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Selective creation of maximum coherence in multi-level Λ systemPraveen Kumarab, Svetlana A. Malinovskayaa, Ignacio R. Solac & Vladimir S. Malinovskyad

a Department of Physics, Stevens Institute of Technology, Hoboken, NJ, USAb Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, TX, USAc Departamento de Quimica Fisica I, Universidad Complutense, Madrid, Spaind Army Research Laboratory, Adelphi, MD, USAAccepted author version posted online: 03 Jun 2013.Published online: 27 Jun 2013.

To cite this article: Praveen Kumar, Svetlana A. Malinovskaya, Ignacio R. Sola & Vladimir S. Malinovsky (2014) Selectivecreation of maximum coherence in multi-level Λ system, Molecular Physics: An International Journal at the InterfaceBetween Chemistry and Physics, 112:3-4, 326-331, DOI: 10.1080/00268976.2013.809166

To link to this article: http://dx.doi.org/10.1080/00268976.2013.809166

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Molecular Physics, 2014Vol. 112, Nos. 3–4, 326–331, http://dx.doi.org/10.1080/00268976.2013.809166

INVITED ARTICLE

Selective creation of maximum coherence in multi-level � system

Praveen Kumara,b, Svetlana A. Malinovskayaa, Ignacio R. Solac and Vladimir S. Malinovskya,d,∗

aDepartment of Physics, Stevens Institute of Technology, Hoboken, NJ, USA; bDepartment of Chemistry and Biochemistry, Texas TechUniversity, Lubbock, TX, USA; cDepartamento de Quimica Fisica I, Universidad Complutense, Madrid, Spain; dArmy Research

Laboratory, Adelphi, MD, USA

(Received 29 April 2013; final version received 20 May 2013)

We consider the creation of the maximum Raman coherence in the six-level � system using optimal control theory. Optimalfields are designed for different initial conditions, resonant, and off-resonant, using the Krotov method including a referencefield into the cost functional. Suppression of the population transfer to the intermediate level is achieved via an additionalfunctional constraint which depends on the system dynamics. We demonstrate that the spectrum of the optimised fields hasmajor contribution from the corresponding resonant frequencies independently of the choice of carrier frequency of theinitial guess field. We also indicate that the pulse train emerges as a solution of the control problem of coherence optimisationin multi-level quantum systems.

Keywords: optimal control theory; coherent population transfer; Raman coherence; pulse train

1. Introduction

Efficient control of population transfer in multi-level sys-tems has been an ultimate goal for some time in a vari-ety of research areas including laser atomic and molecularspectroscopy, photochemistry, biology, and quantum infor-mation processing. During the last few decades, optimalcontrol theory (OCT) [1–6] has been applied successfullyto a broad variety of physical and chemical systems [7–13] and can be considered as a universal methodology forpulse design with specific goals. OCT provides a variationalframework for designing the optimal pulse shape to controlthe dynamics of a quantum system and to reach a desiredphysical objective at the given target time. It may also pro-vide a flexibility when specific constraints are to be system-atically imposed on the evolving system and, therefore, hasa wealth of applications. The equation for the optimal fieldthat emerges from OCT is usually solved numerically in aniterative fashion, starting from an initial guess of the controlfield which gets changed at each iteration step depending onits performance and ideally reaching the optimal solutionof the dynamic equations maximising the overlap with thetarget wave function or density matrix.

It is clearly foreseeable and commonly recognised thatthe application of OCT to a complex quantum system canprovide a very precise optimal field as a desired solution.However, depending on the complexity of the structure ofthe Hamiltonian which governs the system dynamics, thedesigned shape of the field might be extremely complex.Therefore, it might be very difficult or even impossible

∗Corresponding author. Email: vsmalinovsky@gmail.com

to analyse the designed field and extract a control mech-anism. In addition, to make the optimal solution usefulfor potential experimental implementation one should keepin mind the current level of technological development.In essence, the feasibility for the experimental implemen-tation must be taken into account as one of the crucialfactors in OCT methodology. In some cases, the experi-mental scheme may be included in the optimisation pro-cedure and impose additional constrains on the optimalsolution.

In this paper, we apply the OCT algorithm to create aspecific wave function form of the multi-level system whichprovides maximum coherence on several preselected tran-sitions. The considered problem is related to the Ramanspectroscopy and, potentially, imaging based on coherentanti-Stokes Raman scattering (CARS), since maximum co-herence is directly connected to the maximising CARSsignal.

The paper is organised as follows. Section 2 presentsbasic equations of the OCT obtained using the variationalcalculus. Details of the optimal control implementation forthe six-level � system are in Section 3, where the optimalfield equations are derived using the penalty on the energyof the control field. An additional penalty function is appliedto minimise the population of the excited intermediate state.Section 4 provides an illustrative example of the creationof the maximum coherence between selected states in thesix-level system using the formalism of Section 3. FinalSection 5 is the conclusion.

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2. General OCT equations

In this section, we outline an OCT approach to optical pulseshaping to transform the quantum system wave function,|ψ(t)〉, having maximum overlap with the target wave func-tion, |φ(T)〉, at some final time T. The theory of OCT is basedon the variational principle of the cost functional J[ε(t)]which, when it is maximised, provides an optimal controlfield that successfully performs the desired transformation.In order to derive an equation for the control field ε(t) andobtain realistic field amplitude one usually penalises the en-ergy of the field. In some cases, it is also beneficial to addanother requirement to minimise the population of the ex-cited intermediate states throughout the population transferprocess [8,14,15] that can be done by introducing a pro-jection operator P = |ψint(t)〉〈ψint(t)| = ∑

k |k〉〈k|, where|k〉 is the eigenket of the unwanted intermediate state. Obvi-ously, the wave function |ψ (t)〉 must satisfy the Schrodingerequation and it is accomplished through an additional con-straint using the Lagrange multiplier.

All mentioned above requirements lead to a completecost functional of the form

J [ε(t)] = ∣∣〈ψ(T )|φ(T )〉∣∣2 − α(t)∫ T

0dt

[ε(t) − εr (t)

]2

−β

∫ T

0dt〈ψ(t)|P |ψ(t)〉 − 2Re

[ ∫ T

0dt

⟨χ (t)

∣∣∣ ∂

∂t

+ i

�H

∣∣∣ψ(t)⟩]

, (1)

where α(t) is the time-dependent penalty function whichdetermines the significance of the field energy and alsoallows to adjust the envelope of the control field (it can be

crucial for future experimental tests), β is the penalty pa-rameter for the total population of the intermediate state(can be easily generalised for the manifold of states), andεr(t) denotes a reference field. The function |χ (t)〉 is the La-grange multiplier introduced to ensure satisfaction of theSchrodinger equation.

Applying the variational principle to the cost functional,J[ε(t)], with respect to |χ (t)〉, |ψ(t)〉, and ε(t) leads to thefollowing set of equations:

i�∂

∂t|ψ(t)〉 = H |ψ(t)〉 , (2)

∂

∂t|χ (t)〉 = − i

�H |χ (t)〉 + βP |ψ(t)〉 , (3)

Figure 1. Schematic of a six-level system in the � configuration.

ε(t) = εr (t) + 1

α(t)�Im

⟨χ (t)

∣∣∣ ∂H

∂ε(t)

∣∣∣ψ(t)⟩, (4)

and variation with respect to |ψ(T)〉 gives the ini-tial condition for the Lagrange multiplier |χ (T)〉 =|φ(T)〉〈φ(T)|ψ(T)〉. Equations (2) and (3) determine the evo-lution of the initial wave function and the Lagrange multi-plier function which are used in Equation (4) to determinethe optimal field maximising the overlap |〈ψ(T)|φ(T)〉|2.

3. Optimal control equations for the six-level� system

Now we adapt the general optimisation procedure to the six-level � type system shown in Figure 1. The dynamics of thesystem wave function |ψ(t)〉 = ∑6

i=1 ai(t)|i〉, where ai(t) isthe probability amplitude to be in the state |i〉, is governedby the Schrodinger equation with the Hamiltonian of theform

H =

⎛⎜⎜⎜⎜⎜⎜⎝

E1 −μ12εP (t) 0 0 0 0−μ21εP (t) E2 −μ23εS(t) −μ24εS(t) −μ25εS(t) −μ26εS(t)

0 −μ32εS(t) E3 0 0 00 −μ42εS(t) 0 E4 0 00 −μ52εS(t) 0 0 E5 00 −μ62εS(t) 0 0 0 E6

⎞⎟⎟⎟⎟⎟⎟⎠

, (5)

where Ei is the energy of the state |i〉, μij are the dipolemoments, and εP, S(t) are the pump and Stokes fields.

The goal of the optimal control problem is to designthe shape of the pump and Stokes pulses providing a max-imal coherent superposition of the ground state |1〉 and thetwo adjacent states, |4〉 and |5〉 while the population of theintermediate state |2〉 and the two other states, |3〉 and |6〉is minimised. Creation of this maximal superposition pro-vides maximum coherence on |1〉 ↔ |4 〉 and |1〉 ↔ |5〉transitions that will generate maximum CARS signal at the|4〉 ↔ |5〉 transition frequency [16]. To fulfill the optimi-sation goal we chose the population distribution (1/3, 0,0, 1/3, 1/3, 0) as the target and neglect relative phase ar-rangements between states in the target wave function. Here

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we address the resonant and off-resonant excitation of thesingle-photon transitions.

In the Schrodinger representation, Equations (2), (3),and (4) for the Lagrange multiplier vector bi(t), probabilityamplitude vector ai(t), and external fields εP, S(t) take thefollowing form:

i�∂ai(t)

∂t= Hij aj (t) , (6)

∂bi(t)

∂t= − i

�Hij bj (t) + βa2(t)δi2 , (7)

εP (t)

= εrP (t) − 1

2�α(t)· Im

[b�

1(t)μ12a2(t) + b�2(t)μ21a1(t)

],

(8)

and

εS(t) = εrS(t) − Im

[b�

2(t)μ23a3(t) + b�2(t)μ24a4(t)

+ b�2(t)μ25a5(t) + b�

2(t)μ26a6(t) + b�3(t)μ32a2(t)

+ b�4(t)μ42a2(t) + b�

5(t)μ52a2(t) + b�6(t)μ62a2(t)

]/ [2�α(t)] , (9)

where εrP (t) and εr

S(t) are the pump and Stokes referencefields.

4. Optimal control of coherence

To find the optimal pump and Stokes fields which maximisethe cost functional in Equation (1) we solve the set of theoptimal control Equations (6)–(9) using the iterative proce-dure known as the Krotov method [8]. We choose Gaussianform for the pump and Stokes pulses with duration of 500fs as an initial guess and the target time is equal to T =4 ps. The transition frequencies are chosen to be ω21 =0.517 fs−1, ω23 = 0.395 fs−1, and spacing in the four-levelmanifold is ω34 = ω45 = ω56 = 0.02 fs−1.

The optimised results obtained for the resonance excita-tion, when the applied pump ωP and Stokes ωS frequenciesare equal, respectively, to the transition frequencies ω21

and ω24, are shown in Figures 2 and 3. The upper panel inFigure 2 shows the state populations as a function of timeobtained without applying the penalty on the populationof the intermediate state, |2〉 (β = 0). At the target time,T = 4 ps, we obtain maximum coherences |ρ14| ≈ |ρ15| ≈1/3 (ρij = a∗

i aj ) and the equal state populations |a1|2 ≈ |a4|2

≈ |a5|2 ≈ 1/3 while the states |3〉 and |6〉 are almost emptyall the times. We see that the intermediate state population,|a2|2, increases up to about 70% at the middle and then goesto zero at the end of the pulse. Note, that the optimised tran-sition probability defined as P = |〈ψ(T )|φ(T )〉|2, and thefinal optimised cost functional J[ε(t)] reaches nearly 100%after only a few hundred iterations.

Figure 2. Optimised population transfer in the six-level � sys-tem for the resonant initial guess fields without penalty on theexcited state population. The upper panel: population of the states|1〉 − |6〉, negligibly small population of the states |3〉 and |6〉 arenot marked; the lower panel: the optimal pump and Stokes fields.

Optimised pump and Stokes fields (obtained after 1000iterations) are shown in the lower panel of Figure 2. Thespectrum of the optimised pump and Stokes fields is verysimple. The pump spectrum has only one frequency at0.517 fs−1 that corresponds to the transition frequency ω21,

Figure 3. Optimised population transfer in the six-level � sys-tem for the resonant initial guess fields using penalty on the ex-cited state population. The upper panel: population of the states|1〉 − |6〉, negligibly small population of the states |3〉 and |6〉 arenot marked; the middle and lower panel: the optimal pump andStokes fields.

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and the Stokes spectrum contains only two frequencies at0.375 fs−1 and 0.355 fs−1, these lines correspond to thetransition frequency ω24 and ω25, respectively. From thepopulation dynamics shown in Figure 2 and the character-istics of the optimised fields we can extract the mechanismof population dynamics. In essence, we obtained the opti-mised results by using a pair of partially overlapped pulseswhich are applied in the intuitive order and the popula-tion dynamics is controlled by the area of the pulses. Itlooks like the Stokes pulse has a more complex structure.However, a detailed analysis shows that the time-dependentenvelope of the Stokes field is simply the result of interfer-ence between two slightly different frequencies equal to thetransition frequencies ω24 and ω25. The beat frequency isexactly equal to the spacing in the four-level manifold, ω45

= 0.02 fs−1. The ‘pulse-train’ structure of the Stokes fieldemerges automatically from the optimisation procedure andit demonstrates that the pulse train is the optimal solutionfor the selective state excitation in the manifold of closelylocated levels.

Figure 3 shows the optimisation results obtained by ap-plying an additional penalty on the population of the state|2〉, β �= 0. As we see, the population dynamics is very sim-ilar to the previous case, however, now the transient pop-ulation of the state |2〉 is considerably reduced (the upperpanel in Figure 3), in contrast to the unconstrained opti-misation results (Figure 2). Optimised pump and Stokesfields (obtained after 5000 iterations) are shown in the mid-dle and lower panels of Figure 3, respectively. Imposinga penalty on the transient population of state |2〉 providesmore intense pulses in comparison to the unconstrainedoptimisation case (Figure 2). Moreover, a higher numberof iterations are required to reach the final target goal, amaximum value of the optimised cost functional J[ε(t)].

The spectrum of the optimised pump and Stokes fieldsis shown in Figure 4. The pump spectrum (the upperpanel in Figure 4) has three major lines: one at frequency0.517 fs−1 corresponding to the transition frequency ω21

and two others at frequencies 0.537 fs−1 and 0.497 fs−1,respectively. The Stokes spectrum (the lower panel in Fig-ure 4) has two major peaks at 0.375 fs−1 and 0.355 fs−1

corresponding to the transition frequencies ω24 and ω25. Aswe see, a pulse train structure is clearly revealed now inthe optimised fields. It is also notable that the Stokes pulsetrain precedes the pump similar to the well-known coun-terintuitive pulse sequence in STIRAP (Stimulated Ramanadiabatic passage) [14,17]. It is our understanding that thepenalty on the excited state population is responsible forthe counterintuitive pulse sequence arrangement since theoptimised results without this penalty show the intuitivepulse sequence as shown in the lower panel of Figure 2.

Now we consider the sensitivity of the optimisationresults demonstrated previously to the choice of the ini-tial guess field. In particular, we analyse the robustness ofthe optimisation procedure to the initially imposed single-

Figure 4. Spectrum of the optimal fields shown in Figure 3: (a)– pump pulse and (b) – the Stokes pulse.

photon resonance. To do that we repeat the optimisationprocess by using off-resonant initial guess fields. The valueof the single-photon detuning is chosen to be �P, S = ω21 −ωP = ω24 − ωS = � = 0.1 fs−1. The rest of the parametersused in the optimisation process is kept the same as in thecase of resonant initial guess fields.

Figures 5 and 6 illustrate the results without and withpenalty on the excited state population. In both cases, the

Figure 5. Optimised population transfer in the six-level � sys-tem for the off-resonant initial guess fields without penalty on theexcited state population. The upper panel: population of the states|1〉 − |6〉, negligibly small population of the states |3〉 and |6〉 arenot marked; the lower panel: the optimal pump and Stokes fields.

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Figure 6. Optimised population transfer in the six-level � sys-tem for the off-resonant initial guess fields using penalty on theexcited state population. The upper panel: population of the states|1〉 − |6〉, negligibly small population of the states |3〉 and |6〉 arenot marked; the middle and lower panel: the optimal pump andStokes fields.

optimised population dynamics is practically identical tothe results shown in the upper panel of Figures 2 and 3.Here again, we obtained almost perfect overlap with thetarget population distribution at the final time T = 4 psindependently of the penalty parameter. Optimised pumpand Stokes fields are shown in the lower panels of Figures

Figure 7. Spectrum of the optimal fields shown in Figure 6: (a)– pump pulse and (b) – the Stokes pulse.

5 and 6 and closely resemble the optimal fields with theresonant initial condition, Figures 2 and 3.

The spectrum of the pump and Stokes fields inFigure 7 shows a major peak at 0.517 fs−1, which cor-responds to the transition frequency ω21 and two side-bandpeaks at frequencies 0.537 fs−1 and 0.497 fs−1. Two majorpeaks in the Stokes field spectrum in the lower panel ofFigure 7 at frequencies 0.375 fs−1 and 0.355 fs−1 corre-spond to the transition frequencies ω24 and ω25. In essence,by comparing Figures 4 and 7 we can conclude that thespectrum of the optimal fields is independent of the initiallychosen spectral characteristics. However, there are residualtraces corresponding to the central frequency of the initiallychosen guess fields, at frequency 0.417 fs−1 for the pumpand 0.275 fs−1 for the Stokes field. This shift of the centralfrequency of the optimised fields to the single-photon res-onance should be related to the penalty on the pulse energyimposed in the optimisation procedure, since the resonanttwo-photon population transfer is more efficient than theoff-resonant one and it requires less energy in the pulses.

5. Conclusion

In the paper, we have applied OCT to design optimal pulsesequences for the creation of a maximum coherence in thesix-level � system. Using the Lagrange multiplier tech-nique with penalty on the field energy and intermediatestate population we derived a set of control equations forthe optimisation problem. To obtain the numerical solutionof the problem, we have implemented the Krotov methodusing a reference field εr

P (S)(t) in the penalty term appliedto the intensity of the external fields. The performance ofKrotov method was tested and applied to create the maxi-mum coherence in the six-level system. It was shown thatthe OCT procedure is insensitive to the initial field param-eters and generates the optimal fields independently of thechoice of the initial guess fields.

Optimal fields were designed for the resonant and off-resonant initial guess fields to create the maximum coher-ence |ρ14(T)| = |ρ15(T)| at the target time T = 4 ps. Theobtained values of optimised transition probability and costfunctional J[ε(t)] are nearly 100%. The peak amplitude ofthe optimised fields is within reasonable and experimentallyrealisable boundaries. We have demonstrated that the tran-sient population in state |2〉 can be considerably reducedwithout major adjustment of the control fields. Note, thetarget time (T = 4 ps) can be reduced down to 1 ps withalmost no loses in the optimised transition probability andoptimised cost functional J[ε(t)] with some small modifi-cations in the optimal fields.

We have demonstrated that the pulse-train structure ofthe pump and Stokes fields emerges automatically fromthe optimisation procedure. Therefore, it is proven that thepulse train is the optimal solution for a selective state ex-citation in the manifold of closely located levels. We have

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also illustrated the direct connection of the counterintuitiveSTIRAP-type pulse sequence with the constraint on thetransient population of the intermediate state in the multi-level � system.

AcknowledgementsThis research was supported in part by the National Science Foun-dation under Grant No. NSF PHY11-25915 and PHY12-05454.

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