephasing and at low temperature (β=4). We compare initial states with distinctorientations φ(1)

0 = π5 and φ(2)

0 =φ(1)0 + π

2 while r(1, 2)0 =0.05.

following handy form ρ12(t) = f (t) eiφ0 + g(t) e−iφ0 yielding the absolute value

|ρ12(t)| =√

f (t) g(t) e2iφ0 + f (t) g(t) e−2iφ0 + | f (t)|2 + |g(t)|2 (5.74)

with

f (t) =r0

2κ(e−tΛ+κ− + e−tΛ−κ+), (5.75)

g(t) = − r0

2κz e2iθ(e−tΛ+ − e−tΛ−) (5.76)

as well as f (t) and g(t) their complex conjugate. The time evolution of the absolutevalue of the coherence (5.74) clearly depends on the initial phase φ0 in the WCR whereasin the WCL it becomes phase independent |ρ12(t)|= | f (t)| since g(t)=0 because z=0.Thus, initial states characterized by the same amount of coherence |r0| lead to a differentdynamics. This property could be exploited to optimize the decoherence process beforeand after the threshold (see Fig. 5.3 comparing the impact of the phase). Interestingly,the crossover from an oscillatory towards a biexponential damping has been observedfor the spin dynamics in a 2D electron gas [289]. More generally, a biexponentialdecay was found in spins [40, 253, 254, 42], NV centers [255, 256], light-harvestingcomplexes [258, 41] or in QDs [259, 39, 260, 261, 262, 252, 263, 286, 264]. To understandwhich physical mechanisms produce these unorthodox decays, one should investigateeach setup thoroughly. This is done, for example, for GaAs QDs in Chap. 4 of [285] wherethe origins of the non-exponential decay and non-Lorentzian profiles are discussed 24.

24Typically, spin-orbit coupling or electron-phonon/hyperfine interactions and so on. Note that the fulldecay pattern is much richer than what the NTMEs predict in linear response regime, e.g. see Fig. 14 onp. 93 of [285] for an overview of decays at short, intermediate and long times.

87


Figure 5.4 – Bifurcation for a(∆)> aBRMEthr =0.036∆ and a2ndNTME

thr =0.065∆ (left panel, sameparameters as for Fig. 5.3) and temperature-dependence of the thresholds (right panel).

Beyond the threshold, one would like to obtain a time evolution solely driven by thelongest decay time T2− to preserve the coherence as long as possible. To this end, we setthe first term in the solution (5.73) equal to zero yielding the identity

(i∆− κ)/z = e−2i(θ−φ0). (5.77)

To satisfy this condition for a given setup, one should start from an initial state with φ0

being equal to the critical Bloch angle φc = θ − arccos(−κ/z)/2. Using such an initialBloch vector could drastically increase the coherence time being the cardinal resource toperform quantum gate operations [7]. Note that there exist many strategies to preparesuch an initial state [290, 291] and that an iterative algorithm to track optimal Blochvectors has been developed for materials doped with rare-earth ions [292].

T1T2 ratio Beyond the thresholds, the decoherence time T2− is no longer restricted bythe energy relaxation time T1 according to the famous inequality T2− ≤ 2T1, see theFig. 5.4 for an illustration 25. Such a bound violation was already obtained by Laird andco-workers with a fourth-order perturbative master equation [269, 270, 271]. However,its origin is different since they did not recognize the bifurcation process 26. In addition,this fourth-order equation does not ensure the positivity of ρ at all times [273, 275, 277].On the contrary, the NTMEs provide a thermodynamically and statistically safe way to

25The bifurcation point cannot be describe using the standard Markovian microscopic derivations andrequires to use the NTMEs, or better the Bloch equations produced by the generator (5.57), as phenomeno-logical master equations without any reference to an underlying Hamiltonian picture.

26Look, for example, at Laird and co-workers equations (A1)-(A4) in the Appendix of [269] wherethey overlook the case of the Liouville generator’s eigenvalue Λ±=y±

√z2 − ∆2 being real. Thus, they

have only a single decoherence time T2 which is no longer constrained by T1 at high temperature [270]:T2/(2T1)≤ ((1 + k)/(1−

√k)2) with k=ρ22/ρ11 (as the temperature increases, ρ22 gets closer to ρ11).

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5.4. Applications

Figure 5.5 – On the left panel we compare numerically the threshold of the Tg-dependentor dynamically time-averaged NTME (4.28) with the Tg-independent or pre-averagedNTME (4.30) and the BRME (2.35). To this end, we use a bosonic spectral function (2.25)with J(ω)=γω (including an exponential cut-off) and a Tg =0.1 to improve the matchwith the microscopic picture given by the BRME. On the right panel, we compare thethreshold for different Tg values: as Tg increases, the threshold ath goes towards infinityleading to the WCL where no bifurcation takes place.

discuss this phenomenon before or after the bifurcation point 27. Interestingly, a merelyformal nonlinear master equation obtained in [278], being positive but not CP, also notmeet the inequality. On the other hand, the fact that the LSME permits to overcome thisstandard bound seems to contradict the idea that there exists a one-by-one link betweenthe notion of CP and the inequality [102]. However, the bound violation due to thebifurcation is not really one: beyond the threshold we have three different decay ratesassociated to the three different non-zero eigenvalues of the Liouville generator (5.57)requiring to consider the following three decay rate inequalities

Γ2+ + Γ2− ≥ Γ1, Γ2+ + Γ1 ≥ Γ2−, Γ1 + Γ2− ≥ Γ2+ (5.78)

instead of 2Γ2− ≥ Γ1 being equivalent to T2− ≤ 2T1. These inequalities are alwaysrespected by the NTMEs and the LSME which maintain the positivity of the densitymatrix at all times 28; we will see them again in detail in the next section where weinclude pure dephasing.

27The pre-averaged NTME can be used before the threshold instead of the BRME (ensure ρ>0) whereasthe modular-free NTME could replace the LSME after the threshold (ensure convergence to Gibbs state).

28Note that for non-Markovian equations the Bloch inequalities (5.78) can be broken due to the loss ofpositivity of the density matrix [293].

89


T1T2T3 ratios In presence of non-zero pure dephasing the Liouville matrix (5.57) canno longer be used due to the intertwining of the populations and coherence. Moreover,one has to replace the bound 2Γ2≥Γ1 by the Bloch inequalities [293]

Γi + Γj ≥ Γk (5.79)

since the generator is associated to three decay rates with i, j, k=1, 2, 3 (all indices mustbe different from each other). Performing analytical calculations is particularly difficultin this regime for the different master equations and one has to resort to simulations toextract decay rates. Therefore, let us resort to the pre-averaged NTME (3.58) to study theT1T2T3 model numerically without making any reference to an underlying microscopicpicture (in particular for intermediate and large frictions; as previously stated we use aphenomenological master equation to describe the phenomenon). Interestingly, evenin presence of pure dephasing a bifurcation takes places for the NTME but not theLDME as clearly shown in lower plot of Fig. 5.6. Moreover, the Bloch inequalities(5.79) are respected even in presence of the bifurcation as in absence of pure dephasing.Decoherence processes in presence of non-zero pure dephasing are of first importanceto describe the population beating and coherence revival (due to the exchanges takingplace between the diagonal and off-diagonal elements of ρ) which have been observedin photosynthetic complexes [16, 17, 18] or quantum kicked rotors [294, 295].

In summary The LDME is over-restricted by the SA and does not allow to interpretnumerous experimental results. On the contrary, the non-secular terms of the BRME leadat small couplings to an oscillatory exponential decay of the polarizations depending onthe phase of the initial state as well as rescaled Lorentzian lineshapes. Unfortunately,the BRME does not in general preserve the positivity of ρ and leads to a negativeentropy production [83, 84]. In addition, it fails for certain initial transients but, moreimportantly, also in the long-time regime so that it can happen that the steady-statecannot be reached for certain parameters (see the equations (47) and (50) of [128]) whichis of first importance to study the linear response regime. The pre-averaged NTME(3.58) predicts the same unorthodox effects as the BRME prior to the bifurcation but it isfree of its drawbacks. Thus, one could use the NTMEs to explain 2D IR spectroscopymeasurement of vibrational exciton for coupled carbonyl groups of a diketone moleculein chloroform which have been described with BRME (see the Liouville matrix (7)in [296]). Alternatively, one could have applied the CGME (2.64) at the expense ofgiving up the FDT 29, or better, used the time-averaged NTME (4.28) equally carryinga coarse-graining time allowing to recover the LDME in the WCL and respecting theFDT. Beyond the thresholds (5.64)–(5.67), the LSME predicts a biexponential decay ofthe polarization and non-Lorentzian susceptibility profiles. However, it does relax tothe fully mixed state and can only be used in the high temperature limit. In contrast,

29The CGME (2.64) leads to the same generator (5.57) and produces similar phenomena but does not relaxto the Gibbs state so that it is incompatible with the FDT (5.68); see Appendix 9.17 to get the susceptibility.

90

5.4. Applications

Figure 5.6 – From left to right: we increase the pure dephasing |Q11 − Q22| = q =0.1, 0.5, 1.0, 2.0. To perform the simulations, we have used a bosonic spectral function(2.25) for J(ω) = γω (including an exponential cut-off) with β = 4 for different γ’s toobtain the corresponding absorption a(∆). Bottom plot: Comparison of the uprightplot with the LDME with the same set of parameters and |Q11 −Q22|=2.0. Clearly, theLDME displays no bifurcation even in the T1T2T3 model and no coupling between thepopulations and coherence is observed. Note that in the WCL the two equations agree asit should be. As previously stated, we describe the bifurcation point phenomenologically.An intermediate to a large pure dephasing implies that the ratios obtained from thepre-averaged NTME deviate from the LDME prior to the bifurcation contrary to the zeropure dephasing regime described in Fig. 5.4 associated to the T1T2 model. This effect isproduced by the coupling of the populations and coherences.

91


the modular-free NTME (3.66) leads to the correct Gibbs state and can be applied atintermediate and high temperatures. Describing the bifurcation point is not possiblewith the standard microscopic approaches (i.e. the LDME/BRME/CGME and LSME).Thus, for this task one could directly use the phenomenological Bloch equations arisingfrom the Liouville generator (5.57) without any reference to an underlying Hamiltonianpicture. In presence of two decoherence times, the standard bound T2 ≤ 2T1 has to bereplaced by a set of three decay rate ratios in absence (5.78) or presence (5.79) of puredephasing. Producing an ultralong T2− time can be used, for example, to fulfill oneof the famous DiVincenzo’s criteria for quantum information processing [8] requiringlong decoherence times. Finally, in presence of pure dephasing one still observes abiexponential decay with the T1T2T3 model. This regime is of importance since it canlead to the emergence of quantum beats (or beatings) 30 which have been observed inlight harvesting complexes [41] and anions solvated in water [297]. Moreover, puredephasing is responsible for the creation of coherence starting from a diagonal densitymatrix which will be crucial later on when discussing the sudden death as well as thesudden birth of entanglement from an initially unentangled two-qubit system.

5.4.2 Linear response of a simplified light-harvesting complex

Let us briefly anticipate on Chapter 7 using Gilmore and McKenzie’s minimalisticmodel [298, 299, 300] which has been recently used to describe light-harvesting com-plexes (LHCs) [301, 302]. The network of Nc light sensitive pigments (or chromophores)can be modeled [303] as TLSs interacting through a dipole-dipole interactions withHS = 1

2 ∑Nci ε iσz,i + ∑Nc

i>j Jij(σx,iσx,j + σy,iσy,j). Focusing only on two chromophores, thatis a four level system, one can deduce an effective chromophore representation com-posed of only two levels in contact with the two bosonic reservoirs as long as they arenot correlated. This allows to “extract” a single-exciton 2× 2 system Hamiltonian beingnon-diagonal [301, 302] which reads H(1)

S = (ε12/2)σz + J12 σx, where ε12 = ε1 − ε2 whileJ12 arises from the dipole-dipole coupling between the chromophores 1 and 2 in theoriginal model and can be seen as a “tunneling energy”. Using this representation, wecan now apply all the knowledge we acquired when we studied the linear response of asingle qubit with the FDT in Sec. 5.4.1. Note that since the Hamiltonian is non-diagonalthe coupling operator Q automatically acquires diagonal contributions, leading to puredephasing, when performing the calculations in the Hamiltonian basis. Using the T1T2T3

model, we can then model the biexponential decay of the coherence (fast/slow com-ponents and non-Lorentzian susceptibility profiles) observed in LHCs, see Fig. 2 and3 in [41], or in two-dimensional nanostructures [304]. This provides a more plausibleexplanation than the inhomogeneous broadening mechanism proposed in [305] andpredicts a ZPL which could be associated to the one measured in various LHCs 31.

30Quantum beats are the signature of the exchanges taking place between the populations and coherences.31The ZPL is a typical characteristic of chromophores. See, for example, the Fig. 4.12 in the Chapter 4.B

of the book [306] as well as Fig. 6 of [307]. Recently, single-molecule microscopy equally displayed thepresence of a ZPL under weak to intermediate exciton-vibrational couplings [308].

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5.4. Applications

5.4.3 The GKF to study the transport process through a single qubit

Introduction Transport studies of nanoscale devices rapidly evolved in the past decadewith regard to theoretical models and experimental techniques [309, 310]. These de-velopments took place long after the classical transport theory arose in the early 30swith Onsager’s reciprocal relations (ORR) assuming microscopic reversibility undertime-reversal [233, 311]. The derivation of these relations summarized the discoveriesregarding transport phenomena of the previous century for heat/electric conductionand particle diffusion. Later, Casimir [312] proposed a streamlined version of ORRwhich was exploited by De Groot and Mazur [313] as well as by Prigogine [75] in thecontext of thermodynamics of linear irreversible processes. The field of classical trans-port phenomena was widely developed in the past 50 years as depicted by Bird [314]including nonlinear irreversible phenomena [315, 316, 317, 318, 319, 76]. Nowadays, itis possible to probe heat, charge, particle or even spin currents at the molecular levelwhere quantum effects strongly influence the transport properties. Understandingcross effects, such as thermoelectricity in nanowires [320, 321], magnetothermal effectsresponsible for the heat transport by spin [322, 323] or novel symmetry relations [324]is of highest importance to unravel the properties of existing nanostructure as well asbuilding efficient technological devices.

The transport matrix For the case of a quantum system connected to M thermal baths,we arrive for the first order heat flux vector at J(1)H,+=−L · x expressed in terms of thetransport matrix

J1,(1)Q,+

J(1)Q,2,+

...

JM,(1)Q,+

= −

L11 L12 . . . L1M

L21 L22 . . . L2M

......

. . ....

LM1 LM2 . . . LMM

x1

x2

...

xM

. (5.80)

Some of the coefficients Lkl can be shown to be equal remembering that the total heatflux vanishes at the steady-state. More precisely, we can combine the steady heat flux(5.40) with the first order term of the expansion of the heat fluxes (5.44) giving

J(1)Q,+ =M

∑k

Jk,(1)Q,+ = −

M

∑k

M

∑l

Lkl xl =M

∑l

(−

M

∑k

Lkl

)xl =

M

∑l

cl xl = ct · x ≡ 0 (5.81)

so that for all xl 6= 0 we obtain the total flux conservation (TFC) relation

cl = −M

∑k

Lkl = 0 ⇔ Lll = −M

∑k 6=l

Lkl , (5.82)

93


allowing to express the diagonal coefficient as a sum over the off-diagonal one. Furthersimplifications apply when resorting to the well-known reciprocity relations which relyon the existence of symmetries and antisymmetries between the transport coefficients.The ORR [233, 311] states that Lkl =Llk whereas Casimir’s ORR [312] reads Lkl =εkεl Llk

depending on the parity εi with respect to time-reversal (a time-invariant quantityimplies +1 and for all others −1). These relations are linked to the reversibility ofthe underlying microscopic equation of motion. If the ORR applies, it streamlines thedescription of the system because the transport matrix (5.81) becomes symmetric.

For the case of a quantum system coupled to two different heat baths (M=2) we have asystem of two equations which can be simplified using the TFC (5.82) with L11=−L21

and L22 =−L12. Moreover, with the ORR L21 = L12 we have L11 = L22 =−L12 =−L21.Therefore, only a single coefficient (for example L12) is required to capture the transportproperties of the system. The corresponding system of two equations becomes

J1,(1)Q,+ = +L12 (x1 − x2)

J2,(1)Q,+ = −L12 (x1 − x2) (5.83)

so that J1,(1)Q,+ = −J2,(1)

Q,+ as expected. The same can be done for a quantum system incontact with three heat baths, i.e. M = 3. In this case we have for the ORR L12 = L21,L13=L31 and L32=L23 coupled to the TFC relations L11=−(L21 + L31)=−(L12 + L13),L22 =−(L12 + L32) =−(L12 + L23) and L33 =−(L13 + L23) from TFC (5.82). Thus, thesystem of three equations takes the straightforward form

J1,(1)Q,+ = +L12 (x1 − x2) + L13 (x1 − x3)

J2,(1)Q,+ = −L12 (x1 − x2) + L23 (x2 − x3)

J3,(1)Q,+ = −L13 (x1 − x3)− L23 (x2 − x3), (5.84)

where only the three coefficients L12, L13 and L23 are required to get the transport matrix.Note that the sum of the fluxes vanishes. Generally, if the ORR holds, we need

MO =M(M− 1)

2, M = dim(L), (5.85)

coefficients instead of M2 to fully determine L. Hence, we have for M = 2→ MO = 1(e.g. L12), for M = 3 → MO = 3 (e.g. L12, L13 and L23), for M = 4 → MO = 6 (e.g.L12, L13, L14, L23, L24 and L34; details not shown). It is also possible to get a general

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5.4. Applications

expression for the linearized heat flux k in term of differences of affinities, i.e.

Jk,(1)Q,+ =

M

∑l

Lkl (xk − xl) = −M

∑l

Llk (xl − xk), (5.86)

where in the second equality we use the ORR. We could replace the difference (xk− xl)=

−(βk − βl) using the definition of the thermal affinity given in Eq. (5.42). We have seenpreviously that the linearized entropy production is positive in Sec. 5.2. However,from a standard point of view one has to check that the entropy production (5.47) isnon-negative. To do so, one has to control that only small cross-effects take place 32, i.e.

Lkk ≥ 0 and LkkLll ≥ (1/4)(Lkl + Lkl)2. (5.87)

The second condition boils down to LkkLll ≥ L2kl if the ORR holds 33. Bear in mind that

one needs an explicit expression or a numerical value for the coefficients Lkj to check theprevious conditions. In the forthcoming section, we illustrate how to link the first ordercoefficient Lkl to the thermal conductivity κ with the help of Fourier’s law in absence ofmatter flow for a quantum system connected to two heat baths.

Fourier’s law: determination of the thermal conductivity To link the previous On-sager coefficient to a physical observable, we can use Fourier’s law [325] in absenceof matter flow when two baths are connected to a low-dimensional system such asnanotubes or nanowires 34. To this end, we resort to a system initially in equilibriumat the inverse temperature β being in contact with two infinitely extended reservoirs.Each one of these heat reservoirs is in thermal equilibrium but with a different inversetemperature, namely β1 and β2. Under these conditions, we extract a single heat fluxequation from the system of Eq. (5.83) and couple it with the expression of the linearcoefficient obtained in (5.46) in conjunction with Eq. (5.42) defining the affinities yielding

J2,(1)Q,+ = L12(β1 − β2), L12 =

∂

∂β1J2Q,+({β j})

∣∣∣∣∣{β j}=β

, (5.88)

where we expressed the derivative in terms of the inverse temperature instead of theaffinity. The flux J2,(1)

Q,+ arises from the first term of the expansion (5.44); it is proportionalto the Onsager coefficient L12 being the derivative of the flux J2

Q,+ between the systemand the second reservoir. Moreover, the heat flux J2

Q,+ from the second reservoir to the

32The local equilibrium assumption implying a slow variations of the affinities.33Higher order reciprocity relations may be deduced for the nonlinear transport coefficients (see for

example Sec. IV of [250])34For details see the review [326]; mainly Sec. 8 for typical experiments and Sec. 9 for a summary of the

general characteristics of such kind of systems.

95


system can be expressed at the first order using Fourier’s law as follows

J2,(1)Q,+ = κT2∇

(1T

)= −κ∇T (5.89)

to deduce the thermal conductivity κ=κ(T) of the system. To work it out, we expressthe gradient as a finite difference ∇T ≈ ∆T/d using the temperature difference ∆T=

T1 − T2 between the two baths and introduce the characteristic dimension d of the one-dimensional homogeneous quantum system implying that d is a length (for a concreteexample we could use a one dimensional spin-chain, see the system Hamiltonian givenequation (3) of [327]). To be precise, when switching from a gradient to a finite differencedescription, we no longer compute the thermal conductivity (being a material property)but the thermal conductance (being a component property), see for example [216] fordetails. In general the symbol G is often preferred for the thermal conductance whereasκ is privileged for the thermal conductivity. In fact, we should rather talk about aconductance than a conductivity because the thermal properties of a nanotube changewith its length [326]. Accordingly,

J2,(1)Q,+ ≈ −Gκ

∆Td

= −Gκ

d(T1 − T2)

T1T2

T1T2=

GκT1T2

d

(T2 − T1

T1T2

)≈ GκkBT2

d

(1

kBT1− 1

kBT2

)=

Gκ

kBdβ2 (β1 − β2) ≡ L21 (β1 − β2) ,(5.90)

where we have used the approximation T1T2≈T2 under the assumption that the systemis only weakly thermally driven (the temperature of the baths is close to the initialtemperature T of the system). We used the symbol Gκ for the (isopotential) thermalconductance. The last equality originates from the expansion of the heat flux J2

Q,+given in (5.44). Notice that the coefficient L21=L12 is independent of the temperaturesof the connected baths (it is a material property of the system in absence of thermalperturbations) whereas the linearized heat flux J2,(1)

Q,+ depends on them. Accordingly,the Onsager coefficient allows to compute the thermal conductance from the previousexpression

Gκ =d

kBT2 L12 (5.91)

depending on the transport coefficient L12 and T the initial temperature of the system.

Control of the validity of the GKF By computing the thermal conductance (5.91),we can confirm that the definition (5.52) of the GKF is valid. The idea is to perform aseries of simulations to directly compute the transport coefficient L12 given by (5.46)from the heat fluxes at the NESS (to be sure that we have reached the NESS after agiven time evolution, we can check that the condition J1,(1)

Q,+ = −J2,(1)Q,+ is fulfilled). To

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5.4. Applications

Figure 5.7 – Left plot: Comparison of the thermal conductance Gκ given by (5.91) usingthe GKF (5.52) and a direct simulation to compute the transport coefficient L12 (thesimulation parameters are provided in the text). Right plot: Comparison of the thermalconductance Gκ obtained from the GKF for the LDME (3.38), the BRME (2.35), theTg-dependent or dynamically time-averaged NTME (3.53) and Tg-independent or pre-averaged NTME (3.58) using the same parameters as for the left plot apart from thecoupling strength γ1=γ2=0.1.

perform numerically the derivative, we plot the heat flux J2,(1)Q,+ for a fixed β2=βref and

vary β1 permitting to compute the tangent at the point β1 = β2 = β where the heatflux J2,(1)

Q,+ vanishes. To this end, we have chosen a qubit characterized by a diagonalsystem Hamiltonian HS=(0 0; 0 ∆) and connected it to two reservoirs, both modeledby a bosonic spectral function (2.25) with J(ω)=γkω (including an exponential cut-off),through the coupling operators Q1=Q2=(0 1; 1 0) under a coupling strength γ1=0.2and γ2 =0.3. The results are displayed for the LDME (3.38) on the left plot of Fig. 5.7.Clearly, the agreement is nearly perfect supporting the GKF definition (5.52) beingequivalent to (5.37). Note that the different master equations provide nearly the samethermal conductance as can be seen on the right plot of Fig. 5.7 computed using the GKF.This is in contrast to the susceptibility (and decay times) studied in the Sec. 5.4.1 whichare much more sensitive to the nature of the time evolution. However, this does notmean that the transport coefficient are not influenced by the nature of the environment.Looking, for example, at the difference of coupling strength δγ12=γ1 − γ2 between thetwo reservoirs with the pre-averaged NTME (a similar result is obtained for the othermaster equations) shows that one can increase the thermal conductance even for a verysmall δγ12 in the WCR as can be seen in the Fig. 5.8.

Finally, we can examine the thermal conductance produced by the modular free NTME(3.66) and compared it to the LDME (3.38) and the pre-averaged NTME (3.58). We usethe same parameters as previously apart from the frictions for which we use γ1= 0.05and γ2= 0.15. Surprisingly, we see on the Fig. 5.9 that the thermal conductance of the

97


Figure 5.8 – Examination on the impact of difference of coupling strength δγ12=γ1 −γ2 of the two reservoirs: a small difference implies a large increase of the thermalconductance. Note that this increase becomes less and less important for an increasinglylarger difference δγ12.

Figure 5.9 – Left plot: Comparison of the thermal conductance obtained from themodular-free NTME with the LDME and pre-averaged NTME (same parameters asFig. 5.8 apart from γ1= 0.05 and γ2= 0.15). Right plot: Check the dependence on thefriction of the thermal conductance for the modular-free NTME (fixed γ1 = 0.05 andvary γ2).

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5.4. Applications

modular free NTME (3.66) is not exponentially suppressed at low temperatures butconverges towards a fixed values depending on the parameters of the system (systemHamiltonian, coupling operators and frictions). Since we can trust the LDME in the WCLand, according to what we learned from FDT in Sec. 5.4.1 we can rely on the pre-averagedNTME in the WCR, the non-vanishing thermal conductance for low temperatures is mostprobably unphysical. In fact, we found a similar issue with the modular free NTME atlow temperatures for the susceptibility χDD(ω) and the decay time ratio T2/2T1 for weaksystem-environment couplings (see Fig. 5.1 and associated comments). Nevertheless, forhigh temperatures the modular free NTME produces similar results as the LDME, BRMEand the two other NTMEs which suggests to reconsider its regime of applicability.

5.4.4 Implications of the linear response for the modular free NTME

The FDT as well as the GKF told us that we should not use the modular free NTME (3.66)at low temperatures when the system is weakly coupled to the reservoir contrary towhat had been proposed in [146]. But can we use it in another regime? Grabert pointedout that the LDME and this equation do not rely on the same time scale separation;he explained in [221] that the used ”Markovian limit (...) must be distinguished from theweak coupling limit“ since it ”is more closely related to the method used within the frameworkof classical statistical mechanics in order to derive the Fokker-Planck equation“ and furtheroutlined that the equation (3.66) was derived in the aforementioned BML. He clearlydistinguished this equation from a former one he obtained in the WCL [225] whichis nothing else than the LDME itself as we have shown in [231]. Grabert and Talknerexamined the quantum Brownian motion of a heavy particle in contact with a reservoirof lighter particles following this line of thought and obtained in [141] a linearized masterequation which is not of Lindblad type, see equations (1)–(3) therein 35. Grabert makesthe distinction between weakly and overdamped systems, the former being associatedto the WCL whereas the latter are linked to the BML. One can obtain their masterequation from (3.64) by applying the linearization technique of Sec. 4.4 and then usethe additional approximation Kπk(A) ' (1/2){πk, A} which is discussed in Appendix9.19. It is noteworthy that Grabert’s equations [221, 141] have been qualified as quantumFokker-Planck equations by Oppenheim in the context of Brownian motion [328].

The Brownian or small energy limit The LDME (3.38) should be seen as a rate or aBoltzmann kind of equation whereas the modular free NTME (3.66) is rather a Fokker-Planck type of equation. In classical mechanics, there exist general procedures totransform a Boltzmann equation into its Fokker-Planck analogue [329, 330]: one im-

35They use the ratio m/M, with M the mass of the heavy particle and m the mass of the lighter ones, assmallness parameter instead of the coupling strength (suggesting that it can be used for stronger couplings)to perform the Markovian approximation (notice that the detailed calculations appear nowhere in [141]and that we have to live with the cryptic comment “A detailed derivation will be published elsewhere“).

99


poses scattering transition rates being singular and then takes the vanishing limit of aparameter characterizing the intensity of the collisions (such as the collision angle or thetransferred momentum). Let us follow a similar strategy starting from the LDME in itsseemingly nonlinear form (3.33) and separate the zero-frequency contribution from theothers whereby one obtains two terms

J (ω)k (ρ) =

12 ∑

ω 6=0hk(ω) [A(ω)

k, ω

†, Kk,−ω

ρ [A(ω)k, ω, Ψ(ω)

k ]] + hk(0) [A(ω)k, 0

†, Kk,0

ρ [A(ω)k, 0 , Ψ(ω)

k ]], (5.92)

where we have specified that the eigenoperators depend on the characteristic fre-quency 36 of the system Hamiltonian H(ω)

S denoted ω as well as on the Massieu operatorΨ(ω)

k = −(ln ρ + H(ω)S ). We introduced this parameter explicitly to later take the limit

ω→0 to model the fact that the system (e.g. the heavy particle) undergoes only vanish-ingly small quantum jumps and exchanges an infinitesimal amount of energy with thereservoir (e.g. the lighter particles). Thus, each collision displaces the system over aninfinitesimally small distance in the position representation generating an overdampedrelaxation. Let us now define the zero-frequency self-adjoint eigenoperator as beingproportional to the identity

A(ω)k, 0 = α(ω)

k, 0 1 (5.93)

implying that all diagonal elements of the coupling operator Qk are equal to the prefactorα(ω)

k, 0 according to the relation (2.13). Such an idea was already explored to obtain a non-linear “diffusive” quantum master equations by Breuer and Petruccione (see Sec. 6.1.3.at p. 314 of [20]); they start from a Lindblad stochastic process and assume that thetransitions among states become infinitesimally small whereas the number of transitionsin a finite time interval tend to be arbitrary large. For the choice (5.93), the second termin (3.33) vanishes due to the double commutator form and we are allowed to subtractA(ω)

k, 0 from A(ω)k, ω and its adjoint in the first term for the same reason. Moreover, if we define

the following singular operator [77, 78]

∂ω A(ω)k, ω =

A(ω)k, ω − A(ω)

k, 0

ω(5.94)

and its adjoint we get the irreversible contribution

J (ω)k (ρ) = ∑

ω 6=0

hk(ω)

2ω−2 [∂ω A(ω)k, ω

†, Kk,−ω

ρ [∂ω A(ω)k, ω, Ψ(ω)

k ]]. (5.95)

To be able to take the limit of vanishing ω of the previous equation we have to performa renormalization by wisely selecting the spectral function

hk(ω) =γk

2ω2 δω(ω− ω) (5.96)

36It is the smallest energy gap of HS being part of the microscopic Hamiltonian (2.1)–(2.3).

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5.4. Applications

where δω is an enlarged delta-function, for example in the interval [− 1ω , 1

ω ] and γk is afriction associated to the kth reservoir. Moreover, we have to define a set of coarse-grainedcoupling operators

Qk = limω→0

∂ω A(ω)k, ω = lim

ω→0∂ω A(ω)

k, ω

†(5.97)

being self-adjoint; they are linked to the microscopic coupling operator Qk through atwo-fold scaling procedure: a coarse-graining in time followed by a coarse-graining inenergy (or space). Note that in the limit of vanishingly small energy splitting the energyspectrum becomes continuous so that the sum over the frequencies can be replaced byan integral which is eliminated by the delta-function δω(ω− ω) of the spectral function(5.96). So taking (3.33)–(5.97) altogether in the limit ω → 0 leads to

J (0)k (ρ) = − γk

2βk[Qk , Kρ[Qk , ln ρ + βk H(0)

S ]]. (5.98)

This equation has exactly the same structure as the modular free NTME (3.66) exceptfor the operators Qk, appearing in the double commutator, which are not the couplingoperators Qk of the interaction Hamiltonian (2.3). This finding suggests to define for theMDS master equation (3.9)–(3.10) a different scattering operator Xk =(1/2)

√hk(0)Qk

expressed in terms of the coarse-grained coupling operators (5.97) which themselvesdepend on the singular rescaled operator (5.94).

A similar strategy can be applied to the dynamically time-averaged NTME (3.53) tostudy the Brownian motion of a free particle for a single reservoir more concretely,namely for the following dissipative part [77, 78]

J (ρ) =12

∫R

dq∫

Rdν hq(ν)[A†

q,ν, K−νρ [Aq,ν, Ψ]], (5.99)

where Ψ = −(ln ρ + βHS) with HS = P2/2m and P the momentum operator. It isprovided in terms of an additional integral for the momentum q exchanged with thereservoir during the collision process illustrated in the Fig. 4.1. To model the Brownianmotion, we choose for the modified eigenoperators (3.45) the following form 37

Aq,ν = eih X·q

√δωc (ε(p + q)− ε(p)− ν1), (5.100)

where the operator eih X·q plays the role of a generator of momentum transfer and X is

the position operator. The energy ε(p) = p2/2m is defined according to the systemHamiltonian of the free particle and the energy exchanged in each collision is then givenby (ε(p+ q)− ε(p)− ν1). The Gaussian δωc(·) under the square root, previously definedin (4.3), indicates that we do not impose a sharp energy conservation during the collision

37Such kind of Lindblad operators were already proposed by Petruccione and Vacchini, see the generatorgiven in equation (24) or (27) of [331] as well as in (2)–(3) of [332].

101


process as depicted in Fig. 4.1 (a sharp one is recovered in the limit ωc → 0). On theother hand, we have a momentum-dependent peaked spectral function

hq(ν) = | f (q)|2Nνq

(eβνδD(ν− νq) + δD(ν + νq)

), (5.101)

expressed in terms of a linear dispersion relation νq = c|q| associated to the mode qof the reservoir taken such that |q| � 1, the Dirac-delta δD, the number of particlesin the reservoir Nνq = 1/(eβνq − 1) and the form factor f (q) respecting the conditionsf (q) = f (−q)∗ as well as the time-reversal | f (q)|2 = | f (−q)|2 so that the microscopicdetailed balance is fulfilled. To take the grazing collision limit, which is characterizedby a small momentum exchange during each individual collision with the particlesof the reservoir, we use a singular structure factor to eliminate all non-zero momen-tum contributions | f (q)|2 = ∂2

q (δD(q)g(q)). It depends on the regularization functiong(q)= (γνq)/(2δωg(0)) itself expressed in terms of δωg(0)=1/(

√2πωg) yielding, after

performing the calculations provided in the Appendix 9.20, to the nonlinear masterequation

J (ρ) = −12

(i

γ

m[X, Kρ(P)] +

γ

β[X, [X, ρ]]

)= − i

2Dp[X, Kρ(P)]−Dpp[X, [X, ρ]] (5.102)

expressed in terms of the drift Dp = γ/(4m) and diffusion Dpp = γ/(2β) coefficients.This drift/diffusion denomination becomes obvious when we use the approximationKρ(P)≈{P, ρ}/2, discussed in the Appendix 9.19, leading to the CLME (2.46) which inthe classical limit give rise to the Klein-Kramer equation (2.47) fulfilling the well-knownEinstein-Smoluchowski relation [333, 65]. Accordingly, the first terms in (5.102) can beseen as an irreversible drift. Note that for γ=4mγ, with γ an ad-hoc friction, we recoverthe Dp = γ and Dpp = 2mγ/β obtained by Vacchini in [334]. This equation is clearlyof the same type as the previous nonlinear one (5.98) but with concrete operators. Inthis regard, the calculations (5.92)–(5.98) are more a proof of principle to demonstrate thefeasibility of the approach without having to specify the details of the model.

Link to a microscopic picture? Looking back at the equations originating from amicroscopic derivation encountered in the Sec. 2.1.4, can we assign a physical role tothe modular free NTME given either by (3.64)–(3.65) or (5.98)? First of all, we usethe same kind of flat spectral function as in the SCL yielding the LSME (2.43)–(2.44).Interestingly, the modular free NTME (3.65) possesses exactly the same structure in thehigh temperature limit in the sense that both lead to the fully mixed state 38. However,for a high but fixed temperature the modular free NTME (3.65) converges to the correctGibbs state which is more realistic from a thermodynamical point of view. In the samemanner as the pre-average NTME (3.58) can be used instead of the BRME (2.35) for smallbut fixed coupling to handle non-secular contributions, the modular free NTME (3.65)

38The first term in (3.65), which is linear, dominates in the limit βk→0 the second one being nonlinear.

102

5.4. Applications

could replace the LSME (2.44) for moderate to high temperatures. This point is furthersupported by the fact that it provides a proper extension of the NRSME (2.45) used totreat strongly damped systems as well as systems having closely spaced energy level forwhich the LDME cannot be used. Indeed, using the property Kρ(A)≈ (1/2){A, ρ} withthe second term in modular free NTME (3.65) containing Kρ[Q, HS ]≈ (1/2){[Q, HS ], ρ}for a single reservoir one recovers the NRSME (2.45). The same reasoning allows torecover the CLME (2.46) by imposing that Q=X is the position operator and the systemHamiltonian is HS = P2/(2m) + (mω2/2)X2, see Sec. 5 of [146]. This is even moreobvious with the equation (5.102) when considering the Brownian motion of a freeparticle. If compared to the Lindblad equation obtained by Vacchini (see equation (13)in [332]) we notice that the modular free NTME (5.102) is free of the term Dxx[P, [P, ρ]]

which is essential to ensure the positivity of the Lindblad equation in the grazingcollision limit. However, it has been shown recently that the physical role of the positiondiffusion Dxx is “fictitious”: random collisions cannot induce a position diffusion on thesame time scale as the momentum diffusion implying that Dxx is restricted to short timescales and is a byproduct of the coarse-graining in time to get a Lindblad generator (seeequation (36) of [64] and [335]). The position diffusion fictitious effect is observed atthe classical level when deriving the Smoluchowski equation with a coarse-graining intime [65] but absent when avoiding it leading to the Klein-Kramer equation (2.47).

Thus, the modular free NTME (3.65) generalizes the NRSME (2.45) and the CLME(2.46) by ensuring the convergence to the proper Gibbs state as well as maintainingthe positivity of the density matrix. It also avoids a fictitious diffusion present in thestandard Lindblad equation for Brownian motion. Obviously, the modular free NTMEcould be used to discuss overdamped systems for which the reversible motion changeson a similar time scale as the interaction with the reservoir (tH ∼ tS� tR ∼ 0) from aphenomenological point of view. This idea is in accordance with Grabert’s BML to treata heavy particle diffusing in a reservoir of lighter particles [221, 141] earlier mentionedfor the modular free NTME (3.65). However, only environments which are of classicalnature (i.e. characterized by a flat spectrum) can be treated in this way 39. Nevertheless,since the dissipative part contains the system Hamiltonian but is free of the Lindbladeigenoperators 40 it can handle non-adiabatic external perturbations as does the LSME(2.44) in the SCL as pointed out earlier in Sec. 2.1.4. For the same reasons, it could beemployed to study the quantum Zeno effect (see Sec. 3.5.2 in the textbook [20] using theLSME for this purpose). Moreover, contrary to the LSME (2.44) the modular free NTME(3.66) can be applied to study the response to a thermodynamic driving, generatedthrough the exposition to multiple reservoirs at different temperatures, since it leads toa NESS contrary to the LSME. Finally, as discussed in Sec. 5.4.1 it can describe two T2

times beyond the bifurcation at high temperature since it can afford a strong damping.

39For lower temperatures, where the quantum nature of the environment becomes prominent, thegeneralized Feynman-Vernon influence functional proposed in [147, 148, 149] should be used instead.

40The Ak,ω’s are problematic for closely spaced energy levels/continuous spectrum, see Appendix 9.1.

103


Is the environment really structureless in the SCL? In fact, this is not necessarily thecase. If we come back to the bosonic spectral function (2.22) provided earlier during themicroscopic derivation, we neglected one important contribution, namely the spatialcorrelations. To take them into account, we follow the method proposed in [121] andlater developed and exploited in [336, 337, 338, 59]. Thus, the spectral function (2.22)gives in the long time limit

c(j)(ω, rk, rl) =∫ ∞

−∞dt eiωt trRj

(Φ(j)t (rk)Φ(j)(rl) σβ j

) (5.103)

so that the system is coupled to the j reservoir at positions rk and rl leading to a non-diagonal friction matrix. It is obtained under exactly the same assumptions as for astandard microscopic derivation in the Born-Markov limit leading to a homogeneousspectral function c(j)(ω, rk, rl)= c(j)(ω, |rk− rl |). We have seen that for a bosonic reservoir,it is important that the microscopic detailed balance (2.27) or KMS condition (2.28), whichis a purely temporal contribution, is ensured. According to what we have seen for abosonic reservoir (2.25), one could write down the following more explicit formulation

c(j)(ω, |rk − rl |) = c(|rk − rl |) h(j)(ω) (5.104)

containing a spatial c(|rk − rl |) and a temporal h(j)(ω)=∫ ∞−∞ dt eiωth(j)(t) component. For

a bosonic reservoir, we have that h(j)(ω)= 2(nβ j ,ω + 1)J(ω), where nβ j ,ω = 1/(eβ jω − 1)is the bosonic distribution and J(ω) an antisymmetric spectral density so that thedetailed balance condition h(j)(−ω)= eβ jω h(j)(ω) is ensured. For the spatial components,two typical phenomenological forms are used [121], namely an exponential decayc(|rk − rl |) = e−|rk−rl |/ξ or a Gaussian decay c(|rk − rl |) = e−(rk−rl)

2/ξ2provided in terms

of a correlation length ξ ∈ [0, ∞) 41. Since we are interested in the limit ω → 0, weimpose in (5.104) the condition h(j)(0)=γj/(2β j) for a bosonic environment so that theelements c(|rk − rl |) of the friction matrix survive when taking the limit of vanishingenergy splitting leading to the modular free NTME (5.98) starting from a LDME (5.92)whose jth dissipative part then reads

J (0)j (ρ) = −

γj

2β j∑k,l

c(|rk − rl |) [Qk , Kρ[Ql , ln ρ + β jH(0)S ]]. (5.105)

The friction matrix c models the spatial structure of the classical environment and couldprovide a natural “microscopic” interpretation for the dissipative brackets introducedin [205] and later reformulated in the MDS context [77, 78] which are summarized in(3.19)–(3.20). From a phenomenological point of view, this means that the modularfree NTME could be exploited to described extended or macroscopic quantum system

41Two extreme cases should be considered [121], namely (i) a local noise with no correlations for ξ→0so that the irreversible contribution vanishes since c(j)(ω, |rk − rl |)→0 and (ii) a global noise with perfectcorrelations for ξ→∞ so that we recover the spatial-independent spectral function c(j)(ω, |rk− rl |)→ h(j)(ω)in this limit. Examples are provided in [116, 339, 340, 338, 59].

104

5.5. Beyond the FDT: the nonlinear response regime

in hot environments where quantum correlations can matter contrary to the standardintuition [12, 13, 14, 15]. The simplest example one could imagine is to connect each partof the quantum system with a classical spin forming a one-dimensional Ising chain sothat the elements of the friction matrix reads c(|xk − xl |)=exp (ln [tanh(β j Jkl)]|xk − xl |)with Jkl the coupling strength between the spin j and k (see the Sec. III.b of [121] forfurther explanations).

5.5 Beyond the FDT: the nonlinear response regime

So far, we focused our attention to the linear response regime with the FDT and theGKF. However, nonlinear multidimensional spectroscopy has recently been extensivelyapplied to explore a large variety of materials ranging from diatomic molecules to com-plex biomolecules [235, 341, 342]. These time and frequency resolved techniques exploitvibrational or electronic transitions to provide coherent dynamic responses which canbe used to study the detailed evolution of molecules for a large range of space and timescales (down to ∼ 20 fs for electronic spectroscopy). Moreover, they are perfect to probethe Hamiltonian of a complex system and to determine the coupling to its environment.Ultimately, they can allow the reconstruction of molecular potentials by resorting totheoretical models. Nowadays, one of the most popular form of nonlinear spectroscopyis the 2D-Infrared (2D-IR) spectroscopy based on the third order susceptibility leadingto frequency/frequency correlation plots [343, 344]. For this reason, we want to pushfurther the calculation which brought us to the FDT in Sec. 5.2.1 to see to which extent itaffects the nonlinear response.

5.5.1 Polarization and nonlinear response function

The key quantity in multidimensional spectroscopy is the macroscopic polarizationP induced by the application of coherent laser pulses on the quantum setup. If theseapplied fields are weaker than the intramolecular forces of the system, we are allowed toexpand the polarization P(t)= tr(µρε

t ), with µ the dipole operator and ρεt the perturbed

density matrix introduced in the equation (5.18), in terms of the field strength ε as

P(t) = P+ + εP(1)(t) + . . . + εN P(N)(t), (5.106)

where P+ = tr(µρ+) with ρ+ the stationary state of the system. When the system is incontact with a single heat reservoir, ρ+ is equal to the Gibbs state π. We can express thepolarization of order N of the previous expression as a N-folded time-integral (i.e. aN-folded convolution to be precise) of the nonlinear response function S(N), that is

P(N)(t) =∫ t

0dτN

∫ t−τN

0dτN−1 . . .

∫ t−τN−...−τ2

0dτ1 S(N)(τN , τN−1, . . . , τ1)

× E(t− τN) E(t− τN − τN−1) . . . E(t− τN − τN−1 . . .− τ1),(5.107)

105


where τN = t− t1, τN−1 = t1 − t2, . . . τ1 = τN−1 − τN are time intervals expressed interms of the N interaction times ti between the pulses and the sample. To simplify thenotation we have ignored the spatial dependency of the polarization and of the fields.In general, for isotropic bulk medium the even order polarizations such as P(2), P(4), . . .vanish. On the contrary, when considering interfaces in contact with centrosymmetricmaterials, only the even order survives [345, 346, 347]. Moreover, the even orders arealso useful to study the stereochemistry of molecules through their chirality [348, 349].

There exist two well-established approaches to obtain the polarization by computingthe nonlinear response through multitime correlation functions of system observables:Mukamel’s perturbative method 42 and Tanimura’s nonperturbative technique 43. Noticethat Tanimura’s approach is more accurate than Mukamel’s one but it requires highercomputational effort. Besides, we have seen in Sec. 5.2.2–5.2.3 that Mukamel’s techniqueis only valid for vanishing system-reservoir coupling and does not fulfill the FDT.

5.5.2 Third-order susceptibility

The idea is to proceed exactly like we did to obtain the FDT in Sec. 5.2.3 but to considerhigher order contributions. Therefore, we go back to the expansion (5.28) and expandthe generator as an infinite sum

Lεt = L+ ∑

j∈N∗εjE(t)jV×j (5.108)

with the jth extended interaction superoperator

V×j =1

E(t)j

(∂(j)ε Lε

t

) ∣∣∣∣ε=0

:=1

E(t)j L(j)t . (5.109)

For j = 1 we recover the linear interaction superoperator obtained earlier in (5.29). Tofurther expand the perturbed propagator (5.24) we use iteratively (5.27) in conjunctionwith the expansion of the generator (5.108). For the first iteration we obtain a secondorder development

Wεt = Wt + ∑

j1∈N∗j1=1

εj1∫ t

0dt1 E(t1)

j1 Wt−t1V×j1 Wt1

+ ∑j1,j2∈N∗j1+j2=2

εj1+j2∫ t

0dt1

∫ t1

0dt2 E(t1)

j1 E(t2)j2 Wt−t1V×i1 Wt1−t2V×i2 Wε

t2,

(5.110)

42Through the expansion of the density matrix, see Chap. 5 of [235] and Chap. 3 of [344] for closedquantum systems and [350, 237] for open quantum systems.

43Based on a set of hierarchically coupled equations of motion, see Sec.2 of [95] as well as [351] discussingthe connection between the perturbative and nonperturbative equation of motions.

106


where for the infinite sum of the expansion given in (5.108) we have added a constrainton each expansion order. The first term later generates the polarization at the steady-state while the second term leads to the first-order polarization and the third one to thesecond-order polarization. To stop the expansion at the second order, one imposes inthe last propagator Wε

t2of the previous equation ε=0. To reach the third order we have

to insert again the expression (5.27). Repeating the operation N times finally yields

Wεt = W0

t +N

∑n≥1

∑j1,...,jn∈N∗

j1+...+jn=N

εj1+...+jn

×∫ t

0dt1 . . .

∫ tn−1

0dtn E(t1)

j1 . . . E(tn)jnWt−t1V×j1 . . . Wtn−1−tnV×jn W0

tn

(5.111)

which we rewrite using time intervals τi defined for the Nth polarization (5.107) so that

Wεt = W0

t +N

∑n≥1

∑j1,...,jn∈N∗

j1+...+jn=N

εj1+...+jn

×∫ t

0dτn . . .

∫ t−τn−...−τ2

0dτ1 E(t−τn)

j1 . . . E(t− τn − . . .− τ1)jnWτnV×j1 . . . Wτ1V×jn W0

t−τN−...−τ1.

(5.112)

We can rewrite the previous expression in the more compact expansion

Wεt = Wε0

t + Wε1

t + . . . + Wεp

t + . . . + WεN

t , (5.113)

where the power p of εp indicates the propagation order p we are interested in and refersto the constraint N=n= p so that j1 + . . . + jp = p in the second sum expansion (5.112).The previous expansion greatly simplifies if we restrict the interaction superoperator(5.108) to the linear contribution (5.29) yielding

Wεt = W0

t +N

∑n≥1

εn∫ t

0dτn . . .

∫ t−τn−...−τ2

0dτ1

× E(t− τn) . . . E(t− τn − . . .− τ1)WτnV×j1 . . . Wτ1V×j1 W0t−τN−...−τ1

.

(5.114)

We can now compute the contribution of order n, namely the polarization P(n) and thecorresponding response functions S(n). Therefore, we go back to the expression of thepropagator given (5.112) and compute the polarization from the expanded propagator

P(t) = tr (Wεt ρ+) = P(0)(t) +

N

∑n≥1

P(n) (5.115)

107


for which we define the polarization of order n as

P(n) = ∑j1,...,jn∈N∗

j1+...+jn=N

εj1+...+jn P(n)j1 , . . . , jn (5.116)

composed of one or more sub-polarization

P(n)j1 , . . . , jn =

∫ t

0dτn . . .

∫ t−τn−...−τ2

0dτ1

× E(t− τn)j1 . . . E(t− τn − . . .− τ1)

jn S(n)j1 , . . . , jn(τn, . . . , τ1)

(5.117)

in terms of the sub-response functions

S(n)j1 , . . . , jn(τn, . . . , τ1) = tr

(µWτnV×j1 . . . Wτ1V×jn ρ+

). (5.118)

To get the linear response back, we use N=n= p=1 for the full expansion (5.112) or thesimplified form (5.114) because only a single term survives. Thereby,

Wε1

t = ε∫ t

0dτ1 E(t− τ1)W0

τ1V×1 W0

t−τ1, (5.119)

so that the linear polarization extracted from (5.115)–(5.117) now reads

P(1)(t) =1ε

tr(

µWε1

t ρ+

)=∫ t

0dτ1 E(t− τ1) tr

(Wτ1V×1 Wt−τ1 ρ+

)=∫ t

0dτ1 E(t− τ1) tr

(Wτ1V×1 ρ+

)=∫ t

0dτ1 E(t− τ1) S(1)(τ1),

(5.120)

where we have dropped the index ε= 0 of the propagators and used the fact that theprocess is contractive, i.e. the propagated stationary state Wt−τ1 ρ+ is the stationary stateρ+ itself. This expression corresponds to the formulation (5.33) obtained earlier whenwe focused on the linear response regime for the Gibbs state ρ+=π implying that 44

S(1)(τ1) = tr(µWτ1V×1 π

)= β ∂τ1tr

(eL∗τ1 µKπµ

)= β ∂τ1 〈µ(τ1); µ〉π (5.121)

which agrees with the definition of the susceptibility (5.13), denoted earlier by χ(1)AB for

A=B=µ, implying that we recovered the correct FDT 45. Following the same strategy,we obtain the third order response function

S(3)(τ3, τ2, τ1) = ∑j1,j2,j3∈N∗

j1+j2+j3=3

S(3)j1,j2,j3

(τ3, τ2, τ1) (5.122)

44We proceed as previously mainly using the definition of the superoperator (5.29), the properties(5.31)–(5.32) of the instantaneous Gibbs state and the Kubo scalar product (5.10) to get back the FDT.

45We have used the notation S(1)(τ1) instead of χ(1)AB(τ1) to highlight the fact that the expansion is

performed for the average of the dipole operator equal to the macroscopic polarization.

108


where the individual contributions are given by

S(3)j1 , j2 , j3(τ3, τ2, τ1) = tr

(µWτ3V×j1 Wτ2V×j2 Wτ1V×j3 ρ+

). (5.123)

To obtain a tractable expression, we assume that ρ+ = π and restrict ourselves to thefirst order extended interaction superoperator, i.e. by imposing j=1 in (5.109) so thatj1= j2= j3=1 in the third order susceptibility (5.122)–(5.123), meaning that only a singleterm remains 46

S(3)(τ3, τ2, τ1) = β tr(µGτ3V×1 Gτ2V×1 Gτ1LKπµ

)(5.124)

expressed in terms of the Green function superoperator

Gt = θ(t) eLt (5.125)

with θ(t) the step function (zero for t ≤ 0 or else one) typically used in nonlinear spec-troscopy for OQS [237]. Note that LKπµ leads to the FDT for the first order susceptibility.To produce 2D nonlinear spectra being as close as possible to experimental settings, it iscommon to go into the Fourier space for the variable τ3 and τ1 implying that 47

S(3)(ω3, τ2, ω1) = tr(µGω3V×1 Gτ2V×1 Gω1V×1 π

), (5.126)

where we returned back to the more general form βLKπµ→V×1 π, which depends onthe Green function superoperator in the frequency domain Gω =− (L− iω1)−1. Thethird-order susceptibility (5.126) which we just obtained differs from the one derived byMukamel [235, 237] which reads

S(3)(ω3, τ2, ω1) = tr(µGω3V×0 Gτ2V×0 Gω1V×0 π

), (5.127)

expressed in terms of the reversible interaction superoperator (5.16) which is hereV×0 (·) = i[µ, ·] implying that the FDT is not fulfilled as pointed out in Sec. 5.2.2. Usu-ally, Mukamel’s third order susceptibility (5.127) is equally provided in a commutatorform S(3)(ω3, τ2, ω1) = i3 tr(µGτ3 [µ, Gτ2 [µ, Gτ1 [µ, π]]]). This formulation has been usedin conjunction with the Lindblad equation to model optical response in presence oftransport, see Sec. 4 and equations (25)–(27) in [237]. Mukamel’s expression has a simplephysical interpretation in the time-domain 48: we have a sequence of three interactionswith the incoming fields given by the superoperator V×0 , three propagations given byGω3 , Gτ2 and Gω1 while the final signal generation is described by µ. Because V×0 is acommutator and µ acts either from the left or the right we see that it is composed of23 = 8 terms also known as the Liouville space pathways (for N pulses we have 2N

46Obtained using the superoperator (5.29) and (5.31)–(5.32) for the instantaneous Gibbs state.47Since it does not rely on the identities (5.30)–(5.32) one could consider a NESS with π→πk.48Before applying a Fourier transform to go in the frequency domain for τ1 and τ3.

109


different pathways). It is convenient to represent such a process using double Feynmandiagrams (see Fig. 4-6 of [237]). Note that the formulation (5.127) is valid as long asthe duration of the pulses (i.e. the interaction with the sample) are much shorter thanthe delays during which the free propagation takes place (i.e. no field applied). Hence,the N pulses must be temporally clearly separated. This assumption makes sense inthe WCL where the two forms (5.126) and (5.127) agree. However, in the WCR theirreversible interaction given by the superoperator V×1 has to be used instead of thepurely reversible one V×0 . Notably, computing the third order susceptibility with theBRME (2.35), valid in the WCR, should be performed with the extended third ordersusceptibility (5.126) instead of Mukamel’s version (5.127) normally used to study 2D IRspectroscopy as in [352]. We leave this comparison for a future work since it requiresa detailed analysis, including comparison with experiments [353], and also becauseusing the linerization of the NTMEs is a delicate issue when considering higher ordersusceptibilities 49 as we already pointed out in [231]. For this reason, the best strategyto illustrate the presented nonlinear response theory would be to rely on the Lindbladequation proposed by Palmieri and co-workers [70] for light-harvesting complexes. Thisequation is obtained by constructing the Lindblad operators based on the microscopicBRME: this ensures that the system converges to the Gibbs state and that at the sametime the coherences are coupled with themselves and with the populations (i.e. throughnon-secular contributions) without spoiling the positivity of the time evolution.

49Nevertheless preliminary results indicate that for small but finite couplings (γ=0.1) and low tempera-ture (β=5) using the linearized pre-averaged NTME leads to similar result as the BRME.

110

6 Quantum superposition and correla-tions in open quantum systems

Quantum information can equally be stored in coherence or quantum correlations suchas entanglement and discord. Coherence quantifies the degree of superposition of aquantum system 1 and is attached to the off-diagonal element of the density matrix ρ

customarily designed as the polarizations or coherences 2. On the other hand, quantumcorrelations are present when the quantum system is partitioned in subsystems whichcan be physically identified and manipulated separately by means of local operations [7].Entanglement quantifies whether the state of such a composite system is separable [88]while discord (or more precisely the relative entropy of discord) records the degree ofimmunity of a bipartite system to a partial measurement of one of its parts 3. Note thatentanglement [185] does not capture all non-classical correlations, reason why discordbecame popular in recent years [356, 357, 358, 359]. For example, discord is able tocapture some degree of quantumness for systems having zero entanglement 4. Onecan also quantify quantum correlation from a thermodynamical point of view withthe so-called quantum work deficit [362, 363]. Recently, a hierarchy of the main types ofquantumness measures have been proposed [364]

entanglement ⊂ discord ⊂ coherence (6.1)

Thus, coherence can exist in absence of any entanglement (or discord) whereas entangle-ment (and discord) requires generally a non-zero coherence 5. It is also unclear whichone of them is the best resource for quantum computation (i.e. too much entanglementis detrimental, it should “come in the right dose” to be useful [368]). Besides, it was

1A superposition arises from the interferences between simultaneously coexisting states of a quantumsystem (e.g. a spin-1/2 being at the same time in the up and down state as later explained).

2The coherences ρi 6=j taken separately cannot be used to provide a proper measure of the degree ofsuperposition; a measure combining all coherence elements together like the l1 norm [354] is required.

3A zero discord means that the measured sub-part is unaffected by the measurement and is henceclassical; it is asymmetric with respect to the sub-part of the system which is measured [355].

4Some Werner states are not entangled but show discord [357]. In fact, nearly all quantum states displaya non-zero discord [360]. For two qubits, it can be extracted numerically in an efficient manner [361].

5Moreover, there exist many other quantum correlation measures such as the witnessing nonlocalitygiven by the Bell function [365] or the measurement-induced disturbance [366, 367].

111

Chapter 6. Quantum superposition and correlations in open quantum systems

shown that one can convert any degree of coherence (local resource) according to somereference basis to entanglement [369] or discord [370] (non-local resources) through aseries of incoherent operations as well as converting discord into entanglement [371].

We begin by introducing the notion of coherence and entanglement (leaving aside thediscord to keep things straightforward; we discuss it in [372]) and then explain howthey can be quantified following, among others, various measures proposed in the afore-mentioned papers and the reviews [373, 374, 375, 376, 355, 88]. With these tools in handwe will be ready to explore in the forthcoming sections their dynamics with the help ofnumerical simulations. Note that our approach is somewhat naive since we use notionsoriginally developed in the context of completely positive linear dynamics and trans-pose them as such for nonlinear positive dynamics (i.e. without formally proving theirvalidity in this different context). From an experimental viewpoint such an approachis not unusual: it is common to use quantum state tomography [377, 378] to measurethe density matrix of one or more qubits [379, 380, 381] (with an appropriate processingprocedure [382, 383]) from which one evaluates some entanglement measures [384]without specifying the nature of the dynamics. Following this line of thought one couldtake the density matrix produced by the NTMEs (3.53), (3.58) and (3.66), forgetting fora moment how it was produced, and evaluate it with the help of different measuressimilarly as it is done for experimental data. This is supported by the fact that the MDSmaster equations produce a physical time evolution (i.e. it ensures the preservation ofthe hermiticity, trace and positivity of the density matrix) which is compatible with thevarious coherence and entanglement measures. Moreover, nothing prevents that the un-derlying dynamics of the examined open system is nonlinear as discussed in [385, 386].As well, even if linear maps can be accessed experimentally [71, 74] one cannot excludethat the underlying maps are non-CP or nonlinear [387]. From a theoretical point ofview, nonlinear quantum channels have been studied recently, for example, in [388] fortwo-qubit states. Hereafter, after introducing all tools necessary to quantify quantumeffects we will examine a thermodynamically consistent kind of nonlinear quantumchannels given by the NTMEs and compare our result to the LDME and BRME obtainedunder a microscopic derivation. Further analysis will be provided in [372].

6.1 CoherenceThe coherence describes the ability of a system to exist in multiple distinct states at thesame time interfering with each other, i.e. the system is in a superposition of states. Let usconsider a density matrix ρ to describe the state of a quantum system S on a discreteHilbert space HS whose dimension is equal to dim(HS) = dS <∞. Note that ρ has tobe chosen so that it belongs to the space of bounded faithful states D∗(HS), where the∗ indicates that we excluded all states having one or more zero eigenvalues, for thenonlinear MDS master equations 6.

6This issue is similar to the one associated to the zero temperature limit characterized by a pure thermalstate, see Appendix 9.3.

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6.1. Coherence

6.1.1 Coherence of a single qubit

A spin–1/2 is typically described by a 2× 2 density matrix. For a pure state it readsρ = |ψ〉〈ψ| ∈ D(HS) with |ψ〉 ∈ HS a state vector. For example, the normalizedstate vector |ψ+〉=(|0〉+ |1〉)/

√2 belonging to the “up-down” spin basis {|0〉 , |1〉}=

{(1 0)T, (0 1)T} yields the following density matrix

ρ =

(1/2 1/21/2 1/2

), (6.2)

i.e. the spin–1/2 is neither in the up |0〉 nor in the down state |1〉 but instead it is in thesuperposition |ψ+〉. The off-diagonal elements of (6.2) are described as the coherences 7

while the diagonal ones are called the populations. We can diagonalize the previousdensity matrix giving ρ=∑i pi |φi〉〈φi| depending on the orthogonal eigenvectors |φi〉as well as on the eigenvalues pi satisfying ∑i pi =1 and pi∈ [0, 1]. Therefore, it is clearthat the coherence in a given basis does not imply the same one in another basis, i.e. thecoherence is a basis-dependent notion. From a statistical viewpoint, the eigenvalues pi canbe identified with probabilities indicating the degree of occurrence of the eigenvectors|φi〉. In quantum information theory, the superposition of |0〉 and |1〉 characterizes aqubit encoding simultaneously two bits of information in opposition to a classical bitwhich is either equal to 0 or 1 with a given probability 8 and could be described by adiagonal density matrix

ρ =

(1/2 0

0 1/2

). (6.3)

When a spin–1/2 characterized by the pure state (6.2) is weakly coupled to a thermalreservoir at high temperature (β→0) it converges, in general, towards the fully mixedstate (6.3) through a decoherence and thermalization process [20] as displayed in Fig. 6.1.Note that since the coherence is a basis-dependent notion, the equilibrium steady state isgenerically nondiagonal in the energy eigenbasis [389]. For this reason, only a basis-freecoherence measure C(ρ) could provide an univocal estimation of degree of superposition.To model a realistic quantum setup exposed to a noisy environment a statistical mixtureof pure states is required

ρ=∑j

pj∣∣ψj⟩ ⟨

ψj∣∣ , (6.4)

where pj are the normalized weights of (non-necessarily orthogonal) pure state vectors∣∣ψj⟩

observing ∑j pj =1 and pj∈ [0, 1]. The weights pj should not be interpreted as theprobability to find the spin–1/2 in the state

∣∣ψj⟩. However, one can prepare such a mixed

state by combining the pure state |ψ1〉 with a probability p1 and |ψ2〉 with probability

7In NMR one commonly refers to them as the polarizations.8N spins encode simultaneously 2N bits of information instead of only N for the classical case.

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Figure 6.1 – Decoherence process simulated with the LDME and displayed using theBloch sphere representation of a two-level system. The initial state is pure and locatedon the shell whereas the final state located in the center is fully mixed. On the axis, wehave the magnetizations mi = tr(ρσi) with σi the ith Pauli matrix and i= x, y, z.

p2 yielding ρ = p1 |ψ1〉〈ψ1| + p2 |ψ2〉〈ψ2|. Note that if we have only ρ we cannot, ingeneral, reconstruct the individual pure states used to prepare it without ambiguity sincesuch a decomposition is not unique (i.e. a continuous set of decomposition exists [390]).In this averaging procedure, the different pure states will interfere constructively ordestructively so that the degree of coherence of their sum will in general be smaller thanthe one of the individual pure states.

6.1.2 Different measures of coherence

Now that we have an intuitive understanding about what the coherence of a quantumsystem is, it remains to see how to accurately quantify it with scalar quantities [391] andby these means better understand it. The first idea which comes to mind is to computethe von Neumann entropy

S(ρ) = tr(ρ ln (ρ)), (6.5)

or its linearization SL(ρ)=1− tr(ρ2). Both are basis independent quantities thanks totheir unitary invariance which indicates the “information content” of ρ. For a pure stateS(ρ) and SL(ρ) are equal to zero (i.e. perfect knowledge or zero uncertainty about thestate of the system) and strictly positive for a mixed state (i.e. we don’t know in whichstate the system exactly is). Moreover, S(ρ) is equal to ln (dS ), with dS the dimensionof the ρ, for a fully mixed state (i.e. maximal uncertainty). This implies that the “morequantum” a state is, the closer the von Neumann entropy (6.5) is to zero. However, ittreats the diagonal and off-diagonal contributions of ρ on an equal footing and thereforedoes not provide an univocal measure of the quantumness of the state.

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6.1. Coherence

To avoid this issue, one can eliminate the contribution of the diagonal elements of ρ withthe relative entropy quantifying the entropic gain between two density matrices 9

S(ρref || ρ) = tr(ρref(ln ρref − ln ρ)), (6.6)

where ρref is a reference density matrix 10. Alternatively, since we are only interested inthe off-diagonal terms of the density matrix, one could simply define the l1 norm

l1(ρ) = ∑i 6=j|ρij|. (6.7)

Recently, Baumgratz and co-workers [354] showed that the quantities (6.6)–(6.7) satisfya set of axiomatic conditions 11 defining a “good” measure of the coherence. To makea long story short, a good coherence measure C(ρ) must not increase under the actionof an incoherent CP map 12. Thus, if C(ρ)> C(ρ0) the state ρ cannot be reached fromρ0 by sole (classical) incoherent operations so that ρ can be said to be “more coherent”than ρ0. There still remains one problem with the two previous “good” measures (6.6)and (6.7): they are clearly basis-dependent. For this reason, Yao et al. [364] proposedan extension of the relative entropy which is basis-free and equivalent to the relativeentropy of discord. It is however computationally too demanding as the dimension ofthe system increases so we will privilege the relative entropy (6.6) and the l1 norm (6.7)to perform our analysis 13.

6.1.3 Creation of coherence

The question if CP maps are a necessary and sufficient condition (or just a sufficient one)to produce physical maps is still under debate [398, 190, 399, 280, 191, 106]. The samereasoning applies to the measures which have been examined under the lens of CP maps,i.e. (6.6)–(6.7). The aim of Baumgratz and co-workers is to study situations where thecreation of coherence from an incoherent state is forbidden, even probabilistically [354]which can be perfectly understood from the point of view of quantum informationtheory mainly focusing on the Kraus representation of quantum channels [7] . However,for more concrete systems, it is necessary to go beyond this picture and allow thecreation of coherence from an incoherent state under a decoherence process since ithas been measured in plenty of setups. Indeed, observation of coherence revival has

9It measures the distance between two density matrices in analogy to the classical Kullback-Leiblerdistance; note that it does not satisfy the usual property S(ρref, ρ) 6=S(ρ, ρref) of a metric.

10According to Baumgratz and co-workers [354], ρref has to be constructed by retaining only the diagonalelements of ρ.

11These axiomatic constraints have been postulated in analogy to what is required to obtain a “good”entanglement measure discussed in [392].

12CP maps have been presented in Sec. 2.2.5; the LDME (2.38), LKME (2.52), CGME (2.64) are all CP.13Further discussion about Baumgratz’s coherence measures can be found in [393, 394, 395, 370]. A

measure based on the norm of the deviation of the density matrix, which is mainly suited for short-timeswhere the decay process is non-exponential, is discussed in [396, 397].

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been found for a delta-kicked rotor [295], a Rydberg electronic wave packet [400], atrapped atom [200], two electronic states of ozone with an attosecond probe pulse [401],in presence of an artificial Kerr medium [402] as well as for light harvesting complexes assummarized in [403, 201] or in the Sec. 5 of the review [404]. Theoretically, the collapseand revival of quantum coherence have been studied for a harmonic oscillator coupledto a classical fluctuating environment [405]. Note also that symmetry breaking andcoherence revival are most probably linked in one way or another [406, 407] .

In our case, knowing that a measure will not create coherence under the action of aCP time evolution can help us to better define the origin of its generation when usingthe nonlinear master equations which only produce a positive time evolution. Indeed,using a coherence measure which can increase under a CP dynamics would blur ouranalysis. For example, Mukamel’s coherence length measure [408] can increase whenthe dynamics is guided by the LDME overestimating the degree of coherence [409] andshould therefore be avoided.

6.1.4 Three types of decoherence processes

Dissipationless decoherence In this case, the coherences of the system decay whereasthe populations are maintained constant so that no energy is exchanged between thesystem and the environment. Such a phenomena is usually described as a dissipationlessdecoherence process [162] or a pure phase relaxation process [410] which can be associated toenergy-preserving measurements. It takes place when the Hamiltonians (2.1)–(2.3) fulfillthe condition [HS ⊗ 1R, HSR]=∑k[HS , Qk]⊗Φk =0, or equivalently [HS , Qk]=0 for allk breaking the EP presented in Sec. 2.1.5. This is sometimes designed as a quantumnon-demolition interaction [161] in reference to the idea of quantum non-demolitionmeasurements [411, 412, 413, 414]. Typically, one obtains a Lindblad equation satisfyingthis condition for a coupling operator Q = HS such as the one provided in equation(5) of [161] derived in the BM regime. One can even get an exact solution, see e.g. theequations (2.27)–(2.29) in [162] for a bosonic bath and equations (41)–(42) in [415] for afermionic bath.

Dissipative decoherence Such a process differs from the previous one by the fact thatit obeys the condition [HS , Qk] 6=0 for all k. More generally, it fulfills the KMS condition(2.27)–(2.28) and EP (2.48)–(2.49) and thus converges towards a unique thermal steady-state in case the system-environment coupling is weak14. On the other hand, in thehigh temperature limit or for the LKME the steady-state is the fully-mixed state. Theseprocesses are designed as dissipative since energy flows between the system and theenvironment on top of the decay of coherence.

14When the coupling is strong, the system converges towards a noncanonical steady-state [148, 416].

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6.2. Entanglement

Non-ergodic decoherence In case the condition [HS , Qk] 6= 0 for all k is fulfilled butthe EP (2.48)–(2.49) is broken, the system displays a decay of energy and coherence butdoes not end in a unique steady-state (implying e.g. asymptotic entanglement for twoqubits [163, 164] or two harmonic oscillators [165]).

6.2 Entanglement

To study entanglement, let us follow [88] and take a multipartite quantum system S ,composed of N subsystems, so that its discrete Hilbert spaceHS=H1⊗H2⊗ . . .⊗Hp isdecomposed into the tensor product of p factor spacesHi of dimension dim(Hi)=di <∞implying that the dimension of the system is simply dS = ∏

pi=1 di. As previously, we

have that the density matrix ρ ∈ D∗(HS).

6.2.1 Bipartite system

De facto, one simply needs a system composed of two subsystems A and B to storequantum correlations 15. Such a setup is ideal to get a first intuitive understanding ofwhat entanglement is and how it can be characterized. To this end, let us define twodifferent types of separable (or uncorrelated) states. First of all, a pure state is said tobe separable as long as it can be written as a product state |ψ〉= |ψB〉 ⊗ |ψB〉 for a statevector, with |ψA〉 ∈ HA and |ψA〉 ∈ HB, or equivalently

ρ = |ψA〉〈ψA| ⊗ |ψB〉〈ψB| (6.8)

for a density matrix. On the other hand, a mixed state is said to be separable if one canprovide a convex combination of the density matrices of the subsystem A and B 16, i.e.

ρ = ∑i

pi(ρA,i ⊗ ρB,i), (6.9)

where ρA,i ∈ D(HA) and ρB,i ∈ D(HB) are pure states of each subsystem whereas theweights are positive pj≥ 0 and normalized ∑j pj = 1. In this case, the correlations arecharacterized by the classical probabilities pi and are due to an incomplete knowledgeabout the state of the system. Following Werner [185], an entangled state is a state which isnot separable, i.e. it cannot be obtained from the convex combination (6.9). He called sucha state an Einstein-Podolsky-Rosen (EPR) correlated state in reference to the role they areplaying in the EPR paradox. In this regard, a separable state is a classically correlatedstate whose properties can be reproduced with the help of classical mechanics (i.e. itcan be prepared by a set of local operations on the individual subsystems); note that

15In analogy to the famous denomination of Alice and Bob used in cryptography.16The linear combination of two separable states ρ′ and ρ′′, each being in the convex set of separable

states, are such that their linear combination αρ′ + (1− α)ρ′′ with α∈ [0, 1] is still in this convex set.

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a state with zero entanglement can nevertheless display quantum correlations suchas discord [357, 360]. From this perspective, entanglement is clearly a global propertyof the bipartite (or multipartite 17) system. Proving that a convex combination (6.9)doesn’t exist for a bipartite system is a non-trivial task (i.e. there exists, in general, aninfinite number of possible decomposition of ρ). For this reason, different measures ofentanglement have been developed to quantify it with more or less success since noperfect measure valid for an arbitrary dimension has not yet been found.

6.2.2 Quantum operations and entanglement

Before looking at some relevant entanglement measures, it is useful to recall the principleof quantum operations. One has to distinguish local operations, under which the subsys-tems evolve independently from each other 18, from the global operations allowing thegeneration or elimination of correlations between the subsystems since they no longerevolve independently. A special case of global operation is the so-called local operationand classical communication (LOCC) which permits the generation of classical correlationsbetween the subsystems through local transformation (i.e. the subsystems exchangeclassical information 19)

ΦLOCC(ρA ⊗ ρB)=∑j

pj(ρA,j ⊗ ρB,j). (6.10)

It is obvious that any separable state will remain separable under a LOCC meaning aLOCC cannot create entanglement [373]. A proper entanglement measure which doesnot increase under a LOCC is said to be entanglement monotone [417, 418]. On the otherhand, any operation transforming a separable into a non-separable state produced by anexchange of quantum information between the subsystems (e.g. entanglement sudden-birth or revival [88]) is not a LOCC. This idea is very close to the criteria defining a goodcoherence measure according to Baumgratz et al. [354] as we have seen in Sec. 6.1.2.

17We mean by multipartite (or p-partite with p > 2) a system with more than two subsystems. A k-separable state, with k < p, contains k subsystems which are not correlated with the others (it is only“partially” entangled). On the other hand, a p-separable state is “completely” separable and can be writtenas a convex combination of the density matrices of the subsystem. A more detailed classification of“partially separable states” figures in the Definition 5–10 along the Sec. 2 of [88].

18Under local operations neither classical nor quantum correlations are generated.19Typically, a LOCC can be used to measure the state of a pair of qubits A and B initially in a Bell

state. If Alice performs a measurement on the qubit A and transfers this information through a classicalchannel to Bob who measures the qubit B, Bob knows in which Bell state the pair of qubit was prior to themeasurements.

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6.2. Entanglement

6.2.3 Different measures of entanglement

Similarly to the coherence measure given in Sec. 6.1.2, we can exploit the von Neumannentropy (6.5) to estimate the degree of entanglement of a bipartite system through theso-called von Neumann mutual information

IM(ρA, ρB; ρ) = S(ρA) + S(ρB)− S(ρ) (6.11)

provided in terms of the subsystem density matrices arising from the partial tracesρA = trB(ρ) and ρB = trA(ρ). This is nothing else than the relative entropy between theseparable ρA ⊗ ρB and entangled ρ states 20. The problem with this measure is that itcan increase under a LOCC which, as previously mentioned, is not desirable. For thisreason, one should subtract from this quantity all classical correlations (which is exactlywhat is done for the relative entropy of discord, see e.g. equation (12) of [357]; discord ishowever computationally demanding and difficult to use beyond two qubit systems).

As for the coherence measure C(ρ) provided in Sec. 6.1.2, one can define a “good”entanglement measure E(ρ). However, no set of universally accepted axioms is availablefor E(ρ) but a reasonable list of criteria can be established (i.e. it must vanish on the setof completely separable states, be monotone under a LOCC,. . . see [392, 373, 374] fora detailed overview). The idea is the same as for the coherence measure: E(ρ) shouldnot increase under the action of an incoherent CP map. There is however one importantdifference since most of these measures try to determine if a multipartite system isseparable or not: we have to distinguish the measures providing a necessary conditionfor separability (e.g. the negativity), which could lead to false positive results, from theones giving a necessary and sufficient condition for separability (e.g. concurrence) 21.

In analogy with the relative entropy coherence measure (6.6), we have the relative entropyof entanglement [392, 421]

ES (ρ) = minσ⊂S

S(ρ || σ) (6.12)

with S the set of all disentangled (or separable) states. In practice, one has to screen allseparable states σ and find the one displaying the smallest distance (or distinguisha-bility), quantified with the help of the relative entropy S(ρ || σ), from the state ρ forwhich one would like to estimate the degree of entanglement. However, to examinequantum correlation over a wide range of parameters (i.e. different initial states, temper-atures, coupling strengths,...) the relative entropy of entanglement (6.12) is clearly toodemanding in terms of computational resources.

20One also observes that for a separable state ρ the inequalities S(ρA) ≤ S(ρ) and S(ρB) ≤ S(ρ) holdwhereas for an entangled state the contrary is true [419]. This comes from the fact that an entangled stategives more information concerning the total system (and thus a smaller von Neumann entropy) than aboutits individual components as pointed out by Schrödinger [420].

21The negativity and concurrence are later defined.

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A more practical, but nevertheless good measure, is the concurrence exclusively validfor two qubits. It was deduced from the entanglement of formation and was firstobtained for pure states [422] and later for mixed states [423, 424, 425]. The concurrenceis computed by first performing a spin-flip of ρ giving ρ=(σy ⊗ σy)ρ∗(σy ⊗ σy) with thePauli matrix σy (i.e. inducing for each qubit a rotation of π around the y-axis on theirrespective Bloch sphere) and ρ∗ is the complex conjugate of the density matrix. Onethen computes the non-Hermitian matrix R2 = ρρ and obtains the square roots of itseigenvalues denoted ri≥0, with i=1, 2, 3, 4, finally yielding 22

EC(ρ) := C(ρ) = max{0, r1 − r2 − r3 − r4}, (6.13)

where r1 is the largest eigenvalue, ranging from zero for a disentangled state to onefor a maximally entangled two-qubit state (i.e. the Bell states named after Bell and hisfamous inequality). Alternatively, one could compute the eigenvalues ri of the Hermitianmatrix R =

√√ρρ√

ρ and use the same expression (6.13). The idea of concurrence isthat the spin-flip leaves a Bell state unaffected (up to a phase factor) whereas it fullyrotates a separable state to its orthogonal counter-part; all inbetween cases produce ascalar value in (0, 1). For a pure state, R contains a single eigenvalue which is directlythe concurrence. It is crucial that the time evolution leaves ρ positive to be able totrust the concurrence (6.13); luckily all master equations generating a MDS produce apositive dynamics as we have seen in Sec. 3.1.4. Possible extensions of this measure arediscussed, for example, in [426, 427]. Instead of quantifying the degree of entanglement,one could simply checked if the system is entangled or not with a criterion such as theone proposed in [169]; unfortunately, this strategy is not well suited for nonlinear masterequations (one could nevertheless use it for the CGME, BRME, LDME, . . . )

To study more complicated setups than two qubit systems, i.e. any mixed bipartite state,one can use the so-called negativity [418] which readsN (ρ)=(||ρTA ||1− 1)/2 where ρTA isa partial transpose of the bipartite mixed state according to the subsystem A whereas thequantity ||X||1= tr(

√X†X) is the trace norm 23. In practice, one uses ||ρTA ||1=1+ 2|∑j µj|,

where the µj’s are the negative eigenvalue of ρTA , so that N (ρ) = |∑j µj| quantifies byhow much ρTA fails to be positive definite 24. An entangled state is characterized byN (ρ) ∈ (0, 1]; note that for a mixed state the negativity can never exceed the concurrence(6.13). Moreover, this measure gives only a necessary condition for separability contraryto the concurrence as previously stated.

22Both ρ and ρ are non-negative definite so that R2 =ρρ yields real non-negative eigenvalues.23One can get ρTB by transposing ρTA ; it is then possible to compute the equivalent of negativity from the

perspective of the subsystem B.24The negativity relies on the Peres-Horodecki criterion for separability of positive partial transpose

denominated as the positive partial transposed (PPT) criterion [428]. One has to compute for a bipartitesystem given by ρ=∑ij ∑kl pij,kl |i〉〈j|A ⊗ |k〉〈l|B the partial transposed according to one of the subsystemsρTA =∑ij ∑kl pij,kl |j〉〈i|A ⊗ |k〉〈l|B and compute its spectrum. If the PPT-state contains negative eigenvaluesthe system is entangled. It is only a necessary and sufficient condition for separability for systems composedof a pair of qubits or a qubit plus a qutrit [88].

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6.2. Entanglement

6.2.4 Creation of entanglement

For a closed system, a quantum operation is simply an unitary evolution whereas foran open system (i.e. coupled to an environment or undergoing a measurement) it isa non-unitary evolution. The Lindblad operators introduced in Sec. 2.2 are widelyused to model non-unitary CP time evolutions because they come along with a clearphysical interpretation and can be accessed experimentally [71, 387, 74]. Note thatthe algorithms used to extract these operators from spectroscopic data rely on twoconsequences of complete positivity: (i) the measured observables follow an exponentialdecay, (ii) the absence of correlations (or information backflow) between the system andthe environment. As pointed out by Boulant et al. in [387], breaking these requirementscan lead to non-CP maps (in fact, even to nonlinear maps). The non-secular terms areessential to capture the correct physical time evolution in the WCR (e.g. for the initialphase dependence of the dynamics of a single qubit) as well as in the SCL (e.g. for thebiexponential decay of the coherences) as discussed in Sec. 5.4.1 and are appropriatelydescribed by the nonlinear master equations. It is hence not surprising that the samenon-secular terms generating fast oscillations can entangle two qubits initially in aseparable state (6.9) but connected to a common reservoir [44, 45]. Alternatively, onecan transform coherence into entanglement [369] or discord [370] through a series ofincoherent operations. Note that revival and sudden-birth of entanglement have beenmeasured experimentally [429, 43, 430, 202, 203] and can be produced with the help oflocal operations [431]. Taken altogether, these observations indicate that one should gobeyond incoherent CP maps leading to the sole LOCCs to study quantum correlations.Later, we will show that the NTMEs can generate entanglement starting from variousseparable states 25.

6.2.5 Entanglement and non-CP time evolution

The BRME (2.35) leads to a non-necessarily positive dynamics and is consequentlynon-CP; note that a dynamics which is “just” positive is also designed as non-CP. Whathappens if we try to estimation the degree of entanglement of a multipartite systemwith the BRME? Let us look at the enlarged system composed of the multipartite systemof interest S in contact with an ancilla A whose density matrix at time t is given by$t =Λt$0 with Λt = eLt and $0 the initial state. If L=LS ⊗ idA, where LS is the BRMEgenerator while idA is the identity generator (i.e. leaving its subspace unchanged), itis clear that $t can display negative eigenvalues. As long as $0 is separable, one canuse a so-called initial slippage condition to fix this problem [82]. However, if $0 isnon-separable this cannot be done. Imagine now that $t carries negative eigenvaluesbut the density matrix of the reduced system ρt = tr($t) is nevertheless positive definite:

25In fact, even with the LDME we can generate entanglement starting from a degenerated Hamiltonianand appropriate coupling operators as we shall see. In the WCR, one could also use the CGME (2.64)remembering that it does not lead to the Gibbs state in the long time limit.

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in this case, one could still measure the degree of quantum correlations between thesubsystems composing S . However, Anderloni et al. [123] have shown that under thelocal operation LS ⊗ idA an increase of entanglement (originating from the fact that $t

becomes negative definite) can be obtained for the enlarged system S+Awhich is clearlyunphysical. To solve this problem, they applied “a generalized slippage mechanism” or“slippage operation” which further restrict the set of initial states.

The analysis of Anderloni et al. [123] was performed to support the trend that a masterequation has to be CP. For the NTMEs, however, since the dynamics generated bytheir respective evolution superoperator KS (or its enlarged version KS ⊗ idA) remainspositive 26 we are fine following this line of thought since all the argumentation relieson the fact that generator LS not even leads to a positive time evolution. In fact, whatshould have been shown is that a positive (but not a non-positive) generator leads to anincrease of entanglement under a local operation. Moreover, since the KS is a nonlinearevolution superoperator, the usual definition of CP does not apply [187, 188, 189]. Ideally,one should define the notion of CP for the nonlinear maps associated to the nonlinearMDS master equations 27. On the other hand, one could adopt the more pragmaticviewpoint proposed by McCracken [432] by defining a positivity domain in which amap ΦP (i.e. associated to the MDS master equation in our case) is positive so that ittakes initial to final states, both being valid, without having to extend the domain tostates which are not generated in a typical experiment.

6.3 Interaction of multipartite system with a reservoir

Incoherent maps are usually classified in two main classes according to the kind of actionthey have on a multipartite open quantum system 28. A collective map acts simultaneouslyon all parts of the multipartite system and alters the correlations among the subsystemsso that they no longer evolve independently from each others. On the contrary, anindependent map is incapable of correlating the subsystems meaning no information isexchanged between the subsystem through the environment. Hereafter, since we aremainly interested in nonlinear master equations, we use a similar classification butwithout relying explicitly to maps but simply on the nature of the connection betweenthe multipartite system and its environment:

26This can be checked at least numerically for a large set of non-separable initial conditions.27Following the line of thought of [77], one can define an ancilla A which is not coupled to the reduced

system S so that the dynamics of the combined system S+A generates a MDS. Then, when tracing overthe ancilla A, one has to enforce that the dynamics of S still generates a MDS (and therefore ensures thepositivity of the dynamics) by appropriately selecting the Hamiltonians HS ⊗ IA and IS ⊗ HA as wellas the coupling operators QS ⊗ IA and IS ⊗ QA (so that the ergodicity property is preserved). Such acondition may be substituted to the standard notion of CP for nonlinear MDS master equations.

28This nomenclature is mainly inspired from the Sec. 2.2.6 and 3.3.1 of the review [88].

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6.3. Interaction of multipartite system with a reservoir

(i) Independent or local reservoirs act on each part of the system in an individual manner.In general, they cannot correlate the subsystems but can still eliminate quantumcorrelations initially present between the subsystems.

(ii) A common reservoir acts simultaneously on each part of the multipartite systemand can create or destroy quantum correlations.

Hereafter, we briefly examine the role of the initial phases for the case of independentreservoirs (the detailed analysis will be provided in the paper [372]) and then focus on thecommon reservoir to find out how to generate quantum superposition and correlationsthrough an irreversible process. To reach this goal, we have to scrutinise the Lindbladeigenoperators in more details: the idea is to see which microscopic contributions cangenerate non-zero elements in the eigenoperators because the more non-zero elementswe have in a single eigenoperator, the “richer” the interaction with the reservoir is. Thisis, however, not the only microscopic mechanism involved. In fact, beyond the strictWCL the non-secular terms play a central role in this respect enhancing the action of theeigenoperators by “mixing” them.

6.3.1 Uncovering the eigenoperators

Going back to the microscopic description of Sec. 2.1.3, let us reexamine the eigenopera-tors appearing in the LDME (2.38) which we repeat here for convenience 29

ρ=−i[HS , ρ] +M

∑j

N

∑i

∑ω

h(j)i (ω)

((A(j)

i, ω)†ρA(j)

i, ω −12{A(j)

i, ω(A(j)i, ω)

†, ρ})

(6.14)

whose eigenoperators can be written as follows

A(j)i, ω = V

∑EmEn

En−Em=ω

∑k

W(j)ik (V

†QkV)mn |Em〉〈En|

V†, (6.15)

where W(j) is the unitary transformation constructed from the eigenvectors of the jthfriction matrix c(j)(ω) = (c(j)

k, l(ω)), with c(j)k, l(ω) =

∫ ∞−∞ dt eiωt trR(Φ

(j)k, tΦ

(j)l, 0σβ), leading to

the diagonal friction matrix h(j)(ω)= (h(j)i (ω))=W(j) c(j)(ω)(W(j))†. In case the original

friction matrix c(j)(ω) is diagonal, W(j) is simply the identity. The other matrix V comesfrom the system Hamiltonian HS=V†HSV and is used to go into the energy eigenbasis{|Em〉}. In case HS is diagonal, V is the identity matrix. To obtain eigenoperators A(j)

i, ω inthe Hamiltonian basis, one simply does not apply the outermost basis transformation

29Compared to previous Sec. 2.1.3, we added the index j to the eigenoperators A(j)i, ω to better keep track

of the fact that the friction matrix c(j)(ω) is not necessarily diagonal (W(j) is then not equal to the identity).

123


with V in (6.15). In general, for a non-degenerated HS we obtain up to (n2 − n + 1)different eigenoperators, with n=dim(HS ) the dimension of the system S . The term(−n + 1) indicates that we regroup all diagonal eigenoperators into a single one A(j)

i, 0

associated to the frequency ω = 0 (i.e. for pure dephasing); the number of differenteigenoperators depends on the number of nonzero entries in the coupling operatorQk according to (6.15). Thus, for a two-level system, we have at most three differenteigenoperators and for a three-level system a maximum of seven. Looking at theexpression (6.15), one immediately notices that the Lindblad operator A(j)

i, ω carries threedifferent microscopic contributions:

(i) the system coupling operators Qk of the interaction Hamiltonian (2.3).

(ii) the eigenvectors |En〉 of the Hamiltonian HS composing the unitary V.

(iii) the matrix elements of the unitary matrix W(j), issued from the friction matrixc(j)(ω) of the jth reservoir, depending on the reservoir coupling operators Φk ofthe interaction Hamiltonian (2.6) through their time-evolved version Φ(j)

k, t.

To keep it simple, we examine below each of these components for a bipartite system Scomposed of two qubits S1 and S2, whose discrete Hilbert space readsHS=HS1

⊗HS2 ,connected to a common reservoir (i.e. M = 1).

Role of the coupling operators Let us assume that we have a single system andreservoir coupling operators (i.e. N = 1) implying that the friction matrix is diagonal aswell as a diagonal non-degenerated HamiltonianHS . For our two qubit system, we canwrite it either as an independent coupling operator

Q = q1 ⊗ 1+ 1⊗ q2, (6.16)

which is the standard choice for a two-qubit system, or alternatively as a separablecoupling operator

Q = q1 ⊗ q2 (6.17)

and finally a collective coupling operator which can neither take the form of a indepen-dent (6.16) nor of a separable coupling operator (6.17). The two first coupling operatorsare given in terms of two sub-coupling operators q1 and q2

qα =

(aα bα

b∗α cα

)with α = 1, 2, (6.18)

which we parametrize with two real numbers aα and cα as well as a complex number bα

so that it is is Hermitian. Not also that one can express qα in terms of the well-known

124


Pauli matrices providing a more intuitive representation

qα =aα

2(σz + 1)− cα

2(σz − 1) + b∗α(σx − iσy) + bα(σx + iσy). (6.19)

At first glance, the distinction between the two versions (6.16) and (6.17) seems innocentbut it can have drastic consequences on the dynamics. For the independent couplingoperator (6.16) we obtain

Q =

a1 + a2 b2 b1 0

b∗2 a1 + c2 0 b1

b∗1 0 a2 + c1 b2

0 b∗1 b∗2 c1 + c2

(6.20)

whereas for the separable coupling operator (6.17) we get

Q =

a1a2 a1b2 b1a2 b1b2

a1b∗2 a1c2 b1b∗2 b1c2

b∗1 a2 b∗1b2 c1a2 c1b2

b∗1b∗2 b∗1c2 c1b∗2 c1c2

. (6.21)

We right away notice that the anti-diagonal of an independent coupling operator (6.20)will always be composed of zeros for an independent Q contrary to a separable one(6.21). Moreover, these coupling operators can clearly not represent any Q meaningthat there exist plenty of collective coupling operators 30. The standard choice (6.16)or (6.20) is quite restrictive because it ultimately affects the Lindblad eigenoperators(6.15) limiting the possible quantum jumps which the system can undergo through theaction of the irreversible dynamics. The separable form (6.17) or (6.21) is less usualbut one can find a physical origin for it, namely for a nonlinear coupling of the kind(q1 ⊗ 1+ 1⊗ q2)2 = q1 ⊗ q2 with qα =1+

√2qα + q2

α, where qα is given by the generalform (6.19) relying on Pauli matrices. On the other hand, collective coupling operatorsmost probably arise from collective kind of interactions, for example, due to long-rangemagnetic/phonons reservoir interactions [433, 434].

Role of the system Hamiltonian We assume that we have a diagonal friction matrixand a coupling operator of the standard form (6.16). For two non-interacting qubits thesystem Hamiltonian, which is not necessarily diagonal, reads

HS=HS1⊗ 1+ 1⊗ HS2 . (6.22)

30Such a Q can be found by focusing on a Q with the diagonal (1, 1, 1, 0). Looking at the independentform (6.20), we see that c1 + c2 = 0 implies that 1 = a1 + c2 = a1 − c1 which when added to a2 + c1 = 1gives a1 + a2 = 2 contradicting the last relation a1 + a2 = 1. For the separable form (6.21) we havea1a2 = a1c2 = c1a2 =1 which implies c1c2 =1 contradicting the last relation c1c2 =0 meaning that such a Qis collective according to our definitions.

125


On the other hand, in presence of an interaction between the subsystems we have

HS=HS1⊗ HS2 . (6.23)

For example, one can add a pairwise interaction to the system Hamiltonian givingHS=σz ⊗ 1+ 1⊗ σz + κ(σ+ ⊗ σ− + σ− ⊗ σ+) which is non-diagonal and can be recastin the form (6.23). Another source of non-diagonal Hamiltonian can be obtained whenconsidering the Lamb shift arising from the imaginary contribution of the spectralfunction s(j)

i (ω) given earlier in (2.18) and which reads HLS =∑ω ∑i s(j)i (ω)(A(j)

i, ω)† A(j)

i, ω. Ittakes this form following a standard microscopic derivation leading to the LDME (6.14)for a diagonalized friction matrix 31 and is added to the bar system Hamiltonian HS . For aproperly renormalized system Hamiltonian, one should apply an iterative procedure:take HS to compute the Lindblad eigenoperators A(j)

i, ω, construct the Lamb-shift HLS anduse the sum HS + HLS to produce a new set of Lindblad eigenoperators which in turngive rise to a new Lamb-shift and so on until the procedure converges 32. The Lamb-shiftwas exploited by Benatti et al. [45] to entangle two unequal atoms using a non-diagonalKossakowski matrix, see the matrix (20) and the Hamiltonians (15) and (39) therein.

Non-diagonal Hamiltonians, whatever their origin, impact on the structure of the eigen-operators through V arising from the diagonalization of HS and add non-zero entriesin the eigenoperators. To see it clearly, let us compute the eigenoperators for a non-interacting and non-degenerated Hamiltonian (6.22) which, when diagonalized, readsH(V)S =H(V1)

S1⊗1+1⊗H(V2)

S2since V=V1⊗V2 and HSα

=V†α H(Vα)

SαVα. Under this restriction,

the set of frequencies of each subsystem can be treated separately ωSα∈ sp(HSα

) withα = 1, 2. On the other hand, the set of frequencies of the bipartite system is denotedωS ∈sp(HS) for the sake of clarity. Equally, we can write the different set of eigenvectors|En〉= |En1〉 ⊗ |En2〉 and |Em〉= |Em1〉 ⊗ |Em2〉 as well as the associated eigenenergies dif-ferences ωmjnj =Emj − Enj . Consequently, one can compute the Lindblad eigenoperatorin the Hamiltonian basis A(V)

ωS=V† AωSV as follows

A(V)ωS

= ∑ωmn

〈En|V†QV |Em〉 |En〉〈Em|

= ∑ωm1n1

(q(V1)1 )n1m1 |En1〉〈Em1 |︸︷︷︸

A(V1)ωS1

⊗ ∑ωm2n2

δn2m2 |En2〉〈Em2 |︸︷︷︸1

+ ∑ωm1n1

δn1m1 |En1〉〈Em1 |︸︷︷︸1

⊗ ∑ωm2n2

(q(Vj)

2 )n1m2 |En2〉〈Em2 |︸︷︷︸A(V2)

ωS2

,

(6.24)

31See e.g. Sec.5.2.3 of [23]. Note that for a friction matrix which is already diagonal (i.e. here W(j)=1) theLamb-shift can often be left out since it is diagonal in the eigenbasis of the system.

32Personal communication by Robert Alicki.

126


where we have used (q(Vα)α )nαmα = 〈Enα |q(Vα)

α |Emα〉with q(Vα)α =V†

α qαVα and 〈Enα |V†α Vα|Emα〉=

δnαmα for α = 1, 2. It leads to two Lindblad eigenoperators, each in its own subspace.Going now back to the original basis with (V1 ⊗V2)(A(V)

ωS)(V1 ⊗V2)† finally gives

AωS = AωS1⊗ 1+ 1⊗ AωS2

(6.25)

which cannot be rewritten in this form in presence of an interaction between the sub-systems or for a separable coupling operator (6.21) with a non-vanishing anti-diagonal.Finally, a more subtle and unexpected case can couple the subsystems through thereservoir, namely when the eigenenergies of HS are degenerated 33. It can happen inabsence of any direct interaction between the two qubits, in general, for a symmetricsetup given by HS1

=HS2 (i.e. equal qubits).

Role of the friction matrix One immediately notices that a non-diagonal frictionmatrix strongly affects the structure of the eigenoperator (6.15) even for a standardcoupling operator (6.20) and a diagonal non-degenerated Hamiltonian of the form (6.22)for non-interacting subsystems. Such a matrix typically arises from spatial correlationamong the subsystem as shown for non-identical non-overlapping two-level atomsinteracting with a quantized three-dimensional electromagnetic field [116]. Benatti andco-workers discussed non-interacting subsystems connected to a common bath, firstfor two qubits [169] and then for an infinitely long spin chain (i.e. an open many-bodyquantum system) [171], relying on a non-diagonal friction matrix to interpret recentexperimental findings about macroscopic entanglement [435, 436].

6.3.2 Enhancing the eigenoperators’ action with non-secular terms

Looking back at the BRME (2.35), we see that the non-secular contributions (i.e. ω1 6=ω2,sometimes designed as inter-mode transitions [57]) contain mixing gain terms of thekind A(j)

i, ω1ρ(A(j)

i, ω2)† and mixing loss terms of the type (A(j)

i, ω2)† A(j)

i, ω1ρ. Why mixing? Because

each of these terms depends simultaneously on two different eigenoperators mixing theelements of the density matrix. In absence of pure dephasing, these non-secular termscouple the coherences whose decay starts to depend on the initial phase as we haveseen with absolute coherence value (5.74). Besides, in presence of the right quantity ofpure dephasing the populations and coherences are connected so that we can observethe generation of coherence from an incoherent state in a situation where the LDMEwould not produce it, typically for a small but finite coupling strength and at mediumor high temperatures. The four non-secular terms find their origin in the Fourierdecomposition of the time-dependent coupling operators. There are many situationswhere the non-secular terms should be considered, for example photosynthetic excitationenergy transfer [437] or the problem of Cooper-pair pumping [53]. Performing the WCL

33Leading to fewer eigenoperators with multiple non-zero entries.

127


eliminates the fast oscillating non-secular terms and breaks this link. Often, people tryto restore this link a posteriori, typically using a LKME whose Lindblad operators arenot eigenoperators. For example, to model LHCs Palmieri et al. [70] constructed ”byhand“ a LKME whose Lindblad operators matrix elements are all nonzero and selectedthem so that the system converges to the Gibbs state 34. More recently, the BRME wasreconsidered using an empirical ”partial“ SA to preserve the fast-oscillations couplingthe populations and coherence without spoiling the dynamics (see [49, 55, 58] as well asSec. B in [59]). Note that the two LSME (2.43) and (2.44), as well as the CLME (2.46) or theNRSME (2.45), incorporate the coupling operators prior to the Fourier decompositioninto eigenoperators meaning that they contain the same signal as the one generated bythe non-secular terms of the BRME (2.35). The same is true for the modular-free NTME(3.66) which equally contains the non-decomposed coupling operator with the differencethat this equation remains positive compared to the CLME, NRSME and BRME 35.

6.4 Analysis of two qubit systems

Armed with the coherence and entanglement measures of Sec. 6.1.2 and 6.2.3, we are nowready to study the dynamics of two qubits connected either to independent reservoirsor to a common reservoir. Studying entanglement for such kind of setups has beenwidely examined in the literature [376] but rarely from the point of view of Davies’approach since it is generally expected that the LDME in the WCL can neither createcoherence nor entanglement. This is however not entirely true as first observed by Lendiand Wonderen in [170] and we will examine under which conditions the generationof quantum superpositions and correlations is possible for the LDME (for more recentstudies see [438, 439, 440, 441, 442]) as well as for the BRME and the NTMEs 36. To thisend, we focus on the two microscopic elements evoked in Sec. 6.3.1, namely the nature ofthe coupling operator and the system Hamiltonian in conjunction with the non-secularterms since the friction matrix has been already studied elsewhere [116, 169, 171].

6.4.1 Two non-interacting qubits connected to independent reservoirs

Let us first consider two non-interacting qubits connected to local reservoirs to stayas close as possible to the dynamics of individual qubits. Bellomo and co-workersshowed that under these settings the solution to microscopic master equations with andwithout memory is given by the tensor product of the individual qubit’s solutions (see

34They use the Redfield derivation to appropriately select the elements of the Lindblad operators restoringa posteriori a connection to an underlying microscopic picture to go beyond the usual LDME.

35But cannot be used to describe the low friction regime at low temperatures as shown with the help ofthe FDT in Sec. 5.4.1, see e.g. Fig. 5.2 for an illustration of the problem.

36All the results obtained for the pre-averaged NTME (3.58) can be transposed to the dynamically time-averaged NTME (3.53) for a not too large Tg. We use the modular-free NTME (3.66) for intermediate/hightemperature in the strong damping limit (i.e. white noise kind of environment) as we have seen in Sec. 5.4.1.

128

6.4. Analysis of two qubit systems

e.g. equation (10) of [281]). In absence of pure dephasing, we can on top of that considerthe populations and coherences separately as for the single qubit case. Altogether, thisleads to the solutions of the coherences and populations for a two qubit system [372]

ρc(t)= [Ac(t)⊗ Bc(t)]ρ(0)c and ρp(t)= [Ap(t)⊗ Bp(t)]ρ(0)

p , (6.26)

where Xc(t) and Xp(t) are given by the single qubit solution matrices provided in(5.71)–(5.72) for X = A, B while ρc(t) and ρp(t) are, respectively, the coherences andpopulations two-qubit state vectors. Furthermore, the two qubit generator Lα = LA,α ⊗1+ 1⊗LB,α is given by the Kronecker sum of the individual generators LX,α for α = c, pas given in (5.69). We have to add the index X to the single qubit real parameters u→uX,v→ vX, y→ yX and rescaled complex parameter zeiθ→ z → zX to distinguish the qubitA from B. Interestingly, the decay rates associate to the two qubit coherences Liouvillegenerator Lc are simply given by the sum of the eigenvalues (5.63) of the single qubitgenerator (5.69) yielding Λ±,± = −∑X = A, B(yX ±

√zXzX − ∆X). Accordingly, we can have

zero, one or two bifurcations. We provide the detailed expression for the density matrixelements assuming that we initially have a X-state in [372]. We do not want to entermore into details here; we only would like point out that if we prepare the systemin a generalized Bell state [443], whose only non-zero elements are ρ11 = ρ44 = 1 andρ14 = eiφ0 = ρ14 for φ0 ∈ [0, 2π], we obtain a similar kind of initial phase dependence for|ρ14| as in the single qubit case (5.74) but with more complicated f (t) and g(t) functions.Such a dependence on the phases is also observed for the X-state, as defined in [444],which contains two phases ρ14 = rφ0 e

iφ0 = ρ41 and ρ23 = rψ0 eiψ0 = ρ32 for φ0, ψ0 ∈ [0, 2π].

Thus, since the coherences are coupled for zX 6= 0 their decays depend on the initialphases φ0 and ψ0 of the X-state. Moreover, because the concurrence (6.13) for a X-statereads C(ρ) = max(|ρ32| −

√ρ11ρ44, |ρ41| −

√ρ22ρ33, 0) [445, 427] its decay will depend on

the initial phases (but not the concurrence measure itself) as described analytically in[372]. Hereafter, we show numerically that this initial phases dependence equally takesplace for a common reservoir for which it is impossible to provide an analytical solution.But before that, we explore the subtleties of the dynamics for a common reservoir.

6.4.2 Equal qubits connected to a common reservoir

Let us begin with two identical non-interacting qubits coupled in an equal manner to acommon reservoir. Such kind of setup is not necessarily easy to achieve experimentallybut theoretically it is a natural starting point. The system Hamiltonian (6.22) reads

HS=∆(σz ⊗ 1) + ∆(1⊗ σz), (6.27)

where ∆ is the common energy splitting, which is clearly diagonal and degenerated. Tomake it more interesting, let us select the coupling operator (6.16) as

Q = ζ1(σx ⊗ 1) + ζ2(1⊗ σx) (6.28)

129


Figure 6.2 – Left plot: asymptotic entanglement of two identical qubits connectedsymmetrically to a common reservoir under the action of the LDME at low temperatureand friction (β=5 and γ=0.1; ∆ is equal to the largest energy gap of the system). The EP(2.48) is broken since ∃X 6= c1 such that [Aω, X] = [A−ω, X] = 0. Right plot: breakingthe symmetry of the coupling (6.28) restores the EP (2.48) and the convergence to theGibbs state eliminating the created entanglement at long times.

depending on two parameters ζ1 and ζ2 which we use to break the symmetry of thecoupling by varying the parameter δ = ζ1 − ζ2. Firstly, we examine the perfectlysymmetric case with δ=0. Therefore, we look at three different separable initial states

ρ(0)=ρ(0)A ⊗ ρ(0)

B , ρ(0)A = ρ(0)

B =

(ρ(0)

11 ρ(0)12

ρ(0)12 ρ(0)

22

)(6.29)

for ρ(0)11 = 1− ρ(0)

22 and ρ(0)22 = 0.2 as well as the three coherences ρ(0)

12 = 0.17, 0.1 and 0.05.As usually, we take a bosonic spectral function (2.25) with J(ω) = γω (including anexponential cut-off) at low temperature and friction (β=5 and γ=0.1). As can be seenon the left plot of Fig. 6.2, the concurrence obtained by evolving ρ(0) with the LDMEincreases after a short period of time indicating that the two qubits become entangledthrough the action of the common bath (as a double check, we have computed the vonNeumann mutual information (6.11) which showed a similar behaviour). Surprisingly,it then reaches a steady-state value depending on the initial state. The reason is simple:the eigenoperators Aω arising from our choice of system Hamiltonian HS and couplingoperator Q does not fulfill the EP (2.48) presented in Sec. 2.1.5. The same result isobtained for the BRME and the pre-averaged NTME displaying small additional oscilla-tions which does not modify the value of the steady entanglement (not shown). Clearly,in case the EP is broken the nonlinearity alone cannot drive the system towards theGibbs state. Benatti et Floreanini use a similar strategy in [163] (see also [164]) with theLKME to produce such an asymptotic entanglement but without making reference to thedegeneracy or symmetries of the Hamiltonian nor of the symmetries of coupling to the

130


reservoir (interestingly, Isar uses in [165] two identical harmonic oscillators and equallyproduces an asymptotic entanglement). On the other hand, the role of the degeneratedenergy levels leading to an incomplete decoherence preserving entanglement have beenevoked in [446] for times much shorter than the time scale tS of the system. We haveshown here that the asymptotic entanglement has a microscopic origin and takes place inthe WCL as well as in the WCR. Note that when we break the symmetry of the coupling(6.28) with δ 6=0 we restore the EP so that the system converges to the Gibbs state and theconcurrence decays to zero (see right plot in Fig. 6.2). Clearly, the creation (or sudden-birth) of entanglement with the LDME is in itself surprising. The reason is that theHamiltonian (6.27) is degenerated leading to fewer Aω’s with multiple non-zero entriescoupling the subsystems through the reservoir. This is a direct consequence of the factthat the Lindblad eigenoperators take the form (6.15) under a microscopic description.Equal qubits have been recently used to study teleportation with the help of Daviesapproach [442] further supporting the importance of the Hamiltonian’s degeneracy. Ifone choose a non-degenerated Hamiltonian, the LDME is no longer able to generateentanglement contrary to the BRME and the pre-NTME 37.

6.4.3 Unequal qubits connected to a common reservoir

To obtain two qubits which are physically different, we have to modify the systemHamiltonian (6.27) to get distinct energy gaps. The simplest choice is obviously

HS=∆2(σz ⊗ 1) + ∆(1⊗ σz). (6.30)

However, to show that breaking the symmetry of the Hamiltonian is more importantthan breaking the symmetry of the coupling we decided to inspect the action of thecoupling operator (6.28) with ζ1= ζ2=1. Moreover, to see the impact of pure dephasinglet us also study another symmetric coupling operator, namely

Q = (σx + σz)⊗ 1+ 1⊗ (σx + σz). (6.31)

The initial state is separable and taken as previously (6.29) but for ρ(0)12 =0.17. The same

spectral function is used as in Sec. 6.4.2 for β = 5 and γ = 0.1. Looking at Fig. 6.3,one sees that the LDME no longer creates entanglement since the degeneracy of thesystem Hamiltonian has been lifted. On the other hand, we see that the BRME fails topreserve positivity and overestimates the amount of entanglement in comparison tothe pre-averaged NTME. This drawback, discussed in [123], further indicates that thepre-averaged NTME could be a substitute to the BRME in the WCR since it remainspositive at all times. To further check that it can reproduce the same signal as the BRME

37Apart from the situation where a collective coupling operator Q is used in presence of non-zero initialcoherence, see Fig. 6.6 below. It is however hard to imagine in which kind of physical setup one couldencounter this kind of coupling as already mentioned in Sec. 6.3.1.

131


0 2 4 6 80

0.01

0.02

0.03

0.04

C

LDMEBRMENTME

0 2 4 6 8"t

0

0.01

0.02

0.03

min

(eig

(;))

BRMENTME

0 2 4 6 80

0.05

0.1

0.15

0.2 LDMEBRMEBRME (; < 0)NTME

0 2 4 6 8"t

-0.08

-0.06

-0.04

-0.02

0

0.02

BRMENTME

Figure 6.3 – Left plots: In absence of pure dephasing, the LDME generates no entangle-ment whereas the BRME and the pre-averaged NTME yield a very similar entanglementcreation pattern (γ= 0.3, β= 5). Left plots: In presence of pure dephasing, the BRMEfails to preserve positivity (see the smallest eigenvalue of ρ in the lower plot) so thatone cannot trust its concurrence; on the other hand, the positivity of the NTME is notjeopardized (same parameters as for the left plot).

Figure 6.4 – Comparison of the concurrence for the pre-averaged NTME for settingswere the BRME does not fail; the two equations produce a very similar time evolution.Simulations performed for the frictions γ=0.1 and 0.4 as well as the Hamiltonian (6.27)and coupling operator (6.28); all other parameters are the same as for Fig. 6.3.

132


obtained by microscopic means, let us take a different system Hamiltonian

HS=∆2(σz ⊗ 1) + ∆

(12− 0.05

)(1⊗ σz) (6.32)

slightly breaking the symmetry and a non-symmetric coupling operator

Q =

(1

20(σz + 1) + σx

)⊗ 1+ 1⊗

(14(σz + 1) +

120

(σz − 1) +12

σx

)(6.33)

inspired from the Pauli form (6.19). Again, we use the same spectral function as previ-ously but we compare this time two different frictions (γ= 0.1 and 0.4) and the sameinitial state except for ρ(0)

12 = 0.37 selected to avoid that the BRME fails, see Fig. 6.4.The origin of the entanglement’s sudden birth is no longer the symmetry of the sys-tem Hamiltonian but is due to non-secular terms which are absent from the LDMEexplaining why it cannot create entanglement (see Sec. 6.3.2). In addition, increasingthe friction generates a higher amount of entanglement (we are still in the WCR; for astrong coupling one should also include non-Markovian effects to be physically consis-tent [20]). Note that starting from a diagonal initial state, the BRME and pre-averagedNTME will produce coherence which in turn participates to the creation of entangle-ment (whereas this is not the case for the LDME). We will come back to this pointlater on. For the time being, we examine the role of two different types of couplingoperators: the non-symmetric independent Q given in (6.28) and a collective Q obtainedfrom (6.28) by setting the elements of the anti-diagonal to Q41 = Q32 = Q23 = Q14 = 1.All other settings are the same (i.e. spectral function, initial state, Hamiltonian). InFig. 6.5 we see that the collective Q results in two successive creation of entangle-ment, namely at short and at longer times. A similar pattern can be produced usinga separable coupling operator (6.21) instead of a collective one; what matters is thatthe anti-diagonal is non-null. We pointed out earlier that the LDME needs a degen-erated Hamiltonian to create entanglement. However, a collective Q can lead to thecreation of entanglement by the LDME if the system contains initially a non-zero co-herence as displayed in the Fig. 6.6. We use an non-symmetric system HamiltonianHS = 0.5∆(σz ⊗ 1) + 0.7∆(1⊗ σz) and a non-symmetric collective coupling operatorQ = (0.5σx + 0.7σz)⊗1+1⊗ (0.8σx + 0.9σz)+Qad with the 4× 4 matrix Qad containingzero elements apart from its anti-diagonal (Q41, Q32, Q23, Q14)= (2, 1 + i, 1− i, 2). Wevary the element ρ12 in the separable initial state (6.29) with ρ11 = ρ22 =0.5. Althoughvery interesting, it is difficult to understand in which context one could encounter sucha collective coupling operator (i.e. long range interactions? many-body effects?) asearlier mentioned in Sec. 6.3.1. Coming back to more standard settings, namely theHamiltonian (6.32) and the coupling operator (6.33) with a friction γ=0.1, let us nowstudy the impact of the temperature on the creation and destruction of entanglement butonly for the pre-averaged NTME in Fig. 6.7. Clearly, increasing the temperature reducesthe maximal created concurrence and precipitates its subsequent elimination.

133


Figure 6.5 – Comparison of the concurrence produced by the pre-averaged NTME for thenon-symmetric independent coupling operator (6.28), represented by the red curve withpure-dephasing and the yellow curve without, and its collective version represented bythe blue curve (left plot for γ=0.1 and right plot for γ=0.4). We have used a log-scaleto distinguish the creation of entanglement at short and long times.

"t0 5 10 15 20 25 30

C

0

0.01

0.02

0.03

0.04;12 = 0;12 = 0:2;12 = 0:25

Figure 6.6 – Creation of entanglement for the LDME for a collective Q, i.e. whose anti-diagonal contains non-zero elements, in presence of an initial coherence (ρ12 of eachindividual qubit has to be non-zero). We use a non-degenerted Hamiltonian as well as afriction γ=0.1 and an inverse temperature β=5.0.

134


Figure 6.7 – Creation of entanglement and its subsequent destruction at different reser-voir temperatures T using the pre-averaged NTME with the same parameters as inFig. 6.5 but only for a friction γ=0.1.

6.4.4 Sudden-death of the concurrence for a common reservoir

The entanglement sudden-death (ESD) was first recognized in the context of OQS byYu and Eberly [447, 448, 449] and later measured experimentally [429]. Contrary toan asymptotic decay of entanglement towards zero, the sudden-death implies that thedecay is instantly stopped (i.e. it converges to zero in a finite time). For simplifiednoise models one can obtain analytical results, see e.g. equations (50)–(51) in the review[450], indicating that the concurrence (6.13) takes the form C(t) = f (t) e−Γt with f (t)the function responsible for the brutal interruption of the exponential decay e−Γt. Notethat solving analytically the LDME at finite temperature is already a tedious task; asimplified version was solved for a broad class of initially entangled states (i.e. X states)indicating that sudden-death always takes place and that at finite temperature the ESDcannot be avoided [451] contrary to the zero temperature limit [449]. In our case, solvingthe nonlinear master equation is clearly not possible so that we limit ourselves to anumerical analysis of the sudden-death phenomenon comparing the LDME to the BRMEand the NTMEs. To this end, let us take a Bell-state

ρ(0)BS =

12

0 0 0 00 1 1 00 1 1 00 0 0 0

(6.34)

135


"t0 20 40 60 80

C

0

0.2

0.4

0.6

0.8

1LDMEBRMEBRME (; < 0)NTME

0 1 2

0.8

0.9

"t0 20 40 60 80

0

0.2

0.4

0.6

0.8

1. = 0:1. = 0:12. = 0:15. = 0:2. = 0:3

Figure 6.8 – Left plot: comparison of the entanglement sudden-death (ESD) betweenthe LDME, the BRME and the pre-averaged NTME. The non-secular terms lead to adelayed ESD; the NTME should be used instead of the BRME whose density matrixdisplays negative eigenvalues at short time. Right plot: ESD for the pre-averged NTMEfor different frictions γ. For the largest γ a revival of entanglement takes place.

which is a pure state. For the NTMEs, we have to work with the closest initial mixedstate given by the convex combination ρ(0)=(1− α)ρ(0)

BS + απ with the final Gibbs stateπ ∝ exp (−βHS). We perform the simulations at low temperature (β = 5): we firstcompare the LDME with the BRME and the pre-averaged NTME at a friction γ= 0.1for our usual bosonic reservoir and only then focus on the NTME looking at differentfrictions. The results are displayed in the Fig. 6.8. Since we study unequal qubits,the most probable origin of the difference of the ESD between the LDME and theBRME or NTME are the non-secular terms. Note that the positivity of the BRME isjeopardized at short-time as can be seen on the left plot of Fig. 6.8 (one should addan initial slip on top of the modified initial state ρ0 to avoid this problem). We fit 38

the decay of the concurrence (left plot in Fig. 6.8) for the LDME and found that it isexponential C(t) = ae−Γat with a = 0.906 and Γa =−0.082 (R2 = 0.99) whereas for thepre-averaged NTME it is biexponential C(t)= ae−Γat + be−Γbt with a=0.236, Γa =−0.021,b = 0.722 and Γb = −0.267 (R2 = 0.97) reflecting the different dynamics underlyingthese decays. Looking at the right plot of Fig. 6.8, we see that increasing the frictionspeeds-up the sudden-death of the concurrence which is then followed by a revivalof the concurrence. Note also that for the case γ = 0.2 or larger one can no longerfit the decay of the concurrence with a biexponential. One usually distinguishes thesudden-birth of entanglement observed earlier in Sec. 6.4.3 produced by starting froman initially unentangled state from the revival (or rebirth) of entanglement taking placeafter that the initial entanglement has been eliminated 39. It is usually claimed that the

38To perform the fit, we excluded all points which have zero concurrence.39For independent reservoirs, we do not have revival or rebirth of entanglement [372].

136


sudden-birth as well as the revival of entanglement only take place for non-Markovianenvironment [452, 453]. We have seen here that a Markovian environment can equallycreate quantum correlations 40. Measurement performed on a two qubit system in aMarkovian environment displayed entanglement collapse and revival [202] equally asfor a non-Markovian environment [43, 430] supporting our numerical findings.

Let us now examine the famous bipartite X-state 41 which has been found to matter fora large variety of physical situations [445, 427]

ρ(0)X =

a 0 0 w∗

0 b z 00 z∗ c 0w 0 0 d

, (6.35)

where the parameters a, b, c and d are real while w = rφ0 eiφ0 and z = rψ0 e

iψ0 are complex.The X-state structure is usually preserved under incoherent time evolution of Kraustype (see equation (30) in [445] or Sec. 4 of [454]) as well as for Davies–type of environ-ment (see equation (13) in [438]). To check whether this is true in in the WCR, we usethe non-degenerated Hamiltonian (6.32) and two different types of coupling operators,namely without pure dephasing (6.28) and with pure dephasing (6.31). To this end, weuse a screening strategy, i.e. sampling different initial states 42. Before looking at theNTME, we examined the LDME and found that it conserves, as expected, the bipartiteX-state structure in absence and in presence of pure dephasing

ρ(t) =

ρ11 0 0 ρ14

0 ρ22 ρ23 00 ρ32 ρ33 0

ρ41 0 0 ρ44

, (6.36)

where the matrix element with the same color interact with each others. By preserv-ing the X structure, we mean that the zero elements remain zero until the steady-state isreached 43. Note that since the LDME is linear the different coherences decay in an inde-pendent manner (explaining why we used different blue colors for the anti-diagonal).On the other hand, for the pre-averaged NTME (ant the BRME) the X structure is onlypreserved in absence of pure-dephasing (left density matrix below) whereas in presence

40One could not claim it undoubtfully with the sole BRME since it does not preserve positivity in general.41The Bell state (6.34) used earlier is a special X state.42Due to the large number of different X states, we have used a random sampling strategy. To generate

the different density matrices, we have applied the representation given in equation (1) of [441] beingpositive definite under the inequalities (5a)–(5c) equally provided therein.

43The off-diagonal elements present initially fully decay in the long time limit since we respect the EP(2.48) for our choice of Hamiltonian and coupling operators.

137


of pure dephasing (right density matrix below) all elements inherit some coherence

ρ(t) =

ρ11 0 0 ρ14

0 ρ22 ρ23 00 ρ32 ρ33 0

ρ41 0 0 ρ44

ρ(t) =

ρ11 ρ12 ρ13 ρ14

ρ21 ρ22 ρ23 ρ24

ρ31 ρ32 ρ33 ρ34

ρ41 ρ42 ρ43 ρ44

(6.37)

before going back to zero once the Gibbs state is reached. Note that contrary to theLDME, the element of the anti-diagonal are coupled to each other. This is in line withwhat we have observed for a single qubit in the linear response regime, see Sec. 5.4.1and Fig. 5.6, where we have shown that an intermediate amount of pure-dephasingleads to a third decay rate associated to the coupling of the populations and coherences.This fact is remarkable since it has been shown that using an appropriate unitarytransformation one can convert an arbitrary two-qubit state into an X state of the sameentanglement [455, 441]. This means that the changes introduced by the dissipativeevolution of the BRME or the pre-averaged NTME modifies the nature of the entangledstate before leading it to the steady state.

6.4.5 Decay rates of two qubit systems: local and common reservoirs

For two non-interacting qubits connected to local reservoirs, we have seen in Sec. 6.4.1that we can get up to two bifurcations yielding four decoherence rates (two pairs ofrates, each pair typically containing a slow and a fast decay rate). Thus, for N qubitswe could obtain up to 2N decoherence rates in the strong damping regime instead ofonly N in the WCL due to the absence of non-secular terms 44. As for the single qubitcase, one could find an optimal initial two-qubit state favoring the slowest decay timesissued from the different bifurcations analytically, since we have access to the solution ofthe time-evolution (6.26) for X-states [372], or through a search/reinforcement learningalgorithms [456, 457].

When the two qubits are connected to a common reservoir, it becomes harder to computethe decoherence rates analytically. Thus, we checked numerically that multiple bifurca-tions are equally observed; we plotted in Fig. 6.9 the ratios of the decay rates (each rateis divided by the sum of all other rates). We used the unequal qubit Hamiltonian (6.32),non-symmetric coupling operators (6.31) as well as the usual bosonic spectral function(2.25) with J(ω)=γω (including an exponential cut-off) at low temperature (β=5) asso-ciated to the common reservoir. If we on top of that include pure dephasing, the patternof bifurcation becomes more intricate since the populations and coherences intertwinemaking it difficult to define relaxation and decoherence rates (as for a single qubit, see

44As discussed in Sec. 5.4.1, describing the bifurcation events microscopically requires to go to thenon-Markovian regime. One can nevertheless correctly describe the 2N decoherence rates obtained for theN qubit system beyond the thresholds with the LSME (2.44) or modular-free NTME (3.66) in the strongdamping regime for white noise type of reservoirs.

138


Figure 6.9 – Decay rate ratios in black for the LDME and in red for the pre-averagedNTME; we plotted them as a function of the absorption rate a(∆max) of the two qubitsystem’s largest energy gap ∆max. Contrary to the single qubit case, we have twobifurcation events associated to different thresholds (athr=0.0023 and 0.0031) leading toa more complicated decoherence pattern.

Fig. 5.6). Therefore, we consider N2−1 generalized ratios for a N level system 45

N2−1

∑i 6=k

Γi ≥ Γk. (6.38)

For a two qubit system, one simply computes the Liouville matrix, being of dimension16× 16, and extract the real part of its eigenvalues (in fact 15 ratios since the eigenvalueassociated to the steady-state is null). Again, bifurcations are observed (plot not shown).

6.4.6 Impact of the coherence on entanglement for a common reservoir

We mentioned at the beginning of this chapter that there exists a tight link between thecoherence and the entanglement according to the hierarchy (6.1). But does the coherencealways enhance the creation of entanglement or, on the contrary, could coherence spoilits creation? To find it out, let us take the non-degenerated Hamiltonian (6.32) as well asthe non-symmetric coupling operator (6.33) to be sure that the creation of entanglementarises due to the non-secular terms. The idea is to screen a set of separable initial states(6.29) characterized by a different amount of initial coherence. Therefore, we parametrize

45The “-1” stands for the zero eigenvalue associated to the steady-state.

139


-1

0

1corr(r0,Cmax)corr(r0,Cint)

(30;?0)index

10 20 30 40 50 60 70 800

0.5

130=30;max

?0=?0;max

Figure 6.10 – Upper plot: Comparison of the correlation between the radius r0 char-acterizing the initial state and the maximal concurrence corr(r0, Cmax) as well as theintegrated concurrence over time corr(r0, Cint) as a function of the two initial anglesθ0 and φ0. Lower plot: Values for the angles θ0 and φ0. The numbers on the x-axisrepresent the index of the different (θ0, φ0) pairs for a series of r0 comprised in [0,1) usedto compute the quantities corr(r0, Cmax) and corr(r0, Cint). We only represented a portionof the screened states since the phase φ0 has a small impact on the dynamics for theselected settings (we come back to the role of phases later for an entangled X-state).

the initial state of each qubit using the spherical coordinates

ρ(0)A = ρ(0)

B =

(12 (1 + r0 cos θ0)

12 r0 sin θ0 eiφ0

12 r0 sin θ e−iφ0 1

2 (1− r0 cos θ0)

), (6.39)

where r0∈ [0, 1), θ0∈ [0, π] and the phase φ0∈ [0, 2π]. The initial amount of coherence ofthe individual qubits depends on the radius r0 (fully mixed state for r0=0 and pure statefor r0=1, see Fig. 6.1). We perform a series of simulations for different combinations ofr0, θ0 and φ0 to screen the Bloch spheres of each qubit assuming that they are prepared inthe same initial state. To estimate the degree of coherence during the time evolution weuse the l1 norm (6.7) while the degree of entanglement is checked with the concurrence(6.13). We then computed the correlations between a series of radius r0∈ [0, 1) and themaximum as well as normalized integrated concurrence 46, denoted Cmax and Cint, for thedifferent angles θ0 and φ0. Looking at the Fig. 6.10, we first notice that the phase φ0 has nodetectable influence on corr(r0, Cmax) or corr(r0, Cint) for our settings. Besides, the angleθ0 plays a crucial role and indicates that the initial coherence either strongly enhances thecreation of entanglement corr(r0, Cmax)≈corr(r0, Cint)≈1 (i.e. fully correlated) or, on thecontrary, spoils the creation of entanglement corr(r0, Cmax)≈corr(r0, Cint)≈−1 (i.e. fully

46Time-integrated quantum correlation measures have been used in [458, 459, 460].

140


"t0 50 100 150

0

0.05

0.1

0.15

0.2

0.25

"t0 50 100 150

0

0.2

0.4

0.6C, r0 = 0

C, r0 = 0:4

C, r0 = 0:8

l1 norm, r0 = 0

l1 norm, r0 = 0:4

l1 norm, r0 = 0:8

Figure 6.11 – Comparison of the l1 norm and concurrence C for selected radius r0 withthe angles θ0=0 (left plot) and θ0=2.6 (right plot). In the left plot the coherence enhancesthe creation of entanglement while it has exactly the opposite effect in the right plot.

anti-correlated). The transition between these two situations takes place for a smallsubset of θ0’s; representative time evolutions of the concurrence and l1-norm are plottedin the Fig.6.11. One right away notices that the coherence is increased prior to creationof entanglement in the left plot (θ0=0). Moreover, a larger coherence corresponds to alarger entanglement. On the contrary, in the right plot (θ0 = 2.6) a larger coherence isassociated to a lower entanglement. This indicates that the coherence can participateeither constructively or destructively to the creation of entanglement. Moreover, anintermediate amount of coherence is more beneficial than a too large amount accordingto Fig. 6.11. In the next section, we examine more carefully the impact of the initialphases on the concurrence.

6.4.7 Concurrence’s decay in a common reservoir: initial phases’ role

Let us prepare the two qubit system into a mixed entangled X-state (6.35) whose diago-nal elements read ρ0

11 = 0.35, ρ022 = 0.15, ρ0

33 = 0.05, ρ044 = 0.45 while their anti-diagonal

counterpart are ρ014 = 0.3 eiφ0 and ρ0

23 = 0.05 eiψ0 with φ0, ψ0 ∈ [0, 2π]. We use unequalqubits modeled by the asymmetric system Hamiltonian (6.30) and the coupling operatorQ = (σx + qσz ⊗ 1) + (1⊗ σx + qσz) to be able to vary the amount of pure dephas-ing (i.e. q = 0 no pure dephasing, q > 0 pure dephasing). We study the system at lowfriction (γ = 0.01) and high temperature (β = 0.1) with the same spectral function as inSec. 6.4.2 so that the overall irreversible contribution is of the order 0.1 to be in the WCR.Note that in this regime, the different non-secular master equations (NTME’s, BRME orLSME) provide a similar decay pattern. We provide a contour plot of the normalized in-tegrated concurrence Cint for the initial phases φ0 and ψ0 in absence and presence of puredephasing in Fig. 6.12 (for each plot, we normalized with largest Cint(φ0, ψ0) value). Theinitial phases influence non-negligibly the decay of concurrence as we already pointed

141


Figure 6.12 – Comparison of the normalized integrated concurrence Cint starting from anentangled mixed X-state for two non-interacting qubit connected to a common reservoir.Left panel: in absence of pure dephasing. Right panel: with pure dephasing (q = 1.5)we observe a displacement/deformation of the “optimal islands” (the red regions). Weused φ0, ψ0 ∈ [0, 4π] (and not 2π) to better display the periodic pattern of Cint.

out for local reservoirs [372]. Note that the decay pattern is different when considering acommon reservoir because the generation of entanglement takes place. We immediatelynotice that there exists an initial state for (φ0, ψ0)≈ (1.4, 1.9) prolonging the concurrencedecay (in the middle of the red islands) and another one (φ0, ψ0)≈ (3.7, 5.0) precipitatingthe concurrence decay (in the middle of the blue islands). Pure dephasing deform thedecay pattern displacing the position of the optimal islands, namely (φ0, ψ0)≈ (0.3, 2.6)for the prolongation and (φ0, ψ0) ≈ (3.3, 5.9) for the precipitation. From a practicalpoint of view, one should not simulate the full set of (φ0, ψ0) phases but use steepestascent/descent algorithms to reach these states in a minimal number of simulations (inparticular when using non-Markovian master equations which are computationallymore demanding).

6.4.8 Concurrence for a common reservoir: strong damping limit

We discussed in the Sec. 5.4.4 that one could use the modular free NTME (3.66) insteadLSME (2.44) and NRSME (2.45) when the system is strongly coupled to a hot classicalreservoir. To illustrate this idea, let us look at the ESD using the non-degeneratedHamiltonian (6.32) and symmetric coupling operator (6.31). We selected an inversetemperature of β=1, i.e. of the same order as the energy gaps, so that it can be consideredas hot but still finite so that the final state can be distinguished from the fully mixedstate. The initial density matrix is taken as the Bell state (6.34). In the middle plot of

142


"t0 5 10 15 20

hHi

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0LSME3rd NTMENRSMEEquilibrium

"t0.2 0.4 0.6 0.8

C

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1LSME3rd NTMENRSMENRSME (; < 0)

"t0 5 10 15

;12

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018. = 2. = 4. = 6. = 8. = 10

Figure 6.13 – Left plot: comparison of the energy relaxation of the 3rd NTME (3.66), alsodenominated as the modular free NTME, with the LSME (2.44) going towards the fullymixed state and the NRSME (2.45) ending in an approximate Gibbs state (β=1, γ=2.0).Middle plot: ESD for the three equations (β=1, γ=2.0). The NRSME displays negativeeigenvalues. Right plot: Creation of coherence for the 3rd NTME for different γ (β=1).A higher friction leads to a more durable creation of coherence.

"t0 0.5 1 1.5 2 2.5 3 3.5

C

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4LSME3rd NTMENRSMENRSME (; < 0)

"t0 5 10 15

C

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4. = 0:5. = 1:0. = 2:0. = 3:0

Figure 6.14 – Left plot: comparison of the entanglement sudden birth of the 3rd NTME(3.66), the LSME (2.44) and the NRSME (2.45) for γ=0.5 and β=0.2 (the same systemHamiltonian and coupling operator as for the Fig. 6.6 has been used). In black isindicated when the density matrix produced by the NRSME carries negative eigenvalues.Right plot: Sudden birth of entanglement for the 3rd NTME for increasing frictions.

143


Fig. 6.13, obtained for a strong coupling (γ = 2.0), the concurrence decreases extremelyfast and no revival takes place. On the other hand, we observe the creation of thecoherence ρ12, which is initially equal to zero in the Bell state (6.34), breaking the X-state(6.35) in a similar way as observed for the pre-averaged NTME and BRME. Could wefind a set of parameter to create entanglement? If we use the same settings as for thecreation of entanglement for the LDME which led to Fig. 6.6, i.e. a collective couplingoperator, it is possible as can be seen in Fig. 6.14. The entanglement generated bythe “collective” dissipative process survives longer as γ is increased. The idea of amacroscopic robust entanglement in hot and noisy environments is not new [12, 13]. Theinteresting point here is that the collective action of a classical Markovian environmenton the bipartite system, whatever its exact origins, can create entanglement. This factis not only predicted by the phenomenological modular free NTME (3.66) but also bythe LSME (2.44) and NRSME (2.45) which have been formally derived from microscopicsetting. The role of the NTME is once again to provide an equivalent dynamics as thesetwo “micoscopic” master equations but without their respective drawbacks: the LSMEwhich does not lead to the correct steady-state and the NRSME which only partiallyrecovers the correct steady state and does not preserve the positivity of the densitymatrix at all times. Clearly, looking at left plot of Fig. 6.14 one immediately noticesthat the LSME underestimates the concurrence whereas the NRSME overestimatesit. The present approach could equally be applied to a bipartite system composed oftwo harmonic oscillators connected to a common reservoir usually modeled with theLindblad equation. In case the oscillators are identical, one could generate asymptoticentanglement similarly as in [461] but without the fictitious diffusion contribution forsmall as well as for strong couplings using the appropriate NTME. Moreover, one coulddiscuss the phenomenon of stationary entanglement in hot environments in a renewedperspective since it would not require the introduction of a time-dependent interactionbetween the oscillators as proposed in [14].

144

7 Light harvesting complexes

Life is based on three fundamental cornerstones: harvesting energy, carrying out chemi-cal reactions and processing information. Nowadays, this biological machinery is mainlyexplained in terms of molecular biology and biochemistry [1, 2, 3]. Nevertheless, theselast years evidences have accumulated that quantum mechanics may play a non-trivialrole [462] in macroscopic [12] and in biological systems [11, 463] due to their nonequi-librium nature [464, 14]. On one hand quantum uncertainty limits the fidelity of allmolecular mechanism and on the other quantum effects could enhance processes thatare slow or intractable when resorting exclusively to classical ball-and-sticks models[11, 465]. Hence, one should ultimately find out if quantum effects are just byproductsor if they have a true impact on biological functions which can be quantified both theo-retically and experimentally. Light-harvesting complexes (LHCs) [466], which are the keyto harvest energy directly from the sun in the plant kingdom as well as for certain mi-croorganism and animals, are the perfect “toy molecules” to study non-classical effectsat room temperature.

We investigate in this chapter the Fenna-Matthews-Olson (FMO) complex [467], which isone of the simplest LHC in nature. The FMO complex is composed of three subunits, eachone containing an electronic core composed of seven chromophores usually modeled astwo-level systems connected to each other through dipole-dipole interactions. To get atractable description, the seven chromophores are selected as the open system connectedto a heat reservoir representing all irrelevant degrees of freedom (DOF) such as thesurrounding protein scaffold and other organic molecules. After explaining how toreach this simplified but widely used description, we study the energy transfer throughthe FMO subunit. By this means, we disclose the ambivalent role of the coherence forthe energy transfer efficiency. We also examine the energy transport close to equilibriumby applying the GKF demonstrating that it is possible to enhance the transport at roomtemperature for an appropriate coupling with the environment. Finally, we show thatthe bifurcation phenomenon studied for a single qubit equally affects LHCs, using astandard single two-level FMO model, supporting recent experimental findings.

145

Chapter 7. Light harvesting complexes

7.1 FMO subunit: the perfect “toy model”

The biomolecular aggregate theory deals with the quantum mechanical descriptionof molecules embedded in various biological environments composed of water, smallorganic molecules, polymers as well as proteins [468, 469]. Its object is to describebiological processes taking place over a wide range of time-scales [300]. A photosensiblebiomolecular aggregate is generically designed as a pigment-protein complex composedof more than thousand atoms forming so-called chromophores or pigments, which arenon-covalently bounded, encapsulated in a protein shell. LHC, which are at the root ofall photosynthetic processes, enters in this class of biomolecules. Since the electronicdensities of the chromophores embedded in an aggregate do not overlap, they cannotexchange electrons and can therefore be considered as electronically neutral. They areconnected to each other by virtue of mutual Coulomb interactions which are responsiblefor the delocalization of the wave functions over the entire aggregate 1. The LHCcan be optically excited on the femtosecond timescale: an antenna absorbs a photonwhich is transferred to an aggregate where it is projected into a conformal quantumsuperposition leading to the creation of one or more excitons 2 responsible for thecoherent energy transport through the LHC [471]. This process is commonly designed asthe electronic energy transfer (EET). The excitons, subjected to a decoherence process dueto the surrounding environment, are coherently hopping between the chromophoresover timescales ranging from the picoseconds to the nanoseconds until they reach thereaction center (RC). The energy of the excitons is then transferred into chemical cyclestypically producing the adenosine triphosphate (ATP), a key resource for any metabolicprocess, on timescales of the microseconds.

The FMO complex of green sulfur bacteria [472, 467] depicted in Fig. 7.1 is one ofthe best known LHC due to its relatively “small” size compared to LHCs of higherplants. It is a trimeric aggregate which is coupled, on one hand, to a baseplate itselfattached to an ensemble of chlorosomes, playing the role of an antenna harvesting thesunlight. On the other hand it is linked to a RC finally collecting the energy whichhas traveled through the FMO complex 3. Each FMO subunit is composed of sevenbacteriochlorophyll-a (BChla) molecules chromophore molecules 4. The nearest center-

1The wavelength of the light exciting the aggregate has to be much larger than the size of the aggregateitself. For the green sulfur bacterium Prosthecochloris aestuarii [470] a light beam with a wavelength of800 nm, being two times larger than the LHC, is necessary to produce a delocalized excitation [471].

2The Frenkel or molecular exciton is mediated through the Coulomb interaction between the chro-mophores, i.e. no electron is exchanged. We come back more in detail to this point in Sec. 7.2.2.

3See e.g. Fig. 1 of [467], [473] or [474]; the different dimensions the LHC are provided in Fig. 1 of [475].4The BChla are the chromophores of the FMO subunits. In reality, the crystallographic structure

of a FMO subunit displays an eighth BChla located close to the chlorosome baseplate [473, 470]. Thisadditional chromophore is a privileged entrance site compared to previous studies having two independententrance sites, namely the chromophores one and six, and has non-negligible impact on the EET [476, 477,478]. However, the majority of the scientific community focuses on the “seven-site model” using it as a“benchmark model” [479]. For a first exploration with the NTMEs we will stick to this description in orderto put them into perspective with previous works.

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7.2. FMO subunit described as an open quantum system

Figure 7.1 – (a) Structure of a single FMO subunit: in yellow the protein scaffold andin green the chromophores, (b) Sketch of the LHC from the chlorosome antenna tothe RC. (c) Arrangement of the seven chromophores in the FMO subunit highlightedin black. Picture and description reproduced from Cheng, Y.-C. and Fleming, G. R.,Annu. Rev. Phys. Chem., vol. 60, no. 1, pp. 241–262 (2009).

to-center distance inside of a FMO subunit between two chromophores is ∼ 1.1nmwhereas the smallest distance between two chromophores belonging to different FMOsubunits is ∼2.4nm [467]. To study the EET taking place between the antenna and theRC, it is generally enough to focus on a single FMO subunit to capture the main physicalmechanisms producing the EET.

7.2 FMO subunit described as an open quantum system

7.2.1 Relevant and irrelevant DOF

Our aim is to model the excitonic dynamics of a FMO subunit, immersed in a biomolecu-lar solvent, as an open quantum system. To reach this highly coarse-grained descriptiona number of approximations are required. Looking at the Hamiltonian governing thefull dynamics, one distinguishes three parts [480, 467]

Htot = He + Hv + He−v. (7.1)

The first part describes the system of interest as an excitonic Hamiltonian He containingall electronic DOF of the chromophores. The nuclear DOF of the chromophores arealso important, but only indirectly in the present coarse-grained description. Indeed,they are included for one part in the vibrational Hamiltonian Hv, modeling the en-vironment equally containing the DOF of the surrounding proteins and solvent, andfor another part in the excitonic–vibrational interaction Hamiltonian He−v mediatingthe energy and particle exchanges between the environment and the LHC. To keep itsimple, the environment associated to the Hv is modeled as phononic reservoir 5. As

5A phonon is a collective excitation in an elastic periodic lattice of molecules; it is sometimes designedas a quanta of vibrations. Non-secular contributions are crucial to describe phonon reservoirs [20, 199].

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pointed-out by Rebentrost and co-workers [481], the transition energies of the individualchromophores and the Coulomb interaction between the chromophores composing theexcitonic Hamiltonian He depend, in general, on the nuclear coordinates of the chro-mophores and surrounding proteins 6. As far as we are concerned, we choose to neglectthis dependence and focus on the Markovian dynamics of the “excitonic core”. Hereafter,we explain under which approximations we can reach such coarse-grained descriptionleaving out all tedious calculations. Therefore, we extensively use the Chapter 2 and 8of May and Kühn’s textbook [468] as well as the paper [480], the review [404] and theChapter 1 of the book [469].

7.2.2 The excitonic Hamiltonian

A full quantum description of the aggregate relies on the Schrödinger equation dictat-ing the time evolution of the nuclear and electronic DOF of its molecules. It is wellknown that this equation of motion cannot be solved in practice due to the interactionscorrelating the motion of the electrons and nuclei. To reduce the number of DOF onetakes advantage of the fact that the nuclei are much heavier than the electrons so thatthe motion of the nuclei does not produce notable transitions between the differentelectronic states of the chromophores. Thus, the total wave function can be rewrittenas the product of the nuclear and electronic wave-functions. Moreover, it is customaryto “freeze” the motion of the nuclei in an equilibrium geometrical configuration so thatthe coordinates of the nuclei enter only as a parameter in the motion of the electronsallowing to solve the stationary electronic Schrödinger equation. This is commonlydescribed as the Born-Oppenheimer adiabatic approximation [482]. This description is stilltoo complex and one has to apply the Hartree product ansatz to rewrite the total electronicwave-function as the product of the wave-functions of the Nc individual chromophores.To ensure that the electronic transitions of each individual chromophore neither affectthe nuclear positions of the chromophore nor the positions of the rest of the phononicreservoir, it is common to apply the Frank-Condon principle [483, 484] postulating thatthe electronic transitions are instantaneous from the point of view of the nuclear DOF.

Although the electronic and nuclear DOF are no longer correlated, this representation isstill too complicated since multiple chromophores compose a single aggregate. The nextstep consists in describing each chromophore as a two-level system which can either be inthe ground or excited state. During an EET the resonant Coulomb interaction connectingtwo chromophores leads to the de-excitation of a chromophore, called the donor, andthe excitation of a chromophore, designed as the acceptor. Forbidding simultaneousoff-resonant excitation or de-excitation of a donor-acceptor pair, which does not conserveenergy, is obtained under the Heitler-London approximation [485]. The EET between thechromophore m and n can be summarized as follows:

6See the Hamiltonian (5)–(6) in [481] as well as the Hamiltonian (1)–(2) in [479] for the detailed depen-dence on the nuclear coordinates (both papers rely on atomistic simulations).

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(i) an electron in the ground state of the donor m is transferred to its excited statethrough the absorption of a photon.

(ii) the Coulomb interaction transfers the excitation energy from the donor m to theacceptor n promoting an electron from its ground to excited state.

Note that no electron is transferred between the chromophores m and n in this process,only an excitation energy which is called the Frenkel exciton is exchanged. This kindof interaction takes place for pairs of molecules which are non-covalently bounded,i.e. whose electronic densities are not overlapping but are nevertheless interactingthrough a Coulomb or electrostatic force. They are sometimes also denominated asvibrational or molecular exciton [343, 486]. The Frenkel exciton, displaying a static electricdipole moment, has to be distinguished from the charge transfer exciton taking place formolecules having sufficiently overlapping electronic densities permitting an electron-hole type of exchange 7. One has also to distinguish it from a Wannier-Mott excitonrestricted to covalently bounded molecules.

Now, because the intermolecular distance is larger than the dimension of the chro-mophores composing the LHC aggregates, we can simplify the Coulomb interactionby a point-dipole approximation which ignores the shape of the chromophores and onlyretains their relative orientation and distance 8 leading to a dipole-dipole interaction.Clearly, an aggregate composed of Nc two-level systems connected by dipole-dipoleinteraction can absorb up to Nc photons. To further simplify the description, one candecompose the Hamiltonian in the so-called exciton manifolds

He = H(0)e + H(1)

e + H(2)e + . . . (7.2)

so that the absorption of a single photon can be described by the sole HamiltonianH(0)

e + H(1)e . For the subsequent absorption of a second photon, one has to also consider

the H(2)e contribution. It is common to limit the description of the FMO subunit to

the first exciton manifold to study the role of coherence and entanglement amongchromophores and how it impacts energy transport [489, 490, 491, 409, 492]. The one-exciton Hamiltonian to model the electronic core of FMO subunit reads 9

He ∼ H(1)el =

Nc

∑m(E0 + Em) |m〉〈m|+

Nc

∑m 6=n

Jd−dmn , |m〉〈n| , (7.3)

where E0 stands for the zero of the energy, Em is the site energy while Jd−dmn is the coupling

7In this case, an electron in the excited state of the donnor m is transferred to the excited state of theacceptor n, leaving a hole in ground state of the donor m which is filled by an electron of the ground stateof the acceptor n. For a short review on excitons at the nanoscale see, for example, the paper [487].

8This method has already been applied by Förster in 1948 [488]. To more accurately model the coupling,techniques such as the extended dipole approximation or the transition density cube method can be applied,see Chap. 3.5 of [469] for further explanations.

9The detailed calculation leading to the one-exciton Hamiltonian is presented in Chap. 8.2.2 of [468].

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energy given by the dipole-dipole interaction originating from the Coulomb interaction.

The parameters of the one-exciton Hamiltonian (7.3) were obtained experimentally inconjunction with quantum chemistry calculations for the different FMO subunits, seee.g. [493, 494, 495, 476, 496]; for our purpose we will only consider a single subunit anduse the Hamiltonian provided in the equation (13) of [497] given in cm−1 (alternatively,look at (1) in [498] or (31)–(32) of [499] as well as the Fig. 3 of [474]) . It is convenient totransform it into meV using the relation 1meV'8.065cm−1 as well as to subtract the zeroof the energy E0'1516.42meV to obtain a numerically tractable form 10, namely [497]

H(1)e, FMO =

215.0 −104.1 5.1 −4.3 −4.7 −15.1 −7.8−104.1 220.0 32.6 7.1 5.4 8.3 0.8

5.1 32.6 0.0 −46.8 1.0 −8.1 5.1−4.3 7.1 −46.8 125.0 −70.7 −14.7 −61.54.7 5.4 1.0 −70.7 450.0 89.7 −2.5−15.1 8.3 −8.1 −14.7 89.7 330.0 32.7−7.8 0.8 5.1 −61.5 −2.5 32.7 280.0

(7.4)

describing the seven sites of the FMO subunit.

7.2.3 The reservoir and interaction Hamiltonians

We are now ready to come back to the total Hamiltonian (7.1) of the LHC. We devotedthe previous Sec. 7.2.2 to explain under which approximation the system Hamiltoniancan be modeled with the help of the single-exciton manifold Hamiltonian He ∼ H(1)

el

given in (7.3). This representation is particularly appealing since the chromophorecomplex can be described as a network in contact with an environment: an excitonenters essentially from one side of the network and is funneled down to the site incontact with the RC [497]. During its travel, the delocalized excitonic wavepacket isaffected by a certain amount of dissipative and dephasing noise precipitating its collapseas well as avoiding that it undergoes Anderson localization [500, 501]. The nuclear DOFof the chromophores and the surrounding proteins DOF are summarized by an infinitelyextended reservoir [467, 502]

Hv = ∑q

ωq

(b†

q bq +12

)(7.5)

depending on the frequency ωq of the phonon reservoir’s qth mode and the standardboson creation b†

q and annihilation bq operators satisfying the commutation relation[bq, b†

q′ ]=δqq′ . Note that the thermodynamic state of this reservoir cannot be perturbed bythe aggregate since it is only weakly interacting with the electronic part of the aggregate

10Else the differences between the diagonal and off-diagonal coefficients are too large

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through a linear coupling of the form [503, 502, 504]

He−v =Nc

∑m(Qm ⊗Φm) (7.6)

with

Qm = σ+m σ−m = |m〉〈m| (7.7)

Φm = ∑q

(gq,mb†

q + g∗q,mbq

), (7.8)

where gk,m stands for the site-dependent coupling of the site m with the mode q formingthe exciton-vibrational coupling matrix. However, the choice of the system couplingoperator Qm is rather specific. Performing a microscopic derivation of the BRME as doneby Jeske et al. [59], similar to the one we performed earlier in Sec. 2.1.3, indicates thatone has much more freedom in the choice of the system coupling operators 11. Followingthis idea, let us couple all chromophores to a single reservoir using gq,m = ggm fq so that

He−v = g

(∑m

gm |m〉〈m|)⊗∑

q

(fqb†

q + f ∗q bq

)= g (Q⊗Φ), (7.9)

where g stands for the overall system-reservoir coupling strength and fq is the formfactor. There exist no experimental data to characterize the system coupling operator Qin the literature so that it is generally taken as a free parameter. Regardless of this choice,it should always respect the EP (2.48) or (2.49) ensuring that the system converges to aunique steady-state 12.

For the interaction Hamiltonian (7.9), the friction matrix is obviously diagonal and onecan construct the Lindblad eigenoperators following (6.15) where unitary matrix W isjust the identity whereas V is non-trivial and has to be obtained by diagonalizing theexcitonic Hamiltonian (7.3). To perform all analysis, we stay in the Hamiltonian basis.We can then directly obtain the pre-averaged NTME (3.58) as well as the LDME (3.38) andthe BRME (2.35) for a single reservoir and system coupling operator. As previously, westay with the bosonic spectral function (2.25) with J(ω)=γω (including an exponentialcut-off) corresponding to the reservoir Hamiltonian (7.5) and the reservoir couplingoperator Φ in (7.9).

11See Sec. III.A therein where these operators are extended to include off-diagonal terms in the systemcoupling operators for a dimer model including a ground state.

12This is typically the case for the FMO subunit whose Hamiltonian (7.4) is clearly non-degenerated.

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7.2.4 Time and energy scales for the system

For characteristic time and energy scales of biomolecules see Table 1 of [299] or Sec. VI.Cof Chap. 22 in the book [505]. Let us cite here that the Förster or fluorescence resonanceenergy transfer (FRET) from green to red chromophores, mediated by a dipole-dipoleinteraction, has an energy scale of ∼0.2−2meV. On the other hand, the dipole couplingbetween chromophores inside of a LHC of green sulfur bacteria is of the order ∼46− 100meV whereas the coupling between LHC is generally of the order∼0.3−0.6meV.Note that the FMO Hamiltonian reproduced in (7.4) is characterized by dipole couplingsranging from ∼0.1−10meV and site energies which range from ∼3−50meV. This is notso surprising since for this Hamiltonian one only takes into account the subspace of asingle excitation associated to a weak illumination [302]. Note that the room temperature300K corresponds to an energy of ∼25meV (corresponding to 210cm−1 or 6.25THz) 13.The first measurements on the FMO complex were performed at 77K [16] as well as at180K [17] and only later at 277K [506] (as well as for conjugated polymers [507] at 293K).All these measurements displayed a coherent EET. Recently, more precise measurementswere preformed for colloïdal semiconductor nanoplatelets [304] equally displaying“long-lived coherences” indicating that such a phenomena could be ubiquitous in nature.To perform numerical simulations, if we rescale the FMO Hamiltonian (7.4) such that themean site energy (which is of the order of ∼25meV) is around one then the temperatureof the reservoir is of the order T'1 for 300K and T'0.25 for 77K (typical “lab” valueassociated to the condensation of nitrogen from a gas to a liquid phase). In full generality,all model parameters are of the same order [477]. However, since we are working in theWCR we will use phenomenological frictions between 0.1−0.5 according to this scaling.The WCL and the WCR have been commonly applied to model the FMO complex usingthe LDME [489], the LKME [497] as well as the BRME [59]. Presently, our aim is not tofit experimental data, since this would require to go towards the strong coupling regimeand include non-Markovian effects [508, 115, 509], but to perform a qualitative study ofthe quantum correlations as we did in the context of the two-qubit system in Sec. 6.4.

7.2.5 The initial state and the creation of coherence

Exciting a sample containing a FMO complex with a coherent light beam reveals long-lived coherence over two-distinct time-scales: an initial large electronic coherence un-dergoing a fast decay followed by the emergence of a small vibrionic coherence createdthrough the interaction with the environment 14. However, since natural sunlight doesnot provide coherent light-beams, the initial state is most probably incoherent. The pop-ulation is, in general, assumed to be fully localized at a single chromophore in contact

13More precisely, these units are related as follows: 1meV'8.065cm−1'4.135ps'0.242THz. Moreover,the cm−1 units are often privileged over the meV and the THz in the context of LHC.

14The vibrionic coherence arises from the electron–vibrational interactions between the electronic corecomposing the relevant system and the vibrational DOF composing the environment (i.e. nuclear DOF ofthe FMO complex as well as surrounding proteins and solvent DOF) [510].

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7.3. Energy transfer through the FMO subunit

with the baseplate (in the seven chromophores model, the population is either at the siteone or six [495, 473, 479]). Thereby, the coherent EET is most probably a consequence ofthe interaction of the excitonic reduced system with its environment [437, 509], i.e. due tothe vibrionic coherence. However, describing the excitonic dynamic using the LDME perse, obtained in the secular limit, is unable to produce a coherent state from an incoherentone (i.e. for a diagonal friction matrix and a non-degenerated Hamiltonian as it is the casefor the “standard” FMO model [489]). This point follows directly from what we learnedwhen we examined the two qubit system in the Chapter 6. On the contrary, the BRMEcan transform such an incoherent initial state into a coherent one thanks to its non-secularterms [437, 511, 204, 512]. However, we already pointed out that this equation does notpreserve positivity. For this reason, the LDME obtained from the BRME by applyingthe SA is often supplemented by additional contributions such as an anti-Hermitianrecombination and trapping terms [489, 513, 503] or replaced by an ad-hoc LKME con-structed “by hand” to produce the desired effects [497, 514, 70, 515, 466, 516, 517, 518].Clearly, since the pre-averaged NTME (3.58) displays the same dynamics but withoutits drawback, it is the perfect tool to study the creation of coherence which have beenmeasured under natural conditions.

7.2.6 Coherence and entanglement in a FMO subunit

We presented different coherence and entanglement measures in the Sec. 6.1.2 and 6.2.3,respectively. It is noteworthy that for the single-excitation manifold the presence ofcoherence is a necessary and sufficient condition for entanglements [490, 491, 492, 409,519]. The analogue of the relative entropy of entanglement (6.12) proposed in [490]is not a “good measure” since it is not always monotone as pointed out in Sec. II of[520]. On the other hand, the concurrence (6.13) between the site n and m takes fora single-exciton manifold the simple form EC,nm(ρ) = 2|ρnm|, where ρnm is the (m, n)off-diagonal element of the density matrix 15. The concurrence leads to a two-way tanglebetween the site n and m, which reads τnm =4|ρnm|2, used to construct the total tangle

τT =N

∑m,n>m

τnm (7.10)

which is a good measure being at the same time very easy to use [491, 409].

7.3 Energy transfer through the FMO subunit

We adopt in this section three complementary points of view to study the energy flowthrough the FMO subunit. We first study the coherent EET efficiency in LHC focusing

15This link originates from the fact that in the single exciton manifold representation each site representsa pair of two-level systems [491, 409].

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on interrelation between quantum correlations and transfer efficiency far away fromequilibrium generally privileged in the LHC literature. On the other hand, we resortto the more traditional GKF valid close to equilibrium to compute the conductancethrough the FMO subunit. Finally, we examine the interrelation between coherence (orentanglement) and the heat fluxes in the NESS.

7.3.1 The impact of coherence on the energy transfer efficiency

Does the coherence influence the efficiency of the energy transfer, positively as well asnegatively, in the FMO complex? We have now all tools in the hands to answer thisquestion, at least qualitatively since we are working in the WCR. The idea is to comparean incoherent to a coherent dynamics starting from an incoherent initial state with allpopulations localized at the site one (i.e. all ρij =0 apart from ρ11=1)16. The incoherentdynamics is generated by the LDME (3.38) which preserves the initial diagonal state,so that it leads to a classical hopping process between the chromophores sites, whereasthe coherent dynamics is obtained from the pre-averaged NTME (3.58) which is able tocreate the vibrionic coherence starting from an incoherent state which should lead to acoherent hopping process. To measure the efficiency of the energy transfer during therelaxation process, it is customary to compute the trapping time of the site n from 17

〈ttrap,n〉 =∫ tss

0dt (ρnn − ρss

nn) (7.11)

with tss the time at which the steady-state ρssnn is reached, as well as the total trapping

time being simply 〈ttrap〉=∑Ncn 〈ttrap,n〉. As we have seen in the Sec. 7.2.3, the coupling

operator Q is a free parameter of the model. The idea is to screen a wide range of couplingoperators Q, as given in (7.9), by randomly selecting Qmm∈ [0, 3] and performing a timeevolution until the system fully relaxed18. To get reasonable statistics, we have sampled10′000 coupling operators. We measure the degree of entanglement by integrating thetotal tangle measure (7.10) over the full relaxation process

〈τT〉 =∫ ∞

0dt (τT − τss

T ), (7.12)

where τssT is stead-state value of τT, which we normalize according to the maximal

obtained value leading to 〈τT〉. All simulations are performed for β = 1 and γ = 0.5as explained in the previous Sec. 7.2.4. In Fig. 7.2, we see that most of the couplingoperators produce an intermediate amount of entanglement which in itself does notreveal a lot of information. So let us now look at the standard path followed by the

16Again, for the NTME we have to use for the initial state a convex combination with the Gibbs state.17Formula taken from [477]; we subtracted here the steady-state ρss

nn to ensure that the integral notdiverges as tss→∞ contrary to the original expression (i.e. ρnn does generally not vanish at tss).

18Since we go into the Hamiltonian basis, the diagonal coupling operator (7.9) acquires off-diagonalcontributions through the change of basis.

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h=Ti0 0.2 0.4 0.6 0.8 1

0

50

100

150

200

Figure 7.2 – Histogram for the normalized integrated total tangle 〈τT〉 obtained from(7.12) extracted from 10′000 simulations of the pre-averaged NTME for randomly gen-erated coupling operators Q at β = 1 and γ = 0.5. The y-axis provides the number ofQ operator which are associated to a given 〈τT〉 value. Note that the LDME does notgenerate any coherence (or entanglement).

exciton starting at site 1 and jumping towards the site 2 to end at the site 3 where the RCis located [477] and check the efficiency of the energy transfer. Therefore, we extract fromthe simulation the trapping time (7.11) for these sites from the simulations performedwith the LDME and the pre-averaged NTME (as usually, we crosscheck our results withthe BRME leading to similar results) . Note that a larger trapping time at the site 3indicates a better energy transfer efficiency. To estimate the impact of the generatedcoherence on the energy transfer efficiency, we compute the difference of trapping time

δ〈ttrap,n〉= 〈ttrap,n〉2nd NTME− 〈ttrap,n〉LDME

(7.13)

for n = 1, 2, 3. We then order the 10′000 trapping time differences δ〈ttrap,n〉 from thesmallest to the largest value: a negative value indicates that the LDME leads to a moreefficient energy transfer (classical hopping dominates) while if it is positive the pre-averaged NTME produce a higher efficiency (quantum delocalization dominates). Wenow order exactly in the same way the 10′000 normalized averaged total tangles 〈τT〉. Itonly remains to plot together the ordered δ〈ttrap,n〉 and 〈τT〉 which is done in Fig. 7.3. Weimmediately notice that the energy transfer is perfectly correlated with the coherenceat site 1 which is not surprising since all the population starts at site 1. For the site2 and 3 we observe a double effect, namely that a high coherence can either reducethe efficiency of the energy transfer (anti-correlation between 〈τT〉 and δ〈ttrap,n〉) or, onthe contrary, enhance it (correlation between 〈τT〉 and δ〈ttrap,n〉) as it can be seen at thetwo extremities of the plot for the site 2 and 3. One can make the parallel with aninterference phenomenon which either has a constructive or a destructive action. On thecontrary, an intermediate amount of coherence is always beneficial as can be seen in themiddle of the same plots. When examining a single qubit using the FDT in Sec. 5.4.1, we

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Figure 7.3 – Comparison for the pathway 1→ 2→ 3 of the excitonic energy transferefficiency at a given site (red curve), given by the trapping time differences δ〈ttrap,n〉defined from (7.11) and (7.13), and the degree of coherence (blue curve) defined by thenormalized averaged total tangles 〈τT〉 obtained from (7.10) and (7.12). On the x-axis isindicated the index of the 10′000 different coupling operators Q.

already stated that an intermediate amount of pure dephasing couples the populationsto the coherences through the non-secular terms. This explains why the trapping timecan be increased in presence of coherence: namely, because before the energy is fullytrapped it is “transformed” into coherence and vice-versa delaying its trapping. We haveshown for a two qubit system that the non-secular terms break the X-state structure(6.35)–(6.37) so that the populations are in contact with all coherences. The same is truefor the seven site system, namely that none of the coherences remain null during thetime evolution and thus participate to this “population beating”. To be complete, wecarried out the same analysis by connecting at Q11 a hot reservoir (βh =0.9) and at Q33 acold reservoir (βh =1.1) or sink to create a small heat flux between these two sites. Theobserved pattern did not drastically change.

Does the result obtained in Fig. 7.3 mean that the best coupling operator Q is the one“used” by green sulfur bacteria? Not necessarily because biological organisms are neverperfectly optimized, they are just sufficiently enhanced to ensure their survival in aconstantly changing environment. Photosynthetic organisms have to deal with variabledegree of illumination depending on the time of day and on the season. As pointedout in [18], even plants in their maximum growth phase exploit less than half of theexcitation absorbed by their LHC. In fact, most of the time the excess excitation has to beeliminated to avoid irreversible damages. The actual best known protection mechanismsrely on a conformational change favoring the interaction of the LHC with surroundingcarotenoid molecules funneling the excess energy and dissipating it as heat [521, 522].Clearly, such a mechanism is much slower than the typical time scale of the excitonicdynamics of the FMO subunit. Thus, one can reasonably postulate that it modifies thecoupling operator Q switching between the best/worst Q (or even an intermediateone) according to the illumination intensity increasing/reducing the amount of energy

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Figure 7.4 – Top left picture: Sketch of the one-exciton Hamiltonian (7.4) where thedipole-dipole interaction involved in the dominant coherent energy transfer pathway1→2→3 is highlighted in red. Top right picture: Illustration of the FMO subunit withthe seven BChl (in green) where the pathway is also in red. Reproduced from G. D.Scholes et al., Nat. Chem., vol. 3, pp. 763–774 (2011). Bottom picture: Optimization ofthe energy transfer efficiency at site 3, which is given by the trapping time differencesδ〈ttrap,3〉, starting from the an average Q obtained in Fig. 7.3. The yellow dot and linesrepresent the pair of dipole-dipole interactions obtained from experimental data forH(1)

e, FMO := HS so that HS,12 =−0.445 and HS,23 = 0.143. On the other hand, the red dotand lines represent the best modified pair of dipole-dipole interactions is given byHS,12=−0.462 and HS,23=0.025.

157


transferred to the RC 19. Such a versatile coherent energy transport seems much moreplausible than a strict maximization of the energy channeled to the RC.

Would it be possible to improve the results obtained in the previous section by buildingan artificial light harvesting complex, i.e. by changing the Hamiltonian (7.4)? The advantageof manipulating the Hamiltonian contrary to the coupling operators is that it is accessiblethrough experimental measurements. To check this idea, let us again focus on thedominant chromophoric pathway 3→ 2→ 1 (see top pictures of Fig. 7.4) and let usmodify the interaction between the site 3 and 2 on one hand and 2 and 1 on the otherby changing the dipole-dipole interactions (H(1)

el, FMO)23∈ [−1, 1] and (H(1)el, FMO)12∈ [−1, 1]

such that the Hamiltonian remains Hermitian. Moreover, since the Q operator is a freeparameter, let us take an average Q from the simulations performed in the previoussection (i.e. the Q giving the largest and smallest trapping time difference δ〈ttrap,3〉). Wethen recompute the new trapping time difference δ〈ttrap,3〉 for each Hamiltonian.

As depicted in Fig. 7.4, we could improve the energy transfer efficiency. From an en-gineering point of view the “synthetic” Hamiltonian represented by the red dot inFig. 7.4 could be produced by modifying the FMO complex using targeted mutage-nesis [523] and see how it affects the organism as a whole (LHC have been “grafted”onto viruses [524, 525] being much simpler than a bacteria). Note that such a “blind”modification could lead to unexpected deficiencies. Engineering the Hamiltonian aswe did is more compatible with a network of superconducting qubits mimicking thestructure of the FMO complex such as the one proposed in [303]. In such kind of setups,most of the external parameters are under control contrary to a biological organism suchas a green sulfur bacteria.

7.3.2 Optimization of the FMO subunit’s thermal conductance

Restricting the analysis of the FMO subunit to the single-exciton manifold is equivalentto consider a weak illumination corresponding to a slow injection of photons in thesystem [302, 327]. It is therefore reasonable to study the transport of energy throughthe photosynthetic complex using the linear response theory with the GKF (5.52) andthe thermal conductance introduced in Sec. 5.4.3. We proceed exactly as for the singlequbit case and connect two reservoirs to the FMO subunit. We have seen in the previousSec. 7.3.1 that the choice of the coupling operator Q has a strong impact on the coherentenergy transfer efficiency. To check whether the same is true close to equilibrium, in ab-sence of coherence and entanglement, we perform a similar screening by randomly (butin a smaller range) producing 2000 different system coupling operators of the form(7.9) for each reservoir 20 with Q1,m and Q2,m ∈ [0, 2]. Our aim is to find one (or more)

19One can find the best/worst Q for each of the seven sites, i.e. drastically enhancing (or restricting) theenergy flowing towards a given site being, for example, close to carotenoids at a given time.

20To really obtain the best Q one would need much more data; the idea here is to show that it is possibleto have a peak of conductance at physiological temperature, not more.

158


Figure 7.5 – Top left plot: Histogram counting the repartition of the randomly generatedpair of coupling operators (Q1, Q2), abbreviated Qi, according to the reference tempera-ture Tref using the same parameters as in Fig. 7.2. Top right plot: Tref associated to eachpair of coupling operators Qi. Bottom left plot: peak of conductance Gκ for each pairof coupling operators Qi, the largest value being highlighted with a red circle. Bottomright plot: selection of conductances Gκ whose peak is close to Tref = 1; the largestconductance Gκ, highlighted again in red, displays a peak at Tref = 0.55.

coupling operators displaying a peak of thermal conductance, such as the one observedfor a single qubit in e.g. Fig. 5.7, around the physiological temperature (i.e. 300K corre-sponding to T∼1 in our reduced units as discussed in Sec. 7.2.4). Interestingly, most ofthe coupling operators display a peak of conductance at lower temperatures (i.e. aroundT∼0.7 which corresponds to 210K) than the room temperature as can be seen in Fig. 7.5.More precisely, only 0.1% of the total pair of coupling operators (Q1, Q2), associated toeach reservoir and which we abbreviated by Qi, are associated to a temperature largeror equal to 0.95 which correspond to ∼285K. We can also remark that for the highestTref we have the lowest conductance Gκ. Interestingly, the largest Gκ displays a peak atTref = 0.55 (red curve in the bottom right plot in Fig. 7.5).

Which kind of mechanism could use the photosynthetic organism to fine-tune theinteraction of the FMO complex with its environment which we have seen so far that is :(i) selecting an optimal Q to increase/reduce the energy transfer efficiency according toFig. 7.3, (ii) using a pair of coupling operators Qi to maximize/minimize the conductanceaccording to Fig. 7.5? The circadian clock of the bacteria 21 could to some extend play therole of conductor to anticipate which set of coupling operators to use, for example, whenthe sun reaches its highest position at noon to efficiently dissipate the excess of energy.

21The circadian rhythms have been studied, among others, in cyanobacteria [526] closely related to greensulfur bacteria and equally performing photosynthesis.

159


"T0 0.2 0.4 0.6 0.8

hea

t.ux

di,

./J

Q;c

10-8

10-6

10-4

10-2 min (/JQ;c)max (/JQ;c)/JQ;c max l1-norm

"T0.2 0.4 0.6 0.8

coher

ence

l 1-n

orm

0

0.02

0.04

0.06

0.08

0.10

0.12l1-norm for min (/JQ;c)l1-norm for max (/JQ;c)max l1-norm

Figure 7.6 – Role of the steady-state coherence at the NESS. Left plot: δJQ,c = JNTQ,c − JLD

Q,cis the difference of heat flux JQ,c towards the cold reservoir between the pre-averagedNTME and the LDME, Right plot: the corresponding amount of coherence in the NESSquantified with the coherence l1 norm. Clearly, an intermediate amount of coherenceleads to the largest heat flux (i.e. the red curves). The BRME leads to similar results.

7.3.3 Role of the steady-state coherence

Let us now evaluate the impact of the steady-state coherence, generated through theaction of the non-secular terms [204, 60], on the energy transfer through the FMO subunit.Therefore, we used the same settings as in the previous section, i.e. connecting the hotreservoir to the input site to model the baseplate and the cold reservoir to the outputsite to model the RC, to enforce a steady excitonic flux once the system converged to thenon-equilibrium steady-state (NESS). We performed 1′000 simulations, each one witha different pair of randomly generated coupling operators associated to the differentreservoirs, until we reached the NESS 22. We compute then the heat flux differencetowards the reservoir δJQ,c = JNT

Q,c − JLDQ,c between the pre-averaged NTME flux JNT

Q,c andLDME flux JLD

Q,c. We compare the flux difference δJQ,c with coherence l1-norm at theNESS for an increasing temperature difference ∆T = Th − Tc. We show in Fig. 7.6the largest and smallest δJQ,c as well as the one associated to the largest steady-statecoherence. We immediately notice that the biggest increase of heat flux is caused by anintermediate amount of coherence (up to four order of magnitudes for the “best” pair ofcoupling operators). To further improve the energy transfer efficiency far-away fromequilibrium, one could apply machine learning strategies [456] to determine the optimalsystem parameters (such as the pair of coupling operators but also to look for a bettersystem Hamiltonian with optimum dipole-dipole interactions) instead of the brute forcerandom sampling we performed here.

22We stopped the simulation when the change in the average energy was smaller than 10−9. We checkedthat we got the right NESS for the LDME/BRME from the the null space of the Liouville generator.

160

8 Conclusions and outlook

8.1 Why phenomenological nonlinear master equation?

We started our journey with the traditional microscopic formulation of open quantumsystems. Although fundamental, it is clear that such an approach is limited beyond theWCL since the different existing derivations, such as the ones leading to the BRME orthe CLME/NRSME, require approximations which tend to jeopardize the positivity ofthe density matrix, the second law of thermodynamics or the convergence to the correctsteady-state. Performing derivations free of such drawbacks can be counted on thefingers of one hand 1 and generally give rise to master equations 2 which are in practicecomputationally too demanding to perform a detailed analysis requiring to explore alarge space of parameters. For this reason, we decided to use master equations postulatedon the basis of sensible thermodynamical and statistical properties as it is customaryfor classical non-equilibrium system. However, following this strategy requires greatcare. Indeed, it is not because a master equation generates a MDS that it is necessarilyappropriate to study open quantum systems in any regime. To check the relevanceof the different NTMEs we have used the FDT and the GKF providing fundamentalreference observables, namely susceptibilities and transport coefficients, which are well-known and experimentally accessible even for quantum systems. Thereby, we wereable to assign each NTME to a reasonable regime: the pre-averaged and dynamicallytime-averaged NTMEs should be used in the WCR instead of the BRME and CGME,respectively, whereas the modular free NTME is physically meaningful in the strongdamping regime in presence of a classical hot environment and can be substituted to theCLME and NRSME.

1See [89], Sec. 6.3 of the book [23] or [104] discussing dynamical coarse-graining kind of procedures.2Non-Markovian effects, including system-reservoir correlations, have to be taken into account for an in-

terpolation between the weak and strong coupling regimes using e.g. the second-order time-convolutionlessexpansion in the polaron frame [508, 527] or the hierarchical equations of motion method [528, 95].

161

Chapter 8. Conclusions and outlook

8.2 Did we get new insights by using the NTMEs?

We examined a series of reference quantum setups under the lens of the NTMEs. Westarted with a single qubit embedded in a heat reservoir and revealed that, in presenceof non-secular terms, its time evolution depends on the initial phase in the WCR. Wealso showed that in the strong damping regime the coherence undergoes a biexponentialdecay associated to a susceptibility profile which is neither Gaussian nor Lorentzian.This effect is highly ubiquitous in nature, in regards of the numerous experimentsreporting it, but no clear unifying picture has been provided until now. Moreover, itcan be extended to multi-qubit systems. We exploited this property to prolong thecoherence time for an appropriately selected initial state. Besides, we also explainedhow to transfer this knowledge to describe a biexponential decay in a simplified LHC.

As soon as one examines a bipartite (or multipartite) system it is crucial to distinguishcoherence, arising from a superposition of states, from entanglement which is a non-local/global property of the system. We studied these two quantities by comparing theLDME with the pre-averaged NTME/dynamically time-averaged NTME/modular-freeNTME, which are the perfect substitute to the BRME/CGME/LSME, and explainedhow one can generate coherence and entanglement from “scratch”. Therefore, one hasto consider either a degenerated Hamiltonian, non-secular contributions or collectivecoupling operators (or non-trivial friction matrices). Furthermore, we found out that onecan generate asymptotic entanglement at the expense of breaking the EP, using identicalqubits connected to a common reservoir, and this even under the strict microscopicsettings of the WCL. To be complete, we examined the interrelation between coherenceand entanglement. Using a screening strategy, we found that the initial coherence canenhance the creation of entanglement but, and this is more surprising, can also refrainit. Moreover, we showed that the decay of the coherence elements are affected by theinitial phases of the density matrix, starting from an entangled mixed X-state, which inturn can be exploited to accelerate/refrain the decay of entanglement.

Afterwards, we studied the FMO complex which plays a key role in photosynthesis.Using everything we learned so far, we discussed the energy transfer taking place acrossthe electronic core of a FMO subunit close and far-away from equilibrium. We found outthat the transport properties strongly depend on the choice of the coupling operatorswhich are free parameters in our model. Thus, one can control how coherence eitherpromote or restrict the energy transfer during the time evolution or at the NESS. Theseeffects are possibly exploited by plants or bacteria to tune the amount of energy funneledtowards the RC according the degree of illumination and their daily needs in energy.Besides, we have seen that the energy transport efficiency is altered when modifying thedipole-dipole couplings between the chromophores which are encoded in the systemHamiltonian. To reproduce these alterations in the bacteria, one could mutate the FMOsubunit to change the spacing between the chromophores. Alternatively, one couldexploit this knowledge to design artificial light-harvesting devices.

162

8.3. Future perspective

8.3 Future perspective

Let us consider three axes, namely how to further develop the tools we already provided,how to extend the NTMEs to other regimes and finally which other kind of system couldbe studied with them.

Further tool development One could imagine to study the linear response to a slowlyvarying external perturbation for a NESS such as in [529] using the more general nu-merical strategy proposed in Sec. 9.17. Further exploring the role of steady coherencegenerated by the non-secular terms at the NESS, using e.g. machine learning [456] toscreen for optimal parameters, seems also very promising. Alternatively, it could beinteresting to apply the generalized third order susceptibility (5.126) outlined in Sec. 5.5.2to study the FMO subunit in the WCR for which numerous experimental data are avail-able [353]. Such an analysis could already be relevant using the BRME. Ideally, oneshould numerically extract the third order susceptibility for the NTMEs by computingS(3)(ω3, ω2, ω1)=P(3)(ω=ω3 + ω2 + ω1)/(E(ω3)E(ω2)E(ω1)), for external fields beingmonochromatic plan waves, and then perform a Fourier transform to get S(3)(ω3, τ2, ω1)

and compare it to the one obtained from the linearized NTMEs. On the other hand, wecould extend the GKF to capture cross-effects in thermoelectric setups. Therefore, weconsider two reservoirs and extend the transport matrix (5.80) to two different affinitiesx and y, associated respectively to the heat JQ,+ and particles JN,+ fluxes, yielding

J1,(1)Q,+

J1,(1)N,+

J2,(1)Q,+

J2,(1)N,+

= −

Lx1,x1 Lx1,y1 Lx1,x2 Lx1,y2

Ly1,x1 Ly1,y1 Ly1,x2 Ly1,y2

Lx2,x1 Lx2,y1 Lx2,x2 Lx2,y2

Ly2,x1 Ly2,y1 Ly2,x2 Ly2,y2

x1

y1

x2

y2

.

One can simplify this system using an extended TCF property (5.82) and the ORRleading to three distinct transport coefficients Lx1x2 , Ly1y2 and Lx1y2 so that

J1,(1)Q,+

J1,(1)N,+

=

Lx1x2 Lx1y2

Lx1y2 Ly1y2

x1 − x2

y1 − y2

→ J1,(1)

Q,+

J1,(1)C,+ /e

=

Lx1x2 Lx1z2

Lx1z2 Lz1z2

− d

kBT2 ∆T

edkBT ∆φE

,

where we use x1 − x2 = β1 − β2 ≈ − 1kBT2 ∆T and switch from a particle to an electric

current flux using the elementary charge e giving µk = eφE,k with φE,k the electric potentialor voltage linked to kth reservoir and JN,+= JC,+/e with JC,+ the electric current so thaty1 − y2 = β1µ1 − β2µ2 = e (β1φE,1 − β2φE,2) ≈ e

kBT ∆φE and zk = e(βφE − βkφE,k) theelectric affinity. If we decide to stay with chemical potentials, we could describe a

163

Chapter 8. Conclusions and outlook

“thermoexcitonic effect” using the LHC system Hamiltonian HS introduced in Sec. 5.4.2and associated exciton operator NS=(1/2)(∑Nc

i σz,i + 2). Besides, to study an artificialsolar cell one should privilege the representation in terms of electric current. In bothcases, one has to consider an extended KMS condition following e.g. [530] and modifythe GKF and master equations accordingly. To estimate the efficiency of the setup,one computes the figure of merit ZT = L2

x1z2/(Lx1x2 Lz1z2 − L2

x1z2) which is strongly

subordinated to the nature of the studied system [531].

Extending the NTMEs One could first study the action of a spin bath by changingthe spectral function (2.25) as c(j)

k,l (ω) = δk,l 2(1− n(F)β j ,ω

) Jk(ω) with n(F)β j ,ω

= 1/(eβ jω + 1)the Fermi-Dirac distribution and Jk(ω) symmetric to respect the microscopic detailedbalance. Such a change has drastic physical consequences for high temperatures 3. Toinclude short-time memory effects without spoiling the positivity of the time evolution,one could rewrite the time-dependent bosonic spectral function (2.22) by replacing thesinc function (2.23) by δtc(ω±ω)= (

√2π ωtc)

−1 exp (− (ω±ω)2

2ωtc) which is a normalized

Gaussian depending on ωtc = 1/tc with tc = tc/g so that one recovers the standardspectral function in the WCL. One could also use the time-dependent spectral functionand eigenoperators obtained from the time-convolutionless master equations [20, 281]using the link between the seemingly nonlinear LDME (3.33) and the NTMEs. All ofthis can be achieved for a fermionic reservoir. Alternatively, one can use a classicalenvironment at the expense of applying a correction factor to the spectral function toenforce the microscopic detailed balance [532, 533, 534]. To reproduce a time evolutionbeyond the second order perturbative expansion, such as the one generated by thehierarchical equations of motion [528, 95], we can add reaction coordinates (RCS) tothe system [535, 536]. The idea is to couple our system of interest directly to the RCS,modeled by another small quantum system, which in turn is connected to the reservoir.One then simulates the time evolution of this enlarged system and traces over theRCS (usually a two level system or harmonic oscillator). Thus, the system of interestbecomes correlated with the structured environment (RCS plus reservoir) allowing toapply the NTMEs from the weak to strong coupling regime to more accurately model,for example, the FMO complex. Note that the total system remains Markovian.

Possible applications First and foremost, we could study the mentioned thermoex-citonic effect in the FMO complex and compare it to thermoelectic effect of standardsolar cells by using their respective figure of merit. Note that the LHC is just one ofthe biological system where quantum effects have been predicted to have a non-trivialrole. The energy transfer in proteins modeled by the Davydov-Scott Hamiltonian [537]would be equally interesting to study. Let us also cite the avian magnetoreception, thephotoactivation of the rhodopsin involved in the first steps of the vision as well as thelong-range electron tunneling for biological redox reactions and the hydrogen tunnelingfor enzyme catalysis [538].

3Sec. 5.4.2 of [10] from p.229 to 232 for a detailed calculation of the fermionic spectral function.

164

9 Appendix

9.1 Incompatibility of the usual Lindblad eigenoperators withcontinuous energy spectra

To streamline the notation, a single reservoir is connected to the reduced system througha unique coupling operator Q. To clearly see the problem arising when consideringa continuous energy spectrum in conjunction with the eigenoperators Aω, insert theidentity 1 =

∫dEn |En〉〈En| into the arbitrary matrix element 〈Em| Aω |ψ〉 and use the

definition of the eigenoperators

Aω = ∑mn

(Q)mn δKr(En − Em, ω) |Em〉〈En| . (9.1)

The integration has to be performed over a continuous range of energies assuming asmall spacing between the levels yielding

〈Em| Aω |ψ〉 =∫

dEn 〈Em| Aω |En〉〈En|ψ〉

= Qmn

∫dEn δKr(Em − En,−ω)〈En|ψ〉 = Qmn

∫dEn f (En) = 0

(9.2)

because f (En) = δKr(En − Em, ω)〈En|ψ〉 is equal to zero unless En − Em = ω implyingthat one has to integrate over a single point. To avoid that the integral vanishes, we caninsert a ”Gaussian smoothing“ around this single point with the constraint that in a givenlimit the integral equals zero. This is achieved by defining modified eigenoperators [77,78] in the spirit of what is done in [89]. This strategy is applied in Sec. 3.2.2 to produce adynamically time-averaged master equation. On the other hand, for Lindblad operatorswhich are not eigenoperators and do not refer to an underlying microscopic dynamicsthis problem can be avoided. For example, Vacchini and Hornberger model the quantumBrownian motion of a free particle with a phenomenological linear Boltzmann equationof the Lindblad type (see e.g. the equations (2)–(4) of [332]).

165

Chapter 9. Appendix

9.2 Generalized Gibbs state

Usually, a reduced system connected to a heat reservoir relaxes towards the Gibbsstate. However, recent measurements indicate that the final state of certain physicalsetups display a Tsallis distribution being “a possible generalization of the Boltzmann-Gibbs statistics” [207, 209]. Such a distribution was, for example, observed for themotion of cold atoms in dissipative optical lattices [539, 540, 541], driven-dissipative 2Ddusty plasma [542], trapped ions [208], spin-glasses [543] or charged-hadron transverse-momentum [544, 545, 210]. To handle such non-extensive systems, we have to modifythe formulation (3.10) at the level of the Massieu operator (3.11), namely

Ψk → Ψq,k = Sq(ρ)− βk HS , (9.3)

where we introduced Tsallis quantum entropy operator

Sq(ρ) = − lnq (ρ) := − ρq−1 − 1

q− 1, (9.4)

where q ∈ [0, 1] is a parameter quantifying the degree of non-extensitivity, defined fromTsallis quantum entropy

〈Sq(ρ)〉 = tr(ρ Sq(ρ)) = −tr(ρq)− 1

q− 1(9.5)

whose properties are, inter alia, given in [546]. It is equivalent to the linearized form ofthe quantum Rényi entropy as pointed out in [213]. The steady-state associated to thekth reservoir is

πq,k = [1− (q− 1)βk HS ]1

q−1 (9.6)

since Sq(πq,k)=βk HS so that the q-Massieu operator (9.3) vanishes once πq,k is reached.Note that the normalization factor has been omitted since the Massieu operator is insideof a double commutator in the MDS master equation (3.10)–(3.11). We recover the vonNeumann entropy operator in the limit

limq→1

Sq(ρ) = S(ρ) (9.7)

as well as the Gibbs state

limq→1

[1− (q− 1)βk HS ]1

q−1 = e−βk HS . (9.8)

and the normalization drops out as for the usual formulation due to the double com-mutator form insuring the preservation of the trace. Thus, the irreversible term (3.10)with the q-Massieu operator (9.3) will drive the system towards Tsallis distribution (9.6).

166

9.2. Generalized Gibbs state

Note that one should check the positivity of the time evolution (should be fine sinceTsallis entropy is convex as is the von Neumann entropy) and redo all calculations nearequilibrium as presented in Sec. 4.4 (the usual Kubo form is lost and is only restored inthe limit q→ 1). Even if the analytical calculations are laborious one could still computethe susceptibilities and Onsager coefficients numerically. To be sure that we recoverthe LDME in the WCL one could express the non-extensive parameter, for example,as q = e−g/g ∈ [0, 1], with g a parameter, so that for g → 0 we have q → 1 givingback the von Neumann entropy and the Gibbs state. Notice that the same strategycould be applied to treat other types of entropies. The resulting MDS equation could becompared to the master equation derived by Kirchanov in [213] using as starting pointthe linearized Rényi distribution (i.e. Tsallis) instead of the Gibbs distribution. Note thatour approach also permits to use the Rényi entropy itself instead of its sole linearization.

From a more general point of view, there exist many situations where the final state isnot of Gibbs type, in particular in the stronger coupling regime [148, 416] or in presencesystem-environment correlations [535, 536]. From a thermodynamical point of view,one could model such a situation by selecting an appropriate coarse-grained entropySCG(ρ) leading to a coarse-grained Massieu operator Ψk, CG =Sk, CG(ρ)− βk HS driving thedynamics of the system to the sought steady-state. Staying with the von Neumannentropy, one could use S(ρ)→Sk, CG(ρ)= ln (ρ) + ln (πk, CG) shifted by a “coarse-grained”entropy expressed in terms of πk, CG ∝ e−βk HCG leading to a steady-state for the kthreservoir being proportional to e−βk(HS+HCG) no longer equal to the standard Gibbsdepending on the sole system Hamiltonian HS . The best candidate for such a treatmentis the modular-free NTME (3.66) which can be used in the strong damping regime asdiscussed in Sec. 5.4.4.

167

Chapter 9. Appendix

9.3 The zero temperature limit and the NTMEs

A pure thermal state can be reached as β=1/T → 0 if the ground state is not degenerate;the von Neumann entropy 〈S(ρ)〉 is then equal to zero. This is in agreement withthe so-called first formulation of the third law of thermodynamics [547] which can besummarized as: when the temperature of a pure substance approaches the absolute zero, itsentropy approaches zero too, i.e. no uncertainty remains about the different componentsof the substance which are distributed on a perfect lattice. Assuming the reader hasalready looked at the different master equations associated to the MDS (the LDME andthe NTMEs), we briefly discuss the implication of the zero temperature limit for them.

The LDME at a temperature T=0 implies that only the emission terms survive while theabsorption terms are suppressed. The EP (2.48) guarantees, assuming a discrete energyspectrum, that the system relaxes towards a unique ground state (the pure thermalstate). Let use take the LDME in the form (2.40) with a bosonic spectral function (2.25)for a single reservoir (i.e. M = 1) and a single coupling operator (i.e. N = 1) for sakeof simplicity. In this case the Bose-Einstein distribution becomes nβ,ω = 0 since β→∞.Thus, the absorption rate a(ω) vanishes while the emission rate e(ω)=2J(ω) becomestemperature independent yielding

ρ=−i[HS , ρ] + 2 ∑ω≥0

J(ω)

(A†

ωρAω −12{Aω A†

ω, ρ})

. (9.9)

It has been experimentally shown in [548] that the decoherence does not vanish in Aunanowires in the limit T → 0, i.e. it saturates at a finite temperature, so that the LDMEcould be applied to model such a system 1. Concerning the NTMEs, we notice that forthe Massieu operator (3.11) we have ||Ψ|| ∼ || ln ρ||→∞ for pure thermal states so thatthe equation is simply not defined at T=0 (it can only handle faithful states). Thus, onehas first to take the WCL (i.e. the limit of vanishing coupling strength in the long timelimit) giving the LDME and only afterwards impose the zero temperature limit. Thisis entirely in line with the fact that the “zero-temperature reduced density operator does notdescribe a pure state in general, except in the zero-coupling limit” [551]. This also reminds usof the second formulation of the third law, namely the unattainability principle: Anythermodynamic process cannot reach the temperature of absolute zero by a finite number ofsteps and within a finite time. However, we here just discussed the fact that the LDME isdefined at T=0, not how one can reach the absolute zero through a thermodynamicalprocess (which cannot be reached with the NTMEs away from the WCL). A detaileddiscussion on quantum refrigerators is proposed in [552] as well as in [224] indicatingthat one would end up with a zero entropy production. Reaching the absolute zerousing ultracold degenerate quantum gases is discussed in [553]. To avoid unnecessarycomplications, the zero temperature limit will not be considered in the present work.

1However, in general, the debate about the decoherence process at zero temperature is far from beingsettled (see e.g. [549] and the introduction in [550]).

168

9.4. Seemingly nonlinear commutation relation

As a final remark, let us note that the zero temperature limit is essential in the context ofquantum field theory [554] and that care must be taken when incorporating dissipationto it away from the WCL. An interested reader could take a look at the stochasticunravelling of a relativistic Lindblad equation proposed by Breuer and Petruccionein [555], see Eqs. (19)–(25), as well as [556]. Furthermore, the WCL — also known asthe van Hove limit — is discussed in the context of relativistic field-theoretical modelsin [557]. A possible linearization in the low temperature limit of a NTME kind ofequation is discussed in the book [558] which carries a collision term on top of irreversiblecontribution in (9.9). Finally, note that an unravelling of this linearization is provided inthe same book.

9.4 Seemingly nonlinear commutation relation

To obtain a more readable proof, we rename HS → H as well as using a single reser-voir (βk → β) and change accordingly the Bohr frequency index −ωk → ω (alsoinverting the sign for convenience) so that

Kωρ [A−ω, ln ρ + βH] = A−ωρ− eβωρA−ω. (9.10)

This property can be proven by expressing the density matrix as ρ = ∑n pn |pn〉〈pn|where pn and |pn〉 are its eigenvalues and orthonormal eigenvectors. We start from theleft-hand side of Eq. (9.10), define the matrix element of the eigenoperators in the basis ofthe density matrix as 〈pn| A−ω |pm〉 = (A−ω)nm and use the identity [A−ω, H]=ωA−ω

obtained from (3.35) so that

〈pn|Kωρ [A−ω, ln (ρ) + βH] |pm〉 = 〈pn|Kω

ρ [A−ω, ln (ρ)] + Kωρ [A−ω, βH] |pm〉

=∫ 1

0dλ eλβω(A−ω)nm pλ

n (ln pm − ln pn − βω) p1−λm

= −(A−ω)nm pm

∫ 1

0dλ eλβωzλ (ln z + βω) , z =

pn

pm

= −(A−ω)nm pm

∫ 1

0dλ eλ(ln z+βω) (ln z + βω)

= −(A−ω)nm pm

∫ 1

0dλ ∂λ

(eλ(ln z+βω)

)= −(A−ω)nm pm

(e(ln z+βω) − 1

), eln z =

pn

pm

= −(A−ω)nm

(pneβω − pm

)= 〈pn| A−ωρ− eβωρA−ω |pm〉

(9.11)

finally giving the right-hand side of the identity (9.10).

169

Chapter 9. Appendix

9.5 Extended eigenvalue equation

Assuming we select a scattering operator Aν as defined in Eq. (3.45) but for a single reser-voir (we drop the index k to simplify the notations), we prove the following generalizedeigenvalue equation

[Aν, HS ] = νAν + 2ωg2 ∂ν Aν. (9.12)

To do so, we simply insert the definition of the scattering operator Aν given in Eq. (3.45)inside of the commutator [Aν, HS ]. Thereafter, we use [Qt, HS ] = i∂tQt being obtainedby taking the partial time-derivative of the time-evolved coupling operator as definedin (3.47), that is

i∂tQt = i∂t

(eiHS tQe−iHS t

)= i(iHS) Qt + Qt i(−iHS) = [Qt, HS ]. (9.13)

Starting from the left-hand side of the identity (9.12), we use the modified eigenoperatorsAν provided in (3.45), the time-evolved coupling operator (3.47) and the property (9.13)giving altogether

[Aν, HS ] =∫

Rdt eiνt

√δTg(t) [Qt, HS ] =

∫R

dt eiνt√

δTg(t) i∂tQt

= eiνt√

δTg(t) Qt

∣∣∣∣R

− i∫

Rdt ∂t

(eiνt√

δTg(t))

Qt

= −i∫

Rdt eiνt

√δTg(t)

(iν− t

2T2g

)Qt

= ν∫

Rdt eiνt

√δTg(t) Qt + 2ω2

g

∫R

dt (iteiνt)√

δTg(t) Qt

= νAν + 2ω2g ∂ν

∫R

dt eiνt√

δTg(t) Qt = νAν + 2ω2g ∂ν Aν,

(9.14)

where in the second line we integrated by parts noticing that the boundary term vanishesdue to the oscillatory nature of eiνt. From the second to the last line we used theparameter ωg = 1/(2Tg). To reach the last line we applied the definition of Aν as well asthe relation iteiνt = ∂ν(eiνt) and took the partial derivative out from the integral finallygiving the right-hand side of (9.12).

In addition, let us show that the modified eigenvalue equation (9.12) is bounded by aTg-dependent expression, that is

||[Aν, HS ]|| ≤ ν||Aν||+ 2ωg||∂ν Aν|| ≤ C√

Tg +C′√

Tg, (9.15)

where C and C′ are constant. We used the result of (3.49) for the first term and for the

170

9.6. Dynamical time-averaging of the first NTME

second term we computed

||∂ν Aν|| =∫ +∞

−∞dt |t|

√δTg(t) = (2π)1/4 T1/2

g 2∫ ∞

0dt t e−t2/(4T2

g ) = 8(2π)1/4 T3/2g (9.16)

as well as ωg = 1/(2Tg). The result (9.15) can be used to prove that the commutator||[Xk,λ

ω , HS ]|| associated to the scattering operator (3.44) is bounded too.

9.6 Dynamical time-averaging of the first NTME

To uncover the dynamical time-averaging procedure hidden in the irreversible generator3.53, we make use of the definition of the modified eigenoperators Ak,ν given in (3.45)and of the frequency dependent superoperator provided in (3.34). It is also necessaryto introduce the time-evolved coupling operator Qk,t =Ut(Qk) from (3.47) as well time-evolved Massieu operator Ψk,t =Ut(Ψk). The irreversible contribution then becomes

Jk(ρ) =12

∫R


ρ [Ak,ν, Ψk]]

=12

∫R

dt1

∫R

dt2

√δTg,t1 δTg,t2

∫ 1

0dλ

(∫R

dν e−iν(t1−t2−iλβ)hk(ν)

)[Qk,t1 , ρλ[Qk,t2 , Ψk]ρ

1−λ]

=12

∫R

dt1

∫R

dt2

√δTg,t1 δTg,t2

∫ 1

0dλ hk,(t1−t2−iλβ) [Qk,t1 , ρλ[Qk,t2 , Ψk]ρ

1−λ]

=12

∫R

dq∫

Rdr√

δTg,(q+ r2 )

δTg,(q− r2 )

∫ 1

0dλ hk,(r−iλβ) [Qk,q+ r

2, ρλ[Qk,q− r

2, Ψk]ρ

1−λ]

=12

∫R

dq δTg,q

∫R

dr e− ( r

2 )2

2T2g

∫ 1

0dλ hk,(r−iλβ)×

[eiHS q(Qk, r2)e−iHS q, ρλ[eiHS q(Qk,− r

2)e−iHS q, Ψk]ρ

1−λ]

=12

∫R

dqδTg,q

∫R

dre− ( r

2 )2

2T2g

∫ 1

0dλhk,(r−iλβ) eiHS q

([Qk, r

2, ρλ

t−q[Qk,− r2, Ψk,t−q]ρ

1−λt−q ]

)e−iHS q

=∫

Rdq δTg,q Uq

(12

∫R

dr e− ( r

2 )2

2T2g

∫ 1

0dλ hk,(r−iλβ) [Qk, r

2, ρλ


1−λt−q ]︸︷︷︸

Jk(ρt−q)=Jk(U−q(ρt))

),

(9.17)

where δTg,t =δTg(t) and hk,t =hk(t), so that we finally get

Jk(ρt) =∫

Rdq δTg(q)Uq

(Jk(U−q(ρt))

)(9.18)

To reach the fourth line in the calculation (9.17) we have used the change of variablet1 = q+ r/2 and t2 = q− r/2, such that q = (t1 + t2)/2 and r = t1− t2 stand respectively

171

Chapter 9. Appendix

for the mean and the difference, whereas to get to the fifth line we computed the squareroot with the smoothing Gaussian (3.48), e.g.

√δTg(q +

r2)δTg(q−

r2) =

√√√√ 12πT2

gexp

(−(q + r

2

)2+(q− r

2

)2

2T2g

)

=1√

2πTg

√√√√exp

(−2q2 + (r2/2)

2T2g

)= δTg(q) e

− ( r2 )

2

2T2g ,

(9.19)

where we could separate the q and r variables and reconstitute one q-dependent smooth-ing Gaussian δTg(q). To reach the sixth line of the calculation (9.17) we have takenthe two exponential operators outside of the double bracket through the followingtedious calculation by introducing multiple times the identity 1= e−iHS qeiHS q to makeappear the free evolution superoperator U±q and used the notation ρt−q =U−q(ρt) andΨk,t−q =U−t(Ψk,t)=− ln ρt−q + βk HS . More precisely, we get

[eiHS q(Qk, r2)e−iHS q, ρλ

t [eiHS q(Qk,− r

2)e−iHS q, Ψk,t]ρ

1−λt ]

= [eiHS qQk, r2e−iHS q, ρλ

t

(eiHS qQk,− r

2e−iHS qΨk,t −Ψk,teiHS qQk,− r

2e−iHS q

)ρ1−λ

t ]

= eiHS qQk, r2e−iHS qρλ

t

(eiHS qQk,− r


2e−iHS q

)ρ1−λ

t

− ρλt

(eiHS qQk,− r


2e−iHS q

)ρ1−λ

t eiHS qQk, r2e−iHS q

= eiHS q(

Qk, r2e−iHS qρλ

t eiHS qQk,− r2e−iHS qΨk,t 1 ρ1−λ

t e−iHS q

−Qk, r2e−iHS qρλ

t 1Ψk,teiHS qQk,− r2e−iHS qρ1−λ

t e−iHS q

− eiHS qρλt eiHS qQk,− r

2e−iHS qΨk,t 1 ρ1−λ

t eiHS qQk, r2

+ eiHS qρλt 1Ψk,teiHS qQk,− r

2e−iHS qρ1−λ

t eiHS qQk, r2

)e−iHS q

= eiHS q(

Qk, r2

U−q(ρλt ) Qk,− r

2U−q(Ψk,t)U−q(ρ

1−λt )

−Qk, r2

U−q(ρλt ) U−q(Ψk,t) Qk,− r

2U−q(ρ

1−λt )

−U−q(ρλt ) Qk,− r

2U−q(Ψk,t)U−q(ρ

1−λt ) Qk, r

2

+ U−q(ρλt )U−q(Ψk,t) Qk,− r

2U−q(ρ

1−λt ) Qk, r

2

)e−iHS q.

(9.20)

172

9.6. Dynamical time-averaging of the first NTME

Now using U−q(ρλt )=ρλ

t−q we can drastically simplify the previous expression

[eiHS q(Qk, r2)e−iHS q, ρλ

t [eiHS q(Qk,− r

2)e−iHS q, Ψk,t]ρ

1−λt ]

= eiHS q(

Qk, r2ρλ

t−q

(Qk,− r

2Ψk,t−q −Ψk,t−qQk,− r

2

)ρ1−λ

t−q

− ρλt−q

(Qk,− r

2Ψk,t−q −Ψk,t−qQk,− r

2

)ρ1−λ

t−q Qk, r2

)e−iHS q

= eiHS q(

Qk, r2ρλ


1−λt−q − ρλ


1−λt−q Qk, r

2

)e−iHS q

= Uq

([Qk, r

2, ρλ


1−λt−q ]

).

(9.21)

Noticing that for a weak-coupling the macroscopic time is of the order t ∼ ( g−2 t )�q ∼ 1 since g� 1, we can approximate U−q(ρt) = ρt−q ∼ ρt in (9.18) finally giving

Jk(ρt) =∫

Rdq δTg(q)Uq(Jk(ρt)), (9.22)

where the pre-average irreversible contribution appears to be

Jk(ρt) =12

∫R

dr e− ( r

2 )2

2T2g

∫ 1

0dλ hk(r− iλβ)[Qk, r

2, ρλ

t [Qk,− r2, Ψk,t]ρ

1−λt ]. (9.23)

Compared to the MDS master equation (3.16)–(3.17), the two-time correlation function ofthe pre-averaged generator (9.23) leads to a friction matrix which is not positive definitein general so that it does not respect the requirement to generate a MDS. One couldreplace it by the square root of the product of two-time correlation functions to ensurethe positivity of the resulting friction matrix; the corresponding master equation wouldbe similar to the second NTME as given in (3.58).

173

Chapter 9. Appendix

9.7 Seemingly nonlinear commutator identity

We prove in this sequel that

Kρ[A, ln ρ] = [A, ρ] (9.24)

using the eigenbasis of the density matrix {|pn〉}. We start by computing the individualmatrix element of the left-hand side using the definition of the Kubo superoperatorgiven in (4.18). Thus, we get

〈pn|Kρ[A, ln ρ] |pm〉 =∫ 1

0dλ 〈pn| ρλ A ln ρρ1−λ |pm〉 − 〈pn| ρλ ln ρAρ1−λ |pm〉

= 〈pn| A |pm〉∫ 1

0dλ pλ

n p1−λm (ln pm − ln pn)

= Anm pm

∫ 1

0dλ eλ ln z ln z, z =

pn

pm

= Anm pm

∫ 1

0dλ

∂

∂λ

(eλ ln z

)= Anm pm

(eλ ln z − 1

)= Anm (pn − pm) ,

(9.25)

where we used Anm = 〈pn| A |pm〉 and zλ = eλ ln z. The final expression is clearly equalto the right-hand side of (9.24) since

〈pn| A |pm〉 (pn − pm) = 〈pn| [A, ρ] |pm〉 . (9.26)

The identity (9.24) has already appeared in equation (20) of [559], in the context ofentropy production for dynamical semigroup, and was used in equation (5) of [146] andequation (21.3) of [167]. We provided here the proof for sake of completeness and alsoto see the similarities to the property discussed in the Appendix 9.4; both relations areassociated to a seemingly nonlinear contribution.

9.8 Time-averaging procedure applied to the third NTME

What happens if we apply a time-averaging procedure to the modular free (3.65) ex-pressed in terms of the Massieu operator (3.11) and assuming a discrete energy spectrum?In this sequel we show that we obtain the nonlinear equation (3.64) but expressed interms of the Lindblad eigenoperators Ak,ω. The idea is to compute something of the kindlimT→∞(1/T)

∫ T/2

−T/2dt U−t(Jk(Ut(ρt))) for the irreversible generator of the kth reservoir.

This means that we have to go into the interaction picture, apply the time evolution andthen come back to the Schrödinger picture in the weak-coupling scaling limit.

To this end, we have to define with the help of the reversible time evolution superopera-

174

9.8. Time-averaging procedure applied to the third NTME

tor

Ut(·)= eiHS t(·) e−iHS t (9.27)

the following density matrix in the time-rescaled interaction picture

ρτ := eiHSτ/γρτ/γ e−iHSτ/γ, (9.28)

where we used the microscopic time τ= tγ by rescaling the macroscopic or dissipativetime t with γ = g2 with g the coupling strength. Our aim is to take at the end of thecalculation the limit g ∼ 0 to eliminate the fast oscillations (τ is finite for large timet∼∞ and small coupling γ∼ g ∼ 0). Therefore, the nonlinear master equation (3.65)expressed in terms of the Massieu operator (3.11) is put into the interaction picture using(9.27)–(9.28) in order to compute

Jk(ρτ) =γk

βkUτ/γ

([Qk, Kρ[Qk, Ψk]]

)=

γk

βk

∫ 1

0dλ

(Uτ/γ(Qk 1 ρλ

1Qk 1Ψk 1 ρ1−λ)

−Uτ/γ(Qk 1 ρλ1Ψk 1Qk 1 ρ1−λ)

−Uτ/γ(ρλ1Qk 1Ψk 1 ρ1−λ

1Qk)

+ Uτ/γ(ρλ1Ψk 1Qk 1 ρ1−λ

1Qk)

)=

γk

βk[Qk, τ

γ, Kρτ

[Qk, τγ, Ψk, τ

γ]],

(9.29)

where we introduce the identity 1 = e−iHS teiHS t multiple times to obtain with (9.27)the time-evolved version of the coupling operators Uτ/γ(Qk) = Qk, τ

γas well as of the

Massieu operator Uτ/γ(Ψk) = Ψk, τγ=−(ln ρτ + βk HS); such a tedious calculation was

already carefully performed in (9.20)–(9.21) of the Appendix 9.6 so we do not repeat ithere. Note that γk is a finite friction coefficient contrary to γ which is a scaling parameter.Now using the spectral decomposition of the coupling operator (4.1) we obtain

Jk(ρτ) =γk

βk∑

ω1,ω2

e−i(ω1+ω2)τ/γ[Ak,ω1 , Kρτ[Ak,ω2 , Ψk, τ

γ]]. (9.30)

Noticing that for γ∼0 we eliminate the fast oscillations since e−i(ω1 + ω2)τ/γ→δKr(ω1 + ω2)

restricting the frequencies to ω2=−ω1=ω whereas U−t(ρτ)∼ρt allows to go back intothe Schrödinger picture finally giving

Jk(ρt) =γk

βk∑ω

[Ak,−ω, Kρt [Ak,ω, Ψk]] (9.31)

which is equal to (3.64) using the definition of the Massieu operator (3.11) as well as theflat spectral function h(0)=2γk/βk.

175

Chapter 9. Appendix

9.9 Linearization of the perturbed density matrix

We approximate ρδA close to equilibrium by first applying Duhamel’s formula 2

eA − eB =∫ 1

0dλ eλA(A− B)e(1−λ)B (9.32)

for

A = −β(HS − δA) + ln (Z−1δA )1

B = −βHS + ln (Z−1β )1

(9.33)

such that

eA = Z−1δA e−β(HS−δA)

eB = Z−1β e−βHS

A− B = δA + ln (Zβ/ZδA)1

(9.34)

exploiting the fact that eC+D = eCeD is valid when [C, D] = 0 being fulfilled in our casebecause we have C = −β(HS − δA) or −βHS and D = ln (Z−1

δA )1 or ln (Z−1β )1. This

is a consequence of the Baker-Campbell-Hausdorff formula; consult Sec. 2 of [561] forfurther details. Accordingly, we get

ρδA − πk = β∫ 1

0dλ ρλ

δA δA π1−λ = βKπ(δA) + o(||δA||), (9.35)

where we expanded for small δA the density matrix ρδA yielding ρλδAδA = πλδA + o(||δA||)

and resorted to the Kubo-Mori superoperator (4.18) for ρ = π in the last equality. Weignored the term ln (Zβ/ZδA)1 not influencing the final result because it appears insideof the commutator (4.22). Alternatively, one could have directly used the formula (1.8)provided in the Appendix I of [562].

2To prove Duhamel’s formula, simply compute the derivative ∂λ(eλAe(1−λ)B)= eλA(A− B)e(1−λ)B andintegrate both sides

∫ 10 dλ ∂λ(eλAe(1−λ)B) = eA − eB =

∫ 10 eλA(A − B)e(1−λ)B concluding the proof. We

assumed that the operator A and B are bounded; for further details about exponential operator propertiesone could, for example, consult [560].

176

9.10. From the linearized NTME to the LDME for Tg → ∞

9.10 From the linearized NTME to the LDME for Tg → ∞

To prove the relation (4.29), we stay as long as possible in the unfinished time-averagedform and take the limit of infinite Tg at the very end. Moreover, we use a singlereservoir (allowing to drop the index k to simplify the notations). We first provide themain relations required for the proof and only then compute the details.

9.10.1 Main steps of the proof

We start with our first key property

e−λβHS Aν = eλβν S ωgλ (Aν) e−λβHS (9.36)

written in term of the shifted eigenoperator

S ωgλ (Aν) = e2λ2β2ω2

g Aν−νλ,g (9.37)

depending on the modified eigenoperator Aν as defined in (3.45), on the frequency shiftνλ,g = 4λβω2

g and on the parameter λ ∈ [0, 1] originating from the superoperator asgiven in (3.34). Our first property can be used to compute a second one, namely

K−νπ [Aν, X] =

∫ 1

0dλ S ωg

λ (Aν)πλXπ1−λ − e−βν∫ 1

0dλ πλXπ1−λ S ωg

λ (Aν) (9.38)

being close to the result we are after. We notice that for ωg → 0 we have

S ωgλ (Aν)→ Aω (9.39)

according to (9.37); this transition is explicitly illustrated by the calculations leadingfrom (4.2) to (4.9). When taking the limit Tg the integral over the frequencies present inequation (4.4) becomes a sum over the discrete energy spectrum of HS . Using (9.38)-(9.39)altogether with X = K−1

π ρ and Kπ(K−1π ρ) = ρ finally yields

K−ωπ [Aω, K−1

π ρ] = Aωρ− e−βωρAω. (9.40)

Thus, the double commutator simplifies drastically

[A−ω, K−ωπ [Aω, K−1

π ρ]] = [A−ω, A−ωρ− e−βωρAω] (9.41)

proving the relation provided in Eq. (4.29) leading to the LDME (3.37).

177

Chapter 9. Appendix

9.10.2 Detailed calculations

Now that we have outlined the main steps of the proof, we can focus on the first property(9.36) being obtained from the following calculation

e−λβHS Aν =∫ +∞

−∞dt eiνt

√δTg(t) ei(t+iλβ)HSQe−itH

=∫ +∞

−∞dt′ eiν(t′−iλβ)

√δTg(t′ − iλβ) eit′HSQe−i(t′−iλβ)HS

= eλνβ e2λ2β2ω2g

(∫ +∞

−∞dt′ ei(ν−νλ,g)t′

√δTg(t′) Qt′

)e−λβHS

= eλνβ e2λ2β2ω2g Aν−νλ,g e−λβHS = eλνβ S ωg

λ (Aν)e−λβHS ,

(9.42)

where we applied a change of variable t = t′− iλβ giving dt=dt′ leaving the integrationboundaries unchanged. Using this property as well as the definition of the superoperator(3.34) for ρ=π we can prove the second property (9.38) as follows

K−νπ [Aν, X] = K−ν

π (AνX)− K−νπ (XAν)

= Z−1β

∫ 1

0dλ e−λβν

(e−λβHS Aν

)Xe−(1−λ)βHS

− Z−1β

∫ 1

0dλ e−λβνe−λβHSX

(e−(1−λ)βHS A−ν

)†

= Z−1β

∫ 1

0dλ e−λβν

(eλβν e2λ2β2ω2

g A(ν−νλ,g)e−λβHS

)Xe−(1−λ)βHS

− Z−1β

∫ 1

0dλ e−λβνe−λβHSX

(e−(1−λ)βν e2λ2β2ω2

g A−(ν−νλ,g)e−(1−λ)βHS

)†

=∫ 1

0dλ

(e2λ2β2ω2

g A(ν−νλ,g)

)πλXπ1−λ

− e−βν∫ 1

0dλ πλXπ1−λ

(e2λ2β2ω2

g A−(ν−νλ,g)

)=∫ 1

0dλ S ωg

λ (Aν)πλXπ1−λ − e−βν∫ 1

0dλ πλXπ1−λ S ωg

λ (Aν),

(9.43)

where we have used A†ν = A−ν as well as key property (9.36) twice, the second time

with the replacement λ→ (1− λ) finally yielding the second property (9.38).

178

9.11. Linearized equation: decomposition for a N-level system

9.11 Linearized equation: decomposition for a N-level system

We obtained the linearized equation (4.42) in two steps. In the first one we consider

Kk,−ωπk

(B) = ∑n1

∑m1

| pk,n1〉〈 pk,n1 |Kk,−ωπk

(∑

n2,m2

〈 pk,n2 | B | pk,m2〉 | pk,n2〉〈 pk,m2 |)| pk,m1〉〈 pk,m1 |

= ∑n1,m1

∑n2,m2

∫ 1

0dλe−βkωλ | pk,n1〉〈 pk,n1 |π

λk(

Bkn2m2| pk,n2〉〈 pk,m2 |

)π1−λ

k | pk,m1〉〈 pk,m1 |

= ∑n1,m1

∑n2,m2

∫ 1

0dλ e−βkωλ pλ

k,n1p1−λ

k,m1Bk

n2m2δn1 , n2

Kr, k δm1 , m2Kr, k | pk,n1〉〈 pk,m1 |

= ∑n1,m1

∫ 1

0dλ e−(βkω−ln pk,n1

+ln pk,m1)λ Bk

n1m1| pk,n1〉〈 pk,m1 |

=− ∑n1,m1

e−βkω pk,n1 − pk,m1

βkω− ln ( pk,n1 / pk,m1)Bk

n1m1| pk,n1〉〈 pk,m1 |= ∑

n1,m1

f k,ωn1m1

Bkn1m1| pk,n1〉〈 pk,m1 | ,

(9.44)

where we have taken advantage of the decomposition

B = ∑n2,m2

Bkn2m2| pk,n2〉〈 pk,m2 | and Bk

n2m2= 〈 pk,n2 | B | pk,m2〉 (9.45)

as well as the completeness conditions 1 = ∑ni

∣∣ pk,ni

⟩ ⟨pk,ni

∣∣ with i = 1, 2, the delta-Kronecker

⟨pk,η1 | pk,η2

⟩= δη1 , η2

Kr, k with η = n, m as well as the coefficients f k,ωn1m1 defined

in (4.43). To reach the second line we compute the coefficient Bkn2m2

for the operatorB = [Ak,ω2 , K−1

πkρ]; to this end, instead of computing explicitly K−1

πkρ we look at

Kπk ρ = ∑r

∑s

∫ 1

0dλπλ

k | pk,r〉〈 pk,r| ρ | pk,s〉〈 pk,s|π1−λk

=∑r,s

ρrs| pk,r〉〈 pk,s| pk,s

∫ 1

0dλzλ| pk,r〉〈 pk,s|=∑

r,sρrs| pk,r〉〈 pk,s| pk,s

∫ 1

0dλeλ ln z| pk,r〉〈 pk,s|

= ∑r,s

ρrs | pk,r〉〈 pk,s|pk,r − pk,s

ln pk,r/ pk,s=∑

r,sρrs | pk,r〉〈 pk,s| (gk,0

rs )−1

(9.46)

with z =pk,rpk,s

and ρrs = 〈 pk,r| ρ | pk,s〉, being exactly its inverse so that we just have to

invert the function gk,0rs originally defined in (4.44), yielding

Bkn1m1

= 〈 pk,n1 | [Ak,ω2 , K−1πk

ρ] | pk,m1〉 = ∑r,s

gk,0rs ρrs 〈 pk,n1 | [Ak,ω2 , | pk,r〉〈 pk,s|] | pk,m1〉 (9.47)

with gk,0rs provided in (4.44). Inserting (9.47) into (9.44) finally yields after few rearrange-

ments the linearized master equation (4.42) for a discrete energy spectrum.

179

Chapter 9. Appendix

9.12 Generalized perturbation schemes

From the microscopic point of view presented in Sec. 2.1, four main types of perturbationschemes can be designed by perturbing the total Hamiltonian (2.1)–(2.3). The first twoare well known and are used to quantify small fluctuations:

(i) perturbation of the system Hamiltonian HS through an external adiabatic drivingyielding the FDT discussed in Sec. 5.2 in the MDS context.

(ii) perturbation of the reservoir Hamiltonian HR by varying the intensive parame-ters (temperature, chemical potential,. . . ) of the different reservoirs giving the GKFexamined in Sec. 5.3 in the MDS context.

Alternatively, one could also perturb:

(iii) the system coupling operators Qk in the interaction Hamiltonian HSR; this has notyet been used per se to our knowledge. This could break the EP (2.49) so that thesteady-state is no longer unique.

(vi) the reservoir coupling Φk operators in the interaction Hamiltonian HSR, e.g. de-forming the microscopic detailed balance (2.27) to obtain an entropic fluctuationtheorem associated to large deviation principles [86].

One could also combine these different schemes to measure other physical properties.For example, (ii) and (iii) have been merged to obtain multiple NESS distorting the usualcurrent statistics or large deviation principle [160]. On the other hand, combining (i)with (ii) could provide the linear susceptibility for a NESS.

9.13 Explicit expression of Kubo’s superoperator

To evaluate Kubo’s superoperator (4.18), we use the density matrix decompositionρ=∑n pn |pn〉〈pn|, where pn and |pn〉 are its eigenvalues and orthonormal eigenvectors.Thus, its (n, m) component reads

〈pn|Kρ(A) |pm〉 = 〈pn|∫ 1

0dλ ρλ Aρ1−λ |pm〉 = 〈pn| A |pm〉 pm

∫ 1

0dλ zλ, z =

pn

pm

= Anm pm

∫ 1

0dλ (ln z)−1∂λ

(eλ ln z

)= Anm pm

[eλ ln (pn/pm) − 1

ln (pn/pm)

]1

0

= Anm

(pn − pm

ln pn − ln pm

),

(9.48)

180

9.14. Kubo’s property and the reversible time evolution

where the factor associated to the eigenvalues of ρ is bounded (a detailed proof can befound, for example, in [563]), namely

√pn pm ≤

pn − pm

ln pn − ln pm≤ pn + pm

2. (9.49)

For a faithful state, it is comprised in the interval (0, 1]. Moreover, since the superoperatorK−1

ρ is the inverse of Kρ, we immediately get

〈pn|K−1ρ A |pm〉 = Anm

ln pn − ln pm

pn − pm(9.50)

which is perfectly defined for faithful states. In addition, K−1ρ is a completely positive

map (in fact, a Kraus map) even though Kρ is not [226].

9.14 Kubo’s property and the reversible time evolution

In this sequel, a different notation as in the main text is used to stay as general as possible.The link to the reduced and total system is established at the end. Let us now writeKubo’s property (for example given in equation (3.33) of [221] or (2.5) of [560]) in thefollowing form

[A, eB] =∫ λ

1dλ (eB)λ[A, B](eB)1−λ. (9.51)

For B=−βH, where H is an arbitrary Hamiltonian, we multiply both side of the previousequation by tr(e−βH) and use the definition of the Gibbs state ρβ = e−βH/tr(e−βH)

yielding

[A, ρβ] = −β∫ λ

1dλ ρλ

β [A, H]ρ1−λβ = −β

∫ λ

1dλ [ρλ

β Aρ1−λβ , H] = β [H, Kρβ

A]

= −iβ (−i[H, KρβA]) = −iβLrev(Kρβ

A),(9.52)

where we took advantage of the fact that ρβ and H commute to transfer Kubo’s superop-erator Kρβ

as defined in (4.18) inside of the commutator. Finally, to reach the last linewe used the generator of the reversible time evolution associated to the HamiltonianH equal to Lrev(·) = −i[H, ·]. To express the previous property for the total systemevolving under a reversible dynamics, one performs the replacements A→BTot, ρβ→$β

and Lrev→LTot giving

[BTot, $β]=−iβLTot(K$βBTot) (9.53)

181

Chapter 9. Appendix

used in the calculation (5.14). On the other hand, the replacements A→B, ρβ→π andLrev(·)→−i[HS , ·] are required for the reduced system producing the identity

[B, π] = −iβ (−i[HS , Kπ A]) 6= −iβL(Kπ A) (9.54)

with L the unperturbed generator of the reduced time evolution (4.24)–(4.25) whichcomprise the reversible and irreversible contributions.

9.15 Perturbed propagator calculation

The equation (5.27) can be proven by inserting it in the left-hand side of equation (5.26)and recover its right-hand side, namely

∂tWεt = ∂tWt + ∂t

∫ t

0ds Wt−s (Lε

t −L)Wεs

= LWt +∫ t

0ds (LWt−s) (Lε

t −L)Wεs + Wt−t (Lε

t −L)Wεt

= L(

Wt +∫ t

0ds Wt−s (Lε

t −L)Wεs

)+ (Lε

t −L)Wεt

= (L+ Lεt −L)Wε

t = Lεt Wε

t .

(9.55)

From the first to the second line we used the Leibniz integration rule when applying thetime derivative to the integral as well as the propagator time evolution (5.26) for ε = 0and finally the identity W0 = 1.

9.16 Derivative of the instantaneous Gibbs state

To compute π′s appearing in (5.31) we use the definition of the derivative and of the

instantaneous Gibbs state (5.20), that is

π′

s = limε→0

(πεs − π)

ε= lim

ε→0

1ε

(e−β(HS )ε

s

Zεs− e−βHS

Z

)

= limε→0

1ε

(e−β(HS )ε

s

Zεs

+e−βHS

Zεs− e−βHS

Zεs− e−βHS

Z

)

= limε→0

1ε

(e−β(HS )ε

s − e−βHS

Zεs

− e−βHS(

1Zε

s− 1

Z

)),

(9.56)

where the second term of the last line can be simply rewritten as a derivative

e−βHS

ZZ lim

ε→0

(1/Zεs − 1/Z)

ε= π

(Z

ddε

(1

Zεs

) ∣∣∣∣∣ε=0

)(9.57)

182

9.17. Numerical extraction of the susceptibility

so that it vanishes when the generator L is applied to the stationary Gibbs state π

according to (5.31). For the first term of the last line we can use Duhamel’s formula 3

eX − eY =∫ 1

0dλ eλX(X−Y) e(1−λ)Y (9.58)

with X = −β(HS)ε

s = −β(HS + εE(t)B) and Y = −βHS to finally get

π′

s =1Z

limε→0

1ε

(e−λβ(HS )

εs (−βεE(t)B) e−(1−λ)βHS

)= −βE(t)

∫ 1

0dλ

((e−βHS

Z

)λ

B(

e−βHS

Z

)1−λ)

= −βE(t)∫ 1

0dλ πλBπ1−λ = −βE(t)KπB,

(9.59)

where in the last line we have used the definition of the Kubo superoperator (4.18)evaluated at the Gibbs state π.

9.17 Numerical extraction of the susceptibility

The aim of this sequel is to compute the susceptibility of the system without resorting tothe FDT (5.13). To this end, let us focus on the following perturbation

(HS)ε

t = εE(t)µ (9.60)

depending on the electric field E(t) and on the dipole operator µ whose diagonal ele-ments (in the energy eigenbasis) are all zero due to the absence of permanent dipole.Computing the mean value given by (5.33) for A=µ leads to the macroscopic polariza-tion P(t). Expanding this quantity up to the first order gives

P(t) = 〈µ〉ρεt= P+ + εP(1)(t) + o(ε), (9.61)

where the static contribution P+= tr(µρ+) depends on the steady-state ρ+ while

P(1)(t) =∫ t

0ds E(t− s) χ(1)

µµ (s) (9.62)

is the first order polarization. Typically, the steady-state ρ+ is taken as the Gibbs state π

but since we do not specify the explicit form of the first order polarization P(1)(t) onecan also afford a NESS. For our choice of µ the contribution ρ+ generally vanishes. Nowlooking at the Fourier transform of the first order polarization (9.62) one can compute

P(1)(ω) =∫ ∞

0eiωtP(1)(t) = E(ω) χ(1)

µµ (ω) (9.63)

3Already used and proven in Appendix 9.9.

183

Chapter 9. Appendix

thanks to the convolution theorem which immediately gives the susceptibility

χ(1)µµ (ω) =

P(1)(ω)

E(ω). (9.64)

In practice, the electric field E(ω) is known up to a Fourier transform. It is still necessaryto select an appropriate pulse shape to sample the frequency region of interest. A smoothfunction producing a spectrum in the frequency domain with as few zero regions aspossible is often an appropriate choice since E(ω) appears at the denominator of (9.64).Numerically, P(1)(t) can be computed through a linear regression procedure startingfrom the total macroscopic polarization. More precisely, two strategies are possible. Thefirst one is to compute

P(1)(t) = limε→0

P(1)ε (t) = lim

ε→0

P(t)− P+

ε. (9.65)

To perform the limit, the idea is to run independent simulations starting from thesame steady-state ρ+ with different perturbation strength ε. One then plots the linearpolarization in terms of perturbation strength by putting on the x-intercept the increasingvalue of ε and on the y-intercept P(1)

ε (t) at time t. Once the linear region of the curve isisolated (being on the side of ε reaching zero), a linear regression is possible. The pointwhere the line crosses the y-intercept reveals the linear polarization P(1)(t). A practicalstrategy to isolate the linear region is to eliminate points starting from the largest valuesof ε up to the point when the quality of the linear fit reaches some threshold (giving anestimate of the error committed). Alternatively, one can plot the expression

P(1)ε (t)− P+ = εP(1)(t) (9.66)

for increasing ε using independent simulations as for the other approach. On the x-intercept we have ε while on the y-intercept we have the total macroscopic polarizationP(t). The advantage here is that the curve will always cross zero at the origin. Aspreviously, we have to isolate the linear region followed by a linear regression. The slopeextracted from the regression finally gives P(1)(t) at time t (the error is also estimatedthrough the fit quality). In case one would like to compute the susceptibility for a NESS,one has to perform a simulation prior to the application of this scheme to have accessto the steady-state ρ+. Note that if one has access to the generator of the dynamics Lin the Liouville representation 4, ρ+ can immediately be obtained from the eigenvectorassociated to the zero eigenvalue. This avoids performing a full simulation of thedecoherence process (i.e. until the steady state is reached) which can be very timeconsuming.

4Possible if the master equation is linear and the dimension of the system is reasonable. For the NTMEs,one could to linearize the irreversible contribution Jk associated to each reservoir k using the local Gibbsstate πk, which is discussed in Sec. 4.4.1, giving Jk in the form (4.23) to construct the generator L.

184

9.18. Modified Spohn and Lebowitz Lemma calculation

9.18 Modified Spohn and Lebowitz Lemma calculation

To compute the Onsager coefficient following the mentioned Lemma 1 of [37] we onlyhave to recompute (VII.5) therein. It depends on the irreversible generator Jl,β whichis explicitly evaluated at βl = β. This is because in the Onsager coefficient (5.46)one performs the derivative ∂/∂xl and then evaluates it at the equilibrium inversetemperature β. Thereby, we have [77, 78]

Jl,β(πHS) = Jl,β(πλHSπ1−λ) = Jl,β

(∫ 1

0dλ πλHSπ1−λ

)= Jl,β(Kπ(HS))

= Kπ(J ∗l,β(HS)) = Kπ(Jl,βQ ).

(9.67)

In the first line we have used the fact that π and HS commute and the definition of Kobu-Mori equilibrium superoperator (4.18). In the second line, Grabert’s detailed balancewas applied in the form (5.11) for the sole irreversible part Kπ(J ∗l,β(A)) = Jl,β(Kπ(A))

and the heat flux operator (5.49) evaluated at β, i.e. Jl,βQ = J

l,βlQ |βl =β, was introduced.

According to (9.67), we trade in Eq. (VII.3) of Lemma 1 in [37] the operator Jl,βQ π with

Kπ(Jl,βQ ) becoming in our case

Lkl(β) = −tr(

Jk,βQ (t)

1Lk,β

Kπ(Jl,βQ )

). (9.68)

The generator Lk,β comes from the generator L associated to the linearized equation(4.24)–(4.25) but with all reservoirs set at βk =β. Note that, in general, the two versionsJ

l,βQ π and Kπ(J

l,βQ ) disagree. However, for the LDME they match because J

l,βQ commutes

with HS (and π). More precisely, the GKF (5.37) for the LDME (3.38) can be written

Lkl =∫ ∞

0dt tr

(Jk(t)

∫ 1

0dλ πλL∗l,β(HS)π

1−λ

). (9.69)

Using the Heisenberg representation (2.53) of the LDME for HS gives

L∗l,β(HS) = ∑ω

hl,β(ω)([A†l,ω, HS ]Al,ω + A†

l,ω[HS , Al,ω]) = −∑ω

hl,β(ω)ωA†l,ω Al,ω, (9.70)

where we applied [HS , Al,ω] =−ω, naturally leading to the relation [π,L∗l,β(HS)] = 0since [HS , A†

l,ω Al,ω] = 0 according to equation (3.84) of [20]. Clearly, the GKF (9.69)

becomes Lkl =∫ ∞

0 dt tr(

Jk,βQ (t)J l,β

Q π)

matching Spohn and Lebowitz expression (VII.4).

Note also that the detailed balance L∗k,β = K−1π Lk,βKπ and L∗k,β = L(π−1)Lk,βL(π) are

equivalent for the LDME (2.42). The underlying reason for this is that the linearizationof the seemingly nonlinear form of the LDME, appearing in equation (3.33) of Sec. 3.2.1,is the LDME itself thanks to the property (3.36).

185

Chapter 9. Appendix

9.19 Decomposition of the Kubo superoperator

We can rewrite the Kubo superoperator (4.18) to extract a linear contribution, namely [146]

Kρ(A) =12

({A, ρ}+ Kρ(A)

), (9.71)

where we define a purely nonlinear contribution

Kρ(A) =∫ 1

0dλ[[

ρλ, A]

, ρ1−λ]

. (9.72)

To prove this relation, let us isolate Kρ(A) and reorganize the two other terms as follows

Kρ(A) = 2(∫ 1

0dλ ρλ A ρ1−λ

)− (Aρ + ρA) , with

∫ 1

0dλ ρλρ−λ = 1

= 2(∫ 1

0dλ ρλ A ρ1−λ

)−∫ 1

0dλ(

Aρρλρ−λ + ρλρ−λρA)

=∫ 1

0dλ(

2ρλ A ρ1−λ − Aρλρ1−λ − ρλρ1−λ A)

=∫ 1

0dλ(

ρλ A ρ1−λ − Aρλρ1−λ − ρλρ1−λ A + ρλ A ρ1−λ)

, with (1− λ) → λ

=∫ 1

0dλ(

ρλ A ρ1−λ − Aρλρ1−λ −(

ρ1−λρλ A− ρ1−λ A ρλ))

=∫ 1

0dλ((

ρλ A− Aρλ)

ρ1−λ − ρ1−λ(

ρλ A− Aρλ))

=∫ 1

0dλ([

ρλ, A]

ρ1−λ − ρ1−λ[ρλ, A

])=∫ 1

0dλ[[

ρλ, A]

, ρ1−λ]

.

(9.73)

Moreover, Grabert assumed in [141] that at low temperature one can get the identityKπ(A) = (1/2){A, π}, where π is the Gibbs state assuming a single reservoir is con-nected to the system, implying that Kπ(A)=0 to obtain a linear equation. On the otherhand, Öttinger used in [205] that Kρ(A)' (1/2){A, ρ} to obtain the Caldeira-Leggettmaster equation from the nonlinear equation (3.66). Not that none of these two ap-proximations are ”controlled“and should be used with caution (in general one shouldlinearize the equation near equilibrium as we did in Sec. 4.4).

186

9.20. Small momentum exchange limit

9.20 Small momentum exchange limit

Starting from the dynamically time-averaged NTME (5.99) with the spectral function(5.101) in conjunction with the singular structure factor | f (q)|2=∂2

q (δD(q)g(q)) with theregularization function g(q)=(γνq)/(2δωc(0)) given by δωg(0)=1/(

√2πωg) we get

J (ρ) =12

∫R

dq(

∂2q (δD(q)g(q)) Nνq eβνq [A†

q,νq, K−νq

ρ [Aq,νq , Ψ]]

+ ∂2q (δD(q)g(q)) Nνq [A

†q,−νq

, Kνqρ [Aq,−νq , Ψ]]

)=

12

∫R

dq(

δD(q)g(q) ∂2q

(Nνq eβνq [A†

q,νq, K−νq

ρ [Aq,νq , Ψ]])

+ δD(q)g(q) ∂2q

(Nνq [A

†q,−νq

, Kνqρ [Aq,−νq , Ψ]]

))=

12

∫R

dq(

δD(q)g(q)βνq

eβνq [∂q A†q,νq

, K−νqρ [∂q Aq,νq , Ψ]]

+ δD(q)g(q)βνq

[∂q A†q,−νq

, Kνqρ [∂q Aq,−νq , Ψ]]

)=

γ

4βδωg(0)

( [∂q A†

q,νq|q=0, Kρ

[∂q Aq,νq |q=0, Ψ

]]+[∂q A†

q,−νq|q=0, Kρ

[∂q Aq,−νq |q=0, Ψ

]] )=

γ

2βδωg(0)[A0, Kρ[A0, Ψ]].

(9.74)

In the second line we eliminate the integral over the frequencies by applying the twoDirac-delta functions present in the spectral function hq(ν) as defined in (5.101) andintegrate twice by parts neglecting the boundary terms. In the third line we computeNνq = 1/(eβνq − 1)→ 1/(βνq) since we imposed earlier |q| � 1. In the fourth line weapply the partial derivatives and use the fact that [Aq,±νq |q=0, B] = [A†

q,±νq|q=0, B] = 0,

for B an arbitrary operator, since Aq,−νq |q=0 = δωc(0)1 is diagonal. In the fifth line weapply the Dirac-delta δD to eliminate the integral over the momentum as well as thesuperoperator evaluated at νq =0 becoming K0

ρ =Kρ. In the last line we use the short-hand notations A0 = ∂q Aq,νq |q=0 valid for the four terms because evaluated at zeromomentum. Now using the modified eigenoperators (5.100) we can compute an explicitexpression for the Hermitian operator A0 of the previous equation, namely

A0 = ∂q

(e

ih X·q

√δωg

(ε(p + q)− ε(p)− νq

)) ∣∣∣∣∣q=0

=

eih X·q

√δωg

(ε(p + q)− ε(p)− νq

)− 1

√δωg (0)

q

∣∣∣∣∣q=0

,

187

Chapter 9. Appendix

where we use the finite difference representation of the derivative. Now, we compute

A0 =

eih X·q − 1

q

√δωg

(ε(p + q)− ε(p)− νq

)+

√δωg

(ε(p + q)− ε(p)− νq

)−√

δωg(0)

q1

∣∣∣∣∣q=0

=

(iX√

δωg

(ε(p + q)− ε(p)− νq

)) ∣∣∣∣∣q=0

+ 1 ∂q

(√δωg

(ε(p + q)− ε(p)− νq

)) ∣∣∣∣∣q=0

= iX√

δωg(0) + 1 ∂q

(√δωg

(ε(p + q)− ε(p)

qq− c|q|

)) ∣∣∣∣∣q=0

= iX√

δωg(0) + 1 ∂q

(√δωg

(∇pε(p) q− c|q|

))∣∣∣∣∣q=0

= iX√

δωg(0) + 1 ∂q

(√δωg

( pm

q− c|q|))∣∣∣∣∣

q=0

= iX√

δωg(0) + 1

(1

4ω2c

∂q

( pm

q− c|q|)2√

δωg

( pm

q− c|q|)) ∣∣∣∣∣

q=0

= iX√

δωg(0) + 1

(1

4ω2c

∂q

(p2

m2 q2 − 2pm

cq|q|+ c2|q|2)√

δωg

( pm

q− c|q|)) ∣∣∣∣∣

q=0

= iX√

δωg(0) + 1

(1

4ω2c

(p2

m2 2q− 2pm

cq∂|q|∂q− 2

pm

c|q|+ c22|q|∂|q|∂q

)√δωg

( pm

q− c|q|)) ∣∣∣∣∣

q=0

= iX√

δωg(0) + 1

(1

4ω2c

(p2

m2 2q− 2pm

cq2

|q| − 2pm

c|q|+ c22q)√

δωg

( pm

q− c|q|)) ∣∣∣∣∣

q=0

so that A0= iX√

δωg(0). We add and subtract√

δωg

(ε(p + q)− ε(p)− νq

)/q in the first

line to create two derivatives and in the second line we use the definition of the gradientin terms of finite differences (ε(p + q)− ε(p))/q =∇p ε(p) = p/m for ε(p) = p2/2m.To reach the last line we resort to the definition of the absolute value’s derivative∂|q|/∂q= ∂q|q|= q/|q| and then assumed that q2/|q| evaluated at q=0 vanishes 5. Wecan now come back to the calculation of the master equation (9.74) and use the Massieuoperator (3.11) in conjunction with the operator A0 we just computed to rewrite it as

J (ρ) =− γ

2[X, Kρ[X, HS ]]−

γ

2β[X, Kρ[X, ln ρ]] = −1

2

(i

γ

m[X, Kρ(P)] +

γ

β[X, [X, ρ]]

)=− i

2Dp[X, Kρ(P)]− Dpp[X, [X, ρ]] with Dp =γ/(4m) and Dpp =γ/(2β).

We applied the property Kρ[A, ln ρ]= [A, ρ] of Appendix 9.7, the free particle Hamilto-nian HS=P2/(2m) and the fact that [X, P2]=P[X, P] + [X, P]P=2iP.

5This is necessary for our choice of dispersion relation νq = c|q| because the derivative ∂q|q| is notdefined at q=0; such a singularity is avoided when considering a quadratic dispersion νq = cq2 or any otherfunction being differentiable at q=0 so that it is reasonable to make this assumption on a physical ground.

188

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Curriculum VitaeJérôme Flakowski

Address : Bergstrasse 23CH 8105 Regensdorf

B [email protected] of birth : 8.11.1979

Nationality : Swiss and German

Education and Qualifications2010–2016 PhD: Polymer Physics group, Department of Materials (ETHZ).

Supervisor : Prof. Dr. Hans Christian Oettinger.Lectures: Nonequilibrium Systems, Quantum Information, Polymer physics, Transport Phe-nomena, Quantum Chemistry, Molecular Quantum Optics and Spectroscopy.

2009-2010 Assistant: Laboratory of Statistical Biophysics, ITP, EPFL, Lausanne, CH.Wrote a paper in a EPFL-ETHZ collaboration with the Laboratory of Food and Soft Materials.

2007–2009 Master in Physics: EPFL, Laboratory of Statistical Biophysics, Lausanne, CH.Supervisor: Associate Prof. Paolo De Los Rios, Project: Drug-DNA Recognition.

2004–2007 Bachelor in Physics: EPFL, Lausanne, CH. Lectures: Statistical Physics, ProbabilityTheory, Linear Algebra, Basic Electronics, Quantum Physics, Classical/Quantum Optics,Solid State and Plasma Physics, Group Theory, Quantum Field Theory, Biophysics...

2002–2004 Master in Biology: UNIGE, Department of Genetic and Evolution, Geneva, CH.Supervisor: Associate Prof. Jan Pawlowski, Project: Intron Evolution in Actin of Formainifera.

1999–2002 Bachelor in Biology: UNIGE, Geneva, CH. Lectures: Physiology, Embryology, En-docrinology, Cellular Biology, Genetics, Evolution, Biochemistry, Bioinformatics...

1995–1999 Swiss Maturity (High School): Scientific Section, Collège Voltaire, Geneva.

Work experience2010-2016 Teaching Assistant in the Polymer Physics group, ETHZ.

Courses: Thermodynamics, Statistical and Polymer physics, Rheology Lab sessions, Matlab.Writing three papers based on the PhD project (2012, 2014, 2016).

2009–2010 Writing one paper using tools developed during the Physics Master project (2010).2007-2009 Master project in physics: EPFL, Lausanne, CH.

Designed a program allowing the simulation and visualization of molecular system in real-time (OpenGL), for example, interaction of DNA-ligands and proteins fibers.

2004–2005 Programming Project: EPFL, Lausanne, CH.Elaborated a program simulating ants in a basic environment (predator-prey interactions).

2004–2006 Three papers issued from the Master project in Biology (2004, 2005, 2006).2002–2004 Laboratory work: UNIGE, Geneva, CH.

Molecular biology lab technics (DNA/RNA isolation, cloning, PCR,...)

Computing skillsLanguages Pascal, C/C++, Python, OpenGL, Qt Packages GSL, Lapack, libQGLViewerPlatforms Windows, Linux, Solaris Applications Microsoft/OpenOffice, GIMP

Tools Matlab, Mathematica, LATEX Phylogenetic PAUP, MrBayes, PhyML

LanguagesFrench NativeEnglish FluentGerman FluentItalian Basic

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Publications during the PhD2016 Biexponential decay and ultralong coherence of a qubit, Europhys. Lett. v.113(4), 40003,

J. Flakowski, M. Osmanov, D. Taj, and H. C. Öttinger.2014 Time-driven quantum master equations and their compatibility with the fluctuation

dissipation theorem, Phys. Rev. A v.90, 042110, J. Flakowski, M. Osmanov, D. Taj, andH. C. Öttinger.

2012 Stochastic process behind nonlinear thermodynamic quantum master equation. II. Sim-ulation, Phys. Rev. A v.86, 032102, J. Flakowski, M. Schweizer and H. C. Öttinger.

Publications prior to the PhD2010 Understanding amyloid aggregation by statistical analysis of atomic force microscopy

images, Nat. Nanotechnol. v.4, p.423–428, J. Adamcik , J.-M. Jung, J. Flakowski, P. DeLos Rios, G. Dietler and R. Mezzenga.

2006 Tempo and Mode of Spliceosomal Intron Evolution in Actin of Formainifera, J. Mol. Evol.v.63(1), p.30–41, J. Flakowski, I. Bolivar, J. Fahrni and J. Pawlowski.

2005 Actin phylogeny of foraminifera, J. Foramin. Res. v.35(2), p.93–102, J. Flakowski, I.Bolivar, J. Fahrni and J. Pawlowski.

2004 Multigene evidence for close evolutionary relations between Gromia and Foraminifera,Acta Protozool v.43, p.303–311, D. Longet, F. Burki, J. Flakowski, C. Berney, S. Polet,J. Fahrni, J. Pawlowski.

Conferences and Workshops2016 Physics of Biology II, Campus Biotech, Geneva (Switzerland): Poster presentation.2015 Quantum effects in Biological Systems (QuEBS), Florence (Italy): Poster presentation.2013 Mathematical Horizons for Quantum Physics 2, Singapore: Workshop.2013 Materials Day – Today, Tomorrow and Beyond: Conference.2012 6th International Workshop on Nonequilibrium Thermodynamics (IWNET), Røros

(Norway): Poster presentation.2011 Quantum effects in Biological Systems (QuEBS), Ulm (Germany): Poster presentation.2011 Quantum information and foundations of thermodynamic, ETHZ: Workshop.2010 Swiss Soft Days (SSD), 2nd Edition, EPFL: Workshop.2010 Swiss Soft Days (SSD), 1st Edition, Zürich: Workshop.

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thermodynamics of mesoscopic quantum systems: from a single qubit to light harvesting complexes

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