.

CONTRIBUTED TALKS

@QFLM 2017

(Alphabetic order for speakers’ names)

Controlling quantum information in open quantum systems

Victor V. Albert,1 Chao Shen,1 Stefan Krastanov,1 Kyungjoo Noh,1 Barry Bradlyn,2 Martin Fraas,3

R. J. Schoelkopf,1 Mazyar Mirrahimi,1, 4 S. M. Girvin,1 Michel H. Devoret,1 and Liang Jiang1

1Yale Quantum Institute and Departments of Applied Physics and Physics,Yale University, New Haven, Connecticut 06511, USA

2Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08540, USA3Instituut voor Theoretische Fysica, KU Leuven, Leuven 3000, Belgium

4QUANTIC project-team, INRIA of Paris, Paris 75012, France(Dated: November 30, 2016)

Lindbladians, one of the simplest extensions of Hamiltonian-based quantum mechanics, are usedto describe decay and decoherence of a quantum system induced by the system’s environment.While traditionally viewed as detrimental to fragile quantum properties, a tunable environmento↵ers the ability to drive the system toward steady-state subspaces, which can be used to store,protect, and process quantum information. A prominent example of a steady-state subspace is acat code — a subspace spanned by superpositions of well-separated coherent states and protectedfrom certain errors. We report on two schemes designed to process quantum information in catcodes using Hamiltonian-based [1] and holonomic [2] control. Both schemes have been extendedto general steady-state subspaces [3]. While such control schemes can be used for (unitary) gates,they are not su�cient to perform (nonunitary) error correction. We discuss two ways to implementmore general nonunitary maps — quantum channels — in order to drive quantum information backinto a steady-state subspace after an error. The first is a recipe to embed any channel into theinfinite-time evolution generated by a Lindbladian [3]. The second is an e�cient implementationfor state-of-the-art circuit QED experiments based on adaptive control [4].

[1] M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret, New J. Phys. 16,045014 (2014).

[2] V. V. Albert, C. Shu, S. Krastanov, C. Shen, R.-B. Liu, Z.-B. Yang, R. J. Schoelkopf, M. Mirrahimi, M. H. Devoret, andL. Jiang, Phys. Rev. Lett. 116, 140502 (2016).

[3] V. V. Albert, B. Bradlyn, M. Fraas, and L. Jiang, Phys. Rev. X 6, 041031 (2016).[4] C. Shen, K. Noh, V. V. Albert, S. Krastanov, M. H. Devoret, R. J. Schoelkopf, S. M. Girvin, and L. Jiang, arXiv:1611.03463.

BKT phase of exciton polaritons by spatial and temporal correlations

Dario Ballarini,1 Davide Caputo,1, 2 Galbadrakh Dagvadorj,3 Carlos Sanchez Munoz,4

Milena De Giorgi,1 Lorenzo Dominici,1 Kenneth West,5 Loren N. Pfei↵er,5

Giuseppe Gigli,1, 2 Fabrice P. Laussy,6 Marzena H. Szymanska,7 and Daniele Sanvitto1

1CNR NANOTEC—Institute of Nanotechnology, Via Monteroni, 73100 Lecce, Italy2University of Salento, Via Arnesano, 73100 Lecce, Italy

3Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom4Departamento de Fısica Teorica de la Materia Condensada,Universidad Autonoma de Madrid, 28049 Madrid, Spain

5PRISM, Princeton Institute for the Science and Technology of Materials,Princeton Unviversity, Princeton, NJ 08540

6Russian Quantum Center, Novaya 100,143025 Skolkovo, Moscow Region, Russia

7Department of Physics and Astronomy, University College London,Gower Street, London WC1E 6BT, United Kingdom

The Berezinskii–Kosterlitz–Thouless (BKT) theory provides a powerful tool to describe phasetransitions in a wide range of quasi-two-dimensional systems, including superfluid films, high-temperature superconductors, Josephson junction arrays, ultra-cold atomic gases, quantum-Hall bilayers and certain classes of ferroelectrics [1]. Currently, wheter topological ordering isresponsible for the transition to the quasi-ordered phase also in open systems is at the center ofintense investigation [2]. Here, we show the BKT transition of a two-dimensional (2D) gas ofexciton-polaritons through the joint measurement of the first-order coherence both in space andtime. On the other hand, we demonstrate that the topological origin of the transition cannot beproved, in open systems, by looking only at spatial correlations [3]. Indeed, the same behaviorof spatial correlations can be observed for a BKT phase and for a laser, setting these resultsas the first demonstration of a BKT transition in optical systems by temporal correlationsmeasurements [4]. In the BKT phase, we find that spatial and temporal correlations follow apower law decay with the same exponent ↵ 1/4. This value is characteristic for 2D bosons inthermodynamic equilibrium and it is achieved thanks to the long lifetime in high-quality sampleswith small amount of disorder and in a region far away from the excitation spot to avoid thepresence of the exciton reservoir [5]. The experimental findings are reproduced by numericalsolutions of stochastic equations, that further allow us to track vortices in each realisation of thecondensate. The mechanism of vortex-antivortex unbinding below a critical density is shown tobe responsible for the observed decay of correlations both in space and time.

[1] J. M. Kosterlitz, D. J. Thouless, Ordering, metastability and phase transitions in two-dimensionalsystems J. Phys. C: Solid State Phys. 6, 1181 (1973).

[2] E. Altman, L. M. Sieberer, L. Chen, S. Diehl, J. Toner, Two-dimensional superfluidity of excitonpolaritons requires strong anisotropy. Phys. Rev. X 5, 011017 (2015).

[3] G. Roumpos, M. Lohse, W. H. Nitsche, J. Keeling, M. H. Szymanska, P. B. Littlewood, A. Lo✏er, S.Hofling, L. Worschech, A. Forchel, Y. Yamamoto, Power-law decay of the spatial correlation functionin exciton-polariton condensates. PNAS 109, 6467–6472 (2012).

[4] D. Caputo, D. Ballarini, G. Dagvadorj, C. Munoz, M. De Giorgi, L. Dominici, K. West, L. Pfei↵er,G. Gigli, F. Laussy, M. Szymanska,D. Sanvitto, Topological order and equilibrium in a condensateof exciton-polaritons (2016). arXiv:1610.05737.

[5] D. Ballarini, D. Caputo, C. Munoz, M. De Giorgi, L. Dominici, M. Szymanska, W. K., L. Pfei↵er, G.G., F. Laussy, D. Sanvitto, Formation of a macroscopically extended polariton condensate withoutan exciton reservoir (2016). arXiv:1609.05717.

Exact results for dissipative phase transitions in a

nonlinear resonator with one- and two-photon pumping

Nicola Bartolo,⇤ Fabrizio Minganti, Wim Casteels, and Cristiano Ciuti†

Laboratoire Materiaux et Phenomenes Quantiques, Universite Paris Diderot,

Sorbonne Paris Cite, CNRS-UMR7162, 75013 Paris, France

In out-of-equilibrium systems, the competition between Hamiltonian evolution and dissipa-tion can result in dissipative phase transitions [1, 2]. We investigate this kind of phenomenain the steady-state of a general class of driven-dissipative systems, consisting of a nonlinearKerr resonator in the presence of both coherent (one-photon) and parametric (two-photon)driving and dissipation (as sketched in Fig. 1) [3]. We analytically derive the exact steady-state solution via the formalism of the complex P -representation [4, 5]. This allows us toexplore all the photon-density regimes, including the mesoscopic one in which both photonnumbers and quantum correlations are significant. In the thermodynamic limit of a largephoton density, we reveal and characterize the emergence of dissipative phase transitions ofboth first and second order (see Fig. 1).

BARTOLO, MINGANTI, CASTEELS, AND CIUTI PHYSICAL REVIEW A 94, 033841 (2016)

UG

FFFF�������

�F G � �

|0|1|2|3|4

. . . . . . . . .

|n

FIG. 1. Sketch of the considered class of systems. The picturerepresents a photon resonator subject to one-photon losses at rate γ

and coherently driven by a one-photon pump of amplitude F . Theresonator is also subject to a coherent two-photon driving of amplitudeG and two-photon losses at rate η. The strength of the photon-photoninteraction is quantified by U . At the right, we sketch the effects ofthese physical processes on the Fock (number) states |n⟩.

where G is the pump amplitude and ω2 its frequency.Such a two-photon pumping mechanism can be obtained byengineering the exchange of photons between the cavity andthe environment. Recently, this has been realized by couplingtwo superconducting resonators via a Josephson junction[14]. In order to get a time-independent Hamiltonian, weconsider ω2 = 2ωp. Hence, we use the unitary transformationU = e−iωpta†a , which removes the time dependence from theHamiltonian. This allows us to describe the system in thereference frame rotating at the coherent pump frequency ωp.The full Hamiltonian, hence, becomes

H = −$a†a + U

2a†a†aa + F a† + F ∗a + G

2a†a† + G∗

2aa,

(4)

where $ = ωp − ωc is the pump-cavity detuning. For theconsidered system, photon losses are typically appreciable andcannot be neglected [33]. The Markov-Born approximationgives an excellent description of these losses in terms of aLindblad dissipation superoperator D(C) of the form [33,34]

D(C) ρ = 2 C ρ C† − C†C ρ − ρ C†C, (5)

where C is the quantum jump operator corresponding tothe specific dissipation process. Usually, photons are lostindividually to the environment and the jump operator is theannihilation operator a [33]. In addition, we also considertwo-photon losses, which naturally emerge together withthe engineered two-photon pumping [14]. These losses areincluded through the jump operator a2. The resulting Lindbladmaster equation describing the evolution of the the systemdensity matrix ρ is

i∂ρ

∂t= [H,ρ] + i

γ

2D(a) ρ + i

η

2D(a2) ρ, (6)

where γ and η are, respectively, the one- and two-photondissipation rates and H is the one given in Eq. (4).

III. P-REPRESENTATION AND EXACT SOLUTION FORTHE STEADY STATE

The steady-state properties are of central interest in the con-text of out-of-equilibrium quantum systems. These propertiesare encoded in the steady-state density matrix, which is the

solution of Eq. (6) for ∂t ρ = 0. To this purpose, we considerthe P -representation of the density matrix, i.e., we decomposeρ using the overcomplete basis of coherent states |α⟩, such thata |α⟩ = α |α⟩. We use the complex P -representation P (α,β)[35], which is defined by

ρ =!

Cdα

!

C ′dβ

|α⟩ ⟨β∗|⟨β∗|α⟩

P (α,β), (7)

where the closed integration contours C and C ′ must becarefully chosen to encircle all the singularities of the functionP (α,β). Once definition (7) is inserted into Eq. (6), the actionof the annihilation and creation operators on the projector|α⟩⟨β∗| allows one to map the master equation for ρ into acomplex Fokker-Planck equation for P (α,β). Further detailson this procedure are presented in Appendix A. For the caseG = 0, the complex P -representation solution for the steadystate was derived by Drummond and Walls [15] and is given by

Pss(α,β) ∝ e2αβ e−2f/α

α2+2c

e−2f ∗/β

β2+2c∗ . (8)

In Eq. (8), the system parameters are resumed by thedimensionless quantities c = ($ + iγ /2)/(U − iη) andf = F/(U − iη). For the general case corresponding tomaster equation (6), we find

Pss(α,β) = e2αβ

N1

(α2 + g)1+cexp

"− 2f

√g

arctan#√

g

α

$%

× 1(β2 + g∗)1+c∗ exp

"− 2f ∗

√g∗ arctan

#√g∗

β

$%.

(9)

All details on the derivation of Eq. (9) are given in Appendix A.In Eq. (9) we have introduced the dimensionless parameterg = G/(U − iη). We stress that in the limit g → 0 Eq. (9)reduces to Eq. (8), as expected. We note that some particularcases have been considered in [36–38].

The normalization factor N in Eq. (9) ensures that Tr[ρ] =1. By imposing this condition we get

N =!

Cdα

!

C ′dβ e2αβ 1

(α2 + g)1+c

1(β2 + g∗)1+c∗

× exp"− 2f

√g

arctan#√

g

α

$− 2f ∗

√g∗ arctan

#√g∗

β

$%.

(10)

One can Taylor-expand e2αβ and swap the resulting sum withthe integral. The two contour integrals over α and β thusdecouple, leading to

N =∞&

m=0

2m

m!|Fm(f,g,c)|2, (11)

where we have introduced

Fm(f,g,c) =!

C

αm dα

(g + α2)1+cexp

"− 2f

√g

arctan#√

g

α

$%.

(12)

Note that Fm(f ∗,g∗,c∗) = F∗m(f,g,c). Performing the integral

in Eq. (12) requires an appropriate choice of the contour C. In

033841-2

��� � �� ��� ������-�

��-�

�

���

Δ/�

⟨�†�⟩

��� � �� �����-�

���

�

��

���

Δ/�

⟨�†�⟩

G = 0 F = 0

F/U = 0.1 0.3 1 3 10 30

100

300

G/U = 0.1 0.3 1 3 1030

100

300

FIG. 1: Top: A sketch of the considered class of systems: a nonlinear resonator subject to

one- and two-photon drive and dissipation. Bottom: Mean photon number in the resonator in

presence of only one-photon (left) or two-photon (right) driving. For increasing driving

amplitudes, one reaches the high-density regime and an abrupt phase transition eventually

occurs.

[1] E.M. Kessler, G. Giedke, A. Imamoglu, S.F. Yelin, M.D. Lukin, and J.I. Cirac, Phys. Rev. A 86, 012116 (2012).[2] H.J. Carmichael, Phys. Rev. X 5, 031028 (2015).[3] Z. Leghtas, S. Touzard, I.M. Pop, A. Kou, B. Vlastakis, A. Petrenko, K.M. Sliwa, A. Narla, S. Shankar, M.J. Hatridge,

M. Reagor, L. Frunzio, R.J. Schoelkopf, M. Mirrahimi, M.H. Devoret, Science 347, 853 (2015).[4] F. Minganti, N. Bartolo, J. Lolli, W. Casteels, and C. Ciuti, Sci. Rep. 6, 26987 (2016).[5] N. Bartolo, F. Minganti, W. Casteels, and C. Ciuti, Phys. Rev. A 94, 033841 (2016), Editors’ suggestion.

⇤ nicola.bartolo@univ-paris-diderot.fr† cristiano.ciuti@univ-paris-diderot.fr

Driven open quantum systems and Floquet stroboscopic quantum simulation

S. Restrepo,1 J. Cerrillo,1 V. M. Bastidas,2 D. G. Angelakis,2, 3 and T. Brandes1

1Institut fur Theoretische Physik, Technische Universitat Berlin,Hardenbergstr. 36, 10623 Berlin, Germany

2Centre for Quantum Technologies, National Universityof Singapore, 3 Science Drive 2, Singapore 1175433School of Electrical and Computer Engineering,

Technical University of Crete, Chania, Crete 73100, Greece

The study of periodically driven open quantum systems [1, 2] is of utmost interest in diversecommunities ranging from quantum optics [3], circuit QED [4], optomechanical devices [5], andmanybody systems [6]. Most of the previous research has been focused on the Markovian regime,where a Floquet-Markov approach can be used. This consists on deriving a weak coupling Born-Markov master equation in the Floquet basis of the driven system [7]. However, the dynamics ofdriven-dissipative systems in the non-Markovian regime remains unexplored[8]. Due to growinginterest in quantum simulation of open systems by means of driven control, the exploration ofnon-Markovian e↵ects in driven systems constitutes an exciting new avenue of research.

In this contribution, we present an analytic solution to the problem of system-bath dynam-ics under the e↵ect of high-frequency driving, which is valid to all orders in the system-bathcoupling [9]. Our approach provides the time evolution operator of the full system, which goesbeyond usual studies based on weak coupling master equation and Markovian regime. Basedon this solution, we also proposed a method to perform stroboscopic quantum simulation ofnontrivial dissipative systems [10, 11] for strong and weak coupling regimes. We instantiate theresult with the study of the spin-boson model [12] with time-dependent tunneling amplitude.Besides this, we also analyze the class of Hamiltonians that may be stroboscopically accessedfor this example and illustrate the dynamics of system and bath degrees of freedom.

[1] J. H. Shirley, Phys. Rev. 138, B979 (1965).[2] M. Bukov, L. D’Alessio, and A. Polkovnikov, Advances in Physics 64, 139 (2015).[3] I. Carusotto and C. Ciuti, Rev. Mod. Phys. 85, 299 (2013).[4] A. A. Houck, H. E. Tureci, and J. Koch, Nat. Phys. 8, 292 (2012).[5] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).[6] C. Noh, D. G. Angelakis, Report of Progress in Physics 80, 016401 (2016).[7] M. Grifoni and P. Hanggi, Phys. Rep. 304, 229 (1998).[8] J. Cerrillo and J. Cao, Phys. Rev. Lett. 112, 110401 (2014).[9] S. Restrepo, J. Cerrillo, V. M. Bastidas, D. G. Angelakis, and T. Brandes, arXiv:1606.08392 (2016).

To appear in Physical Review Letters.[10] Schindler et al., Nat. Phys. 9, 361 (2013).[11] F. Haddadfarshi, J. Cui, and F. Mintert, Phys. Rev. Lett. 114, 130402 (2015).[12] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger Rev. Mod.

Phys. 59, 1 (1987)

Phase diagram of incoherently-driven strongly correlated photonic lattices

Alberto Biella,

1, 2, ⇤Florent Storme,

1Jose Lebreuilly,

3

Davide Rossini,

2Rosario Fazio,

4, 2Iacopo Carusotto,

3and Cristiano Ciuti

1

1Universite Paris Diderot, Sorbonne Paris Cite, Laboratoire Materiauxet Phenomenes Quantiques, CNRS-UMR7162, 75013 Paris, France

2NEST, Scuola Normale Superiore & Istituto Nanoscienze-CNR, I-56126 Pisa, Italy3INO-CNR BEC Center and Dipartimento di Fisica, Universita di Trento, I-38123 Povo, Italy

4ICTP, Strada Costiera 11, 34151 Trieste, Italy

We explore theoretically the nonequilibrium photonic phases of an array of coupled cavities inpresence of incoherent driving and dissipation [1]. In particular, we consider a Hubbard modelsystem where each site is a Kerr nonlinear resonator coupled to a two-level emitter, which is pumpedincoherently [2]. Within a Gutzwiller mean-field approach, we determine the steady-state phasediagram of such a system. We find that, at a critical value of the inter-cavity photon hopping rate,a second-order nonequilibrium phase transition associated with the spontaneous breaking of theU(1) symmetry occurs. The transition from an incompressible Mott-like photon fluid to a coherentdelocalized phase is driven by commensurability e↵ects and not by the competition between photonhopping and optical nonlinearity. The essence of the mean-field predictions is corroborated byfinite-size simulations obtained with matrix product operators and corner-space renormalizationmethods.

!2 !1 0 1 2!2

!1

0

1

2

Re!Α"

Im!Α"

zJ#0

!0.6

!0.4

!0.2

0

0.2

0.4

!2 !1 0 1 2!2

!1

0

1

2

Re!Α"

Im!Α"

zJ#4

!0.6

!0.4

!0.2

0

0.2

0.4

Figure 1: Contour plot of the steady-state Wigner distribution W (↵) in the Mott-like (left panel) and symmetry

broken phase (right panel). The black dashed contour highlight the region within W (↵) < 0.

[1] A. Biella, F. Storme, J. Lebreuilly, D. Rossini, R. Fazio, I. Carusotto, and C. Ciuti, in preparation.[2] J. Lebreuilly, M. Wouters, and I Carusotto,

Towards strongly correlated photons in arrays of dissipative nonlinear cavities under a frequency-dependent incoherent pump-ing, Comptes Rendus Physique 17, 836 (2016).

⇤Electronic address: alberto.biella@univ-paris-diderot.fr

Critical scaling of the Liouvillian gap for a nonlinear driven-dissipative

resonator

W. Casteels,1 R. Fazio,2, 3 and C. Ciuti4

1TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, Belgium

2ICTP, Strada Costiera 11, I-34151 Trieste, Italy

3NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy

4MPQ, Universite Paris Diderot, CNRS UMR 7162, Sorbonne Paris Cite,

10 rue Alice Domon et Leonie Duquet 75013 Paris, France

In recent years there has been a revival of interest in the physics of driven-dissipative phasetransitions (see for example Refs. [1–6]). This is partly driven by the remarkable experimentalachievements that have been obtained with various photonic platforms such as superconductingcircuits and semiconductor microcavities (see Refs. [7–9] for examples of recent experiments inthis direction). These systems can typically be described by a linear master equation of the form@t⇢ = L⇢, where ⇢ is the reduced density matrix of the system and L the so-called Liouvilliansuperoperator. The steady-state solution L⇢ = 0 corresponds to the zero eigenvalue of theLiouvillian. In general, dissipative phase transitions are expected to occur when the Liouvillianspectral gap closes [10].

We will theoretically explore the critical properties of the Liouvillian gap for a driven-dissipative (Kerr) nonlinear resonator [11]. We show that, by considering a well defined thermo-dynamical limit of large excitation numbers, such a model describes a first-order phase transi-tion. The thermodynamic limit is obtained by letting the nonlinearity going to 0 and the drivingintensity to +1 while keeping constant their product. We determine the exponential vanishingof the complex Liouvillian gap and characterize its finite-size behavior. We show that suchthermodynamical limit of large excitation numbers for one single-mode resonator has a directconnection with the more standard limit of many sites in the driven-dissipative Bose-Hubbardmodel.

[1] H. J. Carmichael, Phys. Rev. X 5, 031028 (2015).[2] J. J. Mendoza-Arenas, S. R. Clark, S. Felicetti, G. Romero, E. Solano, D. G. Angelakis, D. Jaksch,

Phys. Rev. A 93, 023821 (2016).[3] J. Jin, A. Biella, O. Viyuela, L. Mazza, J. Keeling, R. Fazio, D. Rossini, Phys. Rev. X 6, 031011

(2016).[4] R. M. Wilson, K. W. Mahmud, A. Hu, A. V. Gorshkov, M. Hafezi, M. Foss-Feig, Phys. Rev. A 94,

033801 (2016).[5] J. Marino and S. Diehl, Phys. Rev. B 94, 085150 (2016).[6] W. Casteels, F. Storme, A. Le Boite, C. Ciuti, Phys. Rev. A 93, 033824 (2016).[7] J. M. Fink, A. Dombi, A. Vukics, A. Wallra↵, P. Domokos, ArXiv:1607.04892 (2016).[8] M. Fitzpatrick, N. M. Sundaresan, A. C. Y. Li, J. Koch, A. A. Houck, ArXiv:1607.06895 (2016).[9] S. R. K. Rodriguez, W. Casteels, F. Storme, I. Sagnes, L. Le Gratiet, E. Galopin, A. Lemaitre, A.

Amo, C. Ciuti, J. Bloch, ArXiv:1608.00260 (2016).[10] E. M. Kessler, G. Giedke, A. Imamoglu, S. F. Yelin, M. D. Lukin, J. I. Cirac, Phys. Rev. A 86,

012116 (2012).[11] W. Casteels, R. Fazio, C. Ciuti, ArXiv:1608.00717 (2016).

Dynamical many-body phases of the parametrically driven, dissipative Dicke

model

R. Chitra,1 O. ZIlberberg,1 P. Molignini,1 L. Papariello,1 and A. U. J. Lode2

1Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland2Department of Physics, University of Basel, 4056 Basel, Switzerland

The dissipative Dicke model is a paradigmatic model for studying out-of- equilibrium many-body phase transitions. The model describes a coupling between a driven photonic cavity andnumerous two-level atoms and exhibits a continuous phase transition from a normal phase toa super-radiant phase. We study the e↵ect of a time-dependent parametric modulation of thiscoupling, and discover a rich phase diagram as a function of the modulation strength. We findthat in addition to the established normal and super-radiant phases, a new phase with pulsedsuperradiance which we term dynamical normal phase, appears when the system is parametri-cally driven. We also study the closely related model of a BEC condensate interacting with ahigh finesse cavity with a Multi-Configurational time dependent Hartree approach and explicitlyshow how such a dynamical normal phase would manifest itself in this realistic system. Ourresults open the door for the experimental study of dynamically stabilized phases of interactinglight and matter.

[1] R. Chitra and O. Zilberberg, Phys. Rev. A 92, 023815 (2015).[2] P. Molignini, L. Papariello, A. U. J. Lode and R. Chitra, manuscript in preparation (2017).

Dicke phase transition without total spin conservation

Emanuele G. Dalla Torre,1 Yulia Shchadilova,2 Eli Y. Wilner,3 Mikhail D. Lukin,2 and Eugene Demler2

1Department of Physics, Bar-Ilan University, Ramat Gan 5290002, Israel2Department of Physics, Harvard University, Cambridge, MA 02138, U.S.A.3Department of Physics, Columbia University, New York, NY 10027, U.S.A.

We develop a new fermionic path-integral formalism to analyze the phase diagram of open nonequi-librium systems. The formalism is applied to analyze an ensemble of two-level atoms interacting with asingle-mode optical cavity, described by the Dicke model. While this model is often used as the paradig-matic example of a phase transition in driven-dissipative systems, earlier theoretical studies were limitedto the special case when the total spin of the atomic ensemble is conserved1–5. This assumption is notjustified in most experimental realizations6–9. Our new approach allows us to analyze the problem in amore general case, including the experimentally relevant case of dissipative processes that act on eachatom individually and do not conserve the total spin. We obtain a general expression for the positionof the transition, which contains as special cases the two previously known regimes: i) non-equilibriumsystems with losses and conserved spin and ii) closed systems in thermal equilibrium and with the Gibbsensemble averaging over the values of the total spin. We perform a detailed study of di↵erent types ofbaths and point out the possibility of a surprising non-monotonous dependence of the transition on thebaths’ parameters.

𝜅𝜅𝛾𝛾

𝛾𝛾𝛾𝛾

𝛾𝛾

𝛾𝛾𝛾𝛾

𝛾𝛾

𝛾𝛾

𝜅𝜅(i) Dephasing

𝛾𝛾T𝛾𝛾𝜙𝜙

Rate Lindblad𝑎𝑎𝜎𝜎𝑖𝑖𝑧𝑧

𝜎𝜎𝑖𝑖− and 𝜎𝜎𝑖𝑖+(ii) Thermal bath𝛾𝛾t 𝜎𝜎𝑖𝑖− + 𝑡𝑡𝜎𝜎𝑖𝑖+(iii) Generalized

Markovian bath

Single-atom

Dissipative bathCavity decay

Dissipative processes considered in this study.Previous studies mostly focused on the cavitydecay, which conserves the total spin. Ourfermionic path-integral appraoch allows us to con-sider single-atom decay channels, relevant to re-cent quantum-optics experiments.

1 K. Hepp and E. H. Lieb, Annals of Physics 76, 360 (1973).2 Y. K. Wang and F. T. Hioe, Phys. Rev. A 7, 831 (1973).3 C. Emary and T. Brandes, Phys. Rev. E 67, 066203 (2003).4 F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, Phys. Rev. A 75, 013804 (2007).5 D. Nagy, G. Konya, G. Szirmai, and P. Domokos, Phys. Rev. Lett. 104, 130401 (2010).6 M. P. Baden, K. J. Arnold, A. L. Grimsmo, S. Parkins, and M. D. Barrett, Phys. Rev. Lett. 113, 020408(2014).

7 J. Klinder, H. Keßler, M. Wolke, L. Mathey, and A. Hemmerich, Proceedings of the National Academy ofSciences 112, 3290 (2015).

8 J. Klinder, H. Keßler, M. R. Bakhtiari, M. Thorwart, and A. Hemmerich, Phys. Rev. Lett. 115, 230403 (2015).9 S. Roof, K. Kemp, M. Havey, and I. Sokolov, arXiv preprint arXiv:1603.07268 (2016).

Observation of the Photon-Blockade Breakdown Phase Transition

J. M. Fink,1 A. Dombi,2 A. Vukics,2 A. Wallra↵,3 and P. Domokos2

1Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria2Wigner Research Centre for Physics, H-1525 Budapest, P.O. Box 49., Hungary

3Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland

Nonequilibrium phase transitions exist in damped-driven open quantum systems when thecontinuous tuning of an external parameter leads to a transition between two robust steadystates. In second-order transitions this change is abrupt at a critical point [1], whereas in first-order transitions the two phases can coexist in a critical hysteresis domain. Here, we report theobservation of a first-order dissipative quantum phase transition in a driven circuit quantumelectrodynamics system [2]. It takes place when the photon blockade of the driven cavity-atomsystem is broken by increasing the drive power [3]. The observed experimental signature is abimodal phase space distribution with varying weights controlled by the drive strength. Ourmeasurements show an improved stabilization of the classical attractors up to the millisecondrange when the size of the quantum system is increased from one to three artificial atoms. Theformation of such robust pointer states could be used for new quantum measurement schemesor to investigate multiphoton phases of finite-size, nonlinear, open quantum systems.

[1] K. Baumann, C. Guerlin, F. Brennecke and T. Esslinger, Nature 464, 1301 (2010)[2] J. M. Fink, A. Dombi, A. Vukics, A. Wallra↵ and P. Domokos, Phys. Rev. X (in print) (2016)[3] H. J. Carmichael, Phys. Rev. X 5, 031028 (2015)

Observation of a dissipative phase transition in a one-dimensional circuitQED lattice

Mattias Fitzpatrick,1 Neereja M. Sundaresan,1 Andy

C. Y. Li,2 Jens Koch,2 and Andrew A. Houck1

1Department of Electrical Engineering, Princeton University, Princeton, NJ 08540, USA

2Department of Physics and Astronomy,

Northwestern University, Evanston, IL 60208, USA

Condensed matter physics has been driven forward by significant experimental and theoreti-cal progress in the study and understanding of equilibrium phase transitions based on symmetryand topology. However, nonequilibrium phase transitions have remained a challenge, in partdue to their complexity in theoretical descriptions and the additional experimental di�cul-ties in systematically controlling systems out of equilibrium. Here, we present a study of aone-dimensional chain of 72 microwave cavities, each coupled to a superconducting qubit, andcoherently drive the system into a nonequilibrium steady state. We find experimental evidencefor a dissipative phase transition in the system in which the steady state changes dramatically asthe mean photon number is increased. Near the boundary between the two observed phases, thesystem demonstrates bistability, with characteristic switching times as long as 60ms – far longerthan any of the intrinsic rates known for the system. This experiment demonstrates the powerof circuit QED systems for studying nonequilibrium condensed matter physics and paves theway for future experiments exploring nonequilbrium physics with many-body quantum optics.

Emergent equilibrium in many-body optical bistability

M.Foss-Feig,1, 2, 3 P.Niroula,2, 4 J. T.Young,2 M.Hafezi,2, 5

A.V.Gorshkov,2, 3 R.M.Wilson,6 and M.F.Maghrebi2, 3

1United States Army Research Laboratory, Adelphi, MD 20783, USA2Joint Quantum Institute, NIST/University of Maryland, College Park, MD 20742 USA

3Joint Center for Quantum Information and Computer Science,NIST/University of Maryland, College Park, MD 20742 USA

4Department of Physics, Harvard University, Cambridge, MA 02138, USA5Department of Electrical and Computer Engineering andInstitute for Research in Electronics and Applied Physics,University of Maryland, College Park, MD 20742, USA

6Department of Physics, The United States Naval Academy, Annapolis, MD 21402, USA

Many-body systems constructed of quantum-optical building blocks can now be realizedin experimental platforms ranging from exciton-polariton fluids to ultracold gases of Ryd-berg atoms, establishing a fascinating interface between traditional many-body physics andthe driven-dissipative, non-equilibrium setting of cavity-QED. At this interface, the standardtechniques and intuitions of both fields are called into question, obscuring issues as fundamentalas the role of fluctuations, dimensionality, and symmetry on the nature of collective behavior andphase transitions. Here, we study the driven-dissipative Bose-Hubbard model, a minimal de-scription of numerous atomic, optical, and solid-state systems in which particle loss is counteredby coherent driving. Despite being a lattice version of optical bistability—a foundational andpatently non-equilibrium model of cavity-QED—the steady state possesses an emergent equilib-rium description in terms of a classical Ising model. We establish this picture by identifying alimit in which the quantum dynamics is asymptotically equivalent to non-equilibrium Langevinequations, which support a phase transition described by model A of the Hohenberg-Halperinclassification. Numerical simulations of the Langevin equations corroborate this picture, pro-ducing results consistent with the behavior of a finite-temperature Ising model.

Dissipation induced topological states: A recipe

Moshe Goldstein1

1Raymond and Beverly Sackler School of Physics and Astronomy,

Tel-Aviv University, Tel Aviv 6997801, Israel

Nonequilibrium conditions are traditionally seen as detrimental to the appearance ofquantum-coherent many-body phenomena in condensed matter systems, and much e↵ort isoften devoted to their elimination. Recently this approach has changed: It has been realizedthat driven-dissipative Markovian dynamics of the Lindblad type could be used as a resource.By proper engineering of the reservoirs and their local couplings to a system, one may drivethe system towards desired quantum-correlated steady states, even in the absence of internalHamiltonian dynamics.

An intriguing category of nontrivial equilibrium many-particle phases are those which aredistinguished by topology rather than by symmetry. Lately it has been discovered that such abehavior is not a peculiarity of quantum Hall and related phenomena, but applies to a large classof systems, including simple noninteracting band insulators and mean-field superconductors.Natural questions thus arise: which of these topological states can be achieved as the result ofpurely dissipative Lindblad-type dynamics? Moreover, are there new nonequilibrium topologicalphenomena, which have no equilibrium counterpart? Besides the fundamental importance ofthese issues, they may o↵er novel routes to the realization of topologically-nontrivial states inquantum simulators, especially ultracold atomic gases which naturally lend themselves to therequired bath engineering, and also other systems, such as superconducting nanocircuits. Itmay even lead to dissipative topological states featuring fractional excitations and allowing fortopological quantum computation. Recent studies, which concentrated on systems with p-wavesuperconducting correlations, have only provided a partial solution to this problem [1–3].

In this talk I will present a general recipe for the creation, classification, and detection ofstates of the integer quantum Hall and 2D topological insulator type as the outcomes of couplinga system to reservoirs, and show how the recipe can be realized with ultracold atoms and otherquantum simulators. The mixed states so created can be made arbitrarily close to pure states. Iwill discuss ways to extend this construction to other topological phases, including non-Gaussianones, such as fractional quantum Hall states.

[1] S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, Nature Phys. 7, 971 (2011).[2] C. E. Bardyn, M. A. Baranov, E. Rico, A. Imamoglu, P. Zoller, and S. Diehl, Phys. Rev. Lett. 109,

130402 (2012).[3] J. C. Budich, P. Zoller, and S. Diehl, Phys. Rev. A 91, 042117 (2015).

Chirality of topological gap solitons in bosonic dimer chains

D. Solnyshkov, O. Blue, B. Teklu, G. Malpuech1

1Institut Pascal, PHOTON-N2, University Clermont Auvergne, CNRS, Clermont-Ferrand, France

Solitons in non-linear equations on one hand and topologically nontrivial band structures onthe other hand represent two important domains of modern Physics. Gap solitons are formedfrom interacting particles in the bandgaps of periodic lattices. They have been widely studiedin simple 1D lattices, while the works dealing with topologically nontrivial bands are veryscarce [1]. The combination of the topological properties of the quantum fluids (the existenceof topological defects, such as solitons and vortices) with the topology of the lattice can beexpected to create interesting e↵ects.

Fig. 1. Soliton trajectories plotted as theparticle density as a function of positionand time: a,b) Topological gap soliton,oscillating trajectory or free acceleration,depending on the initial soliton position.c,d) oscillating trajectory of an ordinarygap soliton for the same defect.

In this work [2], we demonstrate thata gap soliton appearing in the topologi-cal gap of a bosonic dimer chain (topo-logical gap soliton) exhibits several orig-inal properties, including chirality, ab-sent for ordinary gap solitons appear-ing in simple chains. Such topologi-cal soliton (which is not a Su-Schrie↵er-Heeger soliton and does not requiredynamical modification of the lattice)has a non-trivial sublattice pseudospintexture due to the combination of thepseudospin-anisotropic interactions ande↵ective field due to the dimerization.This chiral pseudospin texture leads tothe chiral behavior of the topologicalgap soliton in presence of a localized po-tential breaking the symmetry betweenthe A and B sites. Such behavior, whichis di↵erent depending whether the topo-logical gap soliton is on the left or onthe right of the defect (Fig. 1, panelsa,b), cannot be observed for ordinarygap solitons (panels c,d), which exhibitsimilar oscillations in both cases. Thepseudospin texture and the possibility

of chiral behavior of the topological gap soliton have passed unnoticed in several previous works,including one where such solitons has been observed experimentally [3].

[1] S. Cheon, T.H. Kim, S.H. Lee, H.W. Yeom, Science 350, 182 (2015).[2] D.D. Solnyshkov, O. Bleu, B. Teklu, G. Malpuech, arXiv:1607.01805 (2016).[3] A. Kanshu et al, Optics Letters 37, 1253 (2012).

Quantum electrodynamics of high-impedance superconducting circuits

Roman Kuzmin,1 Nick Grabon,1 Yen-Hsiang Lin,2 Nitish

Mehta,1 Moshe Goldstein,3 and Vladimir Manucharyan1

1University of Maryland, College Park, USA.

2University of Maryland, College Park, USA.

3University of Tel Aviv, Tel Aviv, Israel.

An electromagnetic resonator is characterized by its resonance frequency and characteristicimpedance. The latter controls the scale of quantum fluctuations of charge and flux circuitvariables. In most superconducting circuits the impedance is close to 50 Ohm and is linked tothe impedance of free space. This low impedance favors small fluctuations of flux (compared toflux quantum) and high fluctuations of charge (compared to charge quantum). If a Josephsonjunction is inserted in such a low-impedance circuit, it’s rich and unusual non-linearity is reducedto a simple quartic in flux term. Novel strongly non-linear e↵ects are expected in high-impedanceJosephson circuits which remained largely unexplored today.

We have fabricated a high-impedance Josephson transmission line consisting of two parallelchains containing over 15,000 Josephson tunnel junctions. Such a transmission line presents anartificial media for 1D photons with highly unusual properties. In particular, the transmissionline shows standing wave resonances corresponding to wave propagation a factor of 100 slowerthan regular light. In other words, the wave impedance of this transmission line exceeds re-sistance quantum (for Cooper pairs), approximately 6.5 kOhm, as opposed to usual value of50 Ohm. In yet another terms, the e↵ective fine structure constant for photons here exceedsa unity. As a result, interaction of photons in our transmission line with a charge is enhancedby at least a factor of 10 compared to conventional low-impedance waveguides. Close exami-nation of resonances reveals signatures of frequency-dependent wave localization predicted fora generic 1D system of photons. Finally, non-linearity of the Josephson junction gives rise toseveral types of interaction terms between all the modes of our transmission line, turning it intoa model system to study out of equilibrium many-body phenomena in fully controlled settings.

In particular, we will describe how our system can be used to perform quantum simulation ofKondo e↵ect and Luttinger liquid tunneling using simple microwave scattering measurements.The trick is to insert a properly tuned ”impurity” circuit into the transmission line and exploremode-mode interaction due to the scattering of the impurity. Our experiment would enablean accurate measurement of the correlation functions of strongly interacting many-body mod-els and time-resolved dynamics of many-body entanglement in quenching experiments. Initialmeasurement results will be disclosed.

Synthetic dimensions in ultracold gases and photonics

Tomoki Ozawa1

1INO-CNR BEC Center and Department of Physics, University of Trento, Povo, Italy

I discuss recent developments of the study of ”synthetic dimensions” in ultracold gases andphotonics. The idea of synthetic dimensions is to identify internal states of an atom or a photoniccavity as extra dimensions, and to simulate higher dimensional lattice models using physicallylower dimensional systems. The concept was originally proposed and experimentally realized inultracold gases [1–5]. I first review the existing theoretical and experimental studies of syntheticdimensions. After discussing some challenges and limitations of the existing methods of syntheticdimensions, I explain our proposals of realizing synthetic dimensions both in ultracold gases [6]and in photonic cavities [7, 8], which overcome some of these limitations. Finally I discuss howthe four dimensional quantum Hall e↵ect can be observed in ultracold gases and photonics usingthe synthetic dimensions [7, 9, 10].

[1] O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein, Quantum Simulation of an Extra Dimension,Phys. Rev. Lett. 108, 133001 (2012).

[2] A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B. Spielman, G. Juzelinas, and M. Lewenstein,Synthetic Gauge Fields in Synthetic Dimensions, Phys. Rev. Lett. 112, 043001 (2014).

[3] M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio,M. Dalmonte, and L. Fallani, Observation of chiral edge states with neutral fermions in syntheticHall ribbons, Science 349, 1510 (2015).

[4] B. K. Stuhl, H. I. Lu, L. M. Aycock, D. Genkina, and I. B. Spielman, Visualizing edge states withan atomic Bose gas in the quantum Hall regime, Science 349, 1514 (2015).

[5] L. F. Livi, G. Cappellini, M. Diem, L. Franchi, C. Clivati, M. Frittelli, F. Levi, D. Calonico, J.Catani, M. Inguscio, and L. Fallani, Synthetic dimensions and spin-orbit coupling with an opticalclock transition, arXiv:1609.04800.

[6] H. Price, T. Ozawa, and N. Goldman, Synthetic Dimensions for Cold Atoms from Shaking a Har-monic Trap, arXiv:1605.09310.

[7] T. Ozawa, H. M. Price, N. Goldman, O. Zilberberg, and I. Carusotto, Synthetic dimensions inintegrated photonics: From optical isolation to four-dimensional quantum Hall physics Phys. Rev.A 93, 043827 (2016).

[8] T. Ozawa and I. Carusotto, Synthetic dimensions with magnetic fields and local interactions inphotonic lattices, arXiv:1607.00140.

[9] H. M. Price, O. Zilberberg, T. Ozawa, I. Carusotto, and N. Goldman, Four-Dimensional QuantumHall E↵ect with Ultracold Atoms, Phys. Rev. Lett. 115, 195303 (2015).

[10] H. M. Price, O. Zilberberg, T. Ozawa, I. Carusotto, and N. Goldman, Measurement of Chernnumbers through center-of-mass responses, Phys. Rev. B 93, 245113 (2016).

Gate tunable Magneto-Plasmon ultrastrongly coupled to LC cavity:Towards Quantum Hall Transport in the ultra strong coupling regime

G.L. Paravicini-Bagliani, G. Scalari, F. Valmorra, J. Keller, C. Maissen, M. Beck and J. Faist1

1ETH Zurich, Institute for Quantum Electronics, Zurich, 8093, Switzerland

Subwavelength cavity photon resonators on two dimensional electron gases (2DEGs) havebeen shown to be a highly flexible platform to study ultra-strong light-matter interaction physicsin the THz.[1] In this work, we introduce a confinement of the 2DEG in one direction. Thisresults in a (magneto-)plasmon excitation that also couples to the cavity. Its dispersion [2]is described by !2

MP = !2p + !2

c where !2p = (nse2)/(2m⇤✏0✏)⇡/W and has been measured in

transmission with THz-TDS shown in Figure 1a). The 2DEG stripe is placed into the near-field of the resonator (see Figure 1b), which produces large vacuum electric field fluctuationsEvac =

p(h!c)/(2✏Vcav) ⇡ 50V/m polarized across the width of the 2DEG stripe. A 2 nm

chromium gate allows to tune the electron density and hence !p (data not shown).

a) Measured magneto-plasmon dispersionin a THz-TDS. b) Sketch of etched GaAs2DEG stripe cross section in a sub-wavelength cavity producing strong vac-uum fluctuations. c) Magneto-Plasmon-Polarition dispersion with !p ⇡ !cav

d) Cavity coupled to cyclotron transition,with !p > !cav

This system shows three very dif-ferent regimes. Figure 1c) shows theregime !p ⇡ !cav, where the collec-tive plasmon resonance ultra-stronglycouples with the subwavelength cavity.Note, the magneto-plasmon polaritonsexist already at zero magnetic field andshow a splitting of approximately 150GHz, which corresponds to a light mat-ter coupling ratio of ⌦/!cav = 15%.Furthermore, both polaritons have anon-zero frequency at zero magneticfield, which is not the case for cyclotronpolaritons [3].

The second interesting regime is!p > !cav, shown in the measurementin Figure 1c. Here, the cavity couplesagain to the single-particle cyclotrontransition, but seems to have stronglyreduced oscillator strength, so that thestrong coupling regime is not reachedhere. This is clearly di↵erent to thethird and well known regime !cav <<

!p where normalized light-matter coupling ratios ⌦/!cav of nearly 100% have been observed forcyclotron-polaritons [4].

Outlook: The new geometry of the sample allows to place a Hall bar entirely into the cavity.In this case, electrons at the Fermi energy contributing to Quantum Hall transport are at thesame time ultra strongly coupled to the cavity.

[1] G. Scalari, C. Maissen, D. Turcinkova, D. Hagenmller, S. De Liberato, C. Ciuti, C. Reichl, D. Schuh,W. Wegscheider, M. Beck, J. Faist, Science, 335, 1323 (2012)

[2] F. Stern, Physical Review Letters, 18, 546 (1967)[3] D. Hagenmuller, S. De Liberato and C. Ciuti, Physical Review B, 81, 235303 (2010)[4] C. Maissen, G. Scalari, F. Valmorra, M. Beck, J. Faist, Physical Review B, 90, 205309 (2014)

Experimental Measurement of the Berry Curvature from Anomalous

Transport

M. Wimmer,1, 2 H. M. Price,3 I. Carusotto,3 and U. Peschel2

1Erlangen Graduate School in Advanced Optical Technologies (SAOT), 91058 Erlangen, Germany2Institute of Solid State Theory and Optics,

Abbe Center of Photonics, Friedrich-Schiller-Universitat Jena,Max-Wien-Platz 1, 07743 Jena, Germany

3INO-CNR BEC Center and Department of Physics,University of Trento, via Sommarive 14, 38123 Povo, Italy

Geometrical properties of energy bands underlie fascinating phenomena in a wide-rangeof systems, including solid-state materials, ultracold gases and photonics. Most famously, localgeometrical characteristics like the Berry curvature can be related to global topological invariantssuch as those classifying quantum Hall states or topological insulators. Regardless of the bandtopology, however, any non-zero Berry curvature can have important consequences, such as inthe semiclassical evolution of a wave packet [1]. We experimentally demonstrate for the firsttime that wave packet dynamics can be used to directly map out the Berry curvature [2]. To thisend, we use optical pulses in two coupled fibre loops to study the discrete time-evolution of awave packet in a 1D geometrical charge pump, where the Berry curvature leads to an anomalousdisplacement of the wave packet under pumping. This is both the first direct observation of Berrycurvature e↵ects in an optical system, and, more generally, the proof-of-principle demonstrationthat semi-classical dynamics can serve as a high-resolution tool for mapping out geometricalproperties.

[1] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).[2] M. Wimmer, H.M. Price, I. Carusotto and U. Peschel, arXiv:1609.09412.

Probing a dissipative phase transition via dynamical optical hysteresis

S.R.K. Rodriguez,1 W. Casteels,2 F. Storme,2 N. Carlon Zambon,1 I. Sagnes,1

L. Le Gratiet,1 E. Galopin,1 A. Lemaıtre,1 A. Amo,1 C. Ciuti,2 and J. Bloch1

1Centre de Nanosciences et de Nanotechnologies,

CNRS, Univ. Paris-Sud, Universite Paris-Saclay,

C2N—Marcoussis, 91460 Marcoussis, France

2Laboratoire Materiaux et Phenomenes Quantiques,

Universite Paris Diderot, Sorbonne Paris Cite and CNRS,

UMR 7162, 75205 Paris Cedex 13, France

Optical bistability — the existence of two stable states with di↵erent photon numbers forthe same driving conditions — is a general feature of driven nonlinear systems described withinthe mean-field approximation (MFA). Beyond the MFA, a quantum treatment predicts that thesteady-state of a nonlinear cavity is always unique. The origin of this apparent contradiction wasnoted by Bonifacio and Lugiato [1], and by Drummond and Walls [2]: quantum fluctuations (thelost feature in the MFA) trigger switching between states and the exact solution corresponds toa weighted average over the two metastable states.

Recently, the physics emerging from fluctuations in nonlinear resonators is receiving re-newed theoretical [3–5] and experimental [6–9] interest in photonics. Nonlinear photonic systemsprovide new opportunities for studying quantum many-body phases [10–13], critical phenom-ena [5, 13–15], and dissipative phase transitions [16].

In this contribution, we experimentally probe a dissipative phase transition by measuringthe dynamic optical hysteresis of semiconductor microcavities. Scanning the driving powerup and down at decreasing speeds, we observe the progressive closure of the hysteresis cycle.The hysteresis area exhibits a temporal double power law decay with experimentally retrievedexponents in agreement with calculations including quantum fluctuations only. Probing di↵erentlaser-cavity detunings and photon-photon interactions, we show that the algebraic decay evolvestowards a single power law when the photon number becomes very large, i.e. when approachingthe thermodynamic limit. This algebraic behavior characterizes a dissipative phase transition.The present experimental approach is promising for exploring critical phenomena in photoniclattices.

[1] R. Bonifacio and L. A. Lugiato, Phys. Rev. Lett. 40, 1023 (1978).[2] P. D. Drummond and D. F. Walls, J. Phys. A 13, 725 (1980).[3] W. Casteels, F. Storme, A. Le Boite, and C. Ciuti, Phys. Rev. A 93, 033824 (2016).[4] J. J. Mendoza-Arenas et al., Phys. Rev. A 93, 023821 (2016).[5] M. Foss-Feig et al., arXiv:1611.02284 (2016).[6] R. Vijay, M. H. Devoret, and I. Siddiqi, Rev. Sci. Instrum. 80, 111101 (2009).[7] H. Abbaspour et al., Phys. Rev. B 92, 165303 (2015).[8] J. M. Fink, A. Dombi, A. Vukics, A. Wallra↵, and P. Domokos, Phys. Rev. X 7, 011012 (2017).[9] M. Fitzpatrick, N. M. Sundaresan, A. C. Y. Li, J. Koch, and A. A. Houck, Phys. Rev. X 7, 011016

(2017).[10] M. J. Hartmann, F. G. Brandao, and M. B. Plenio, Nature Phys. 2, 849 (2006).[11] A. D. Greentree, C. Tahan, J. H. Cole, and L. C. Hollenberg, Nature Phys. 2, 856 (2006).[12] D. G. Angelakis, M. F. Santos, and S. Bose, Phys. Rev. A 76, 031805 (2007).[13] R. M. Wilson et al., Phys. Rev. A 94, 033801 (2016).[14] H. J. Carmichael, Phys. Rev. X 5, 031028 (2015).[15] W. Casteels, R. Fazio, and C. Ciuti, Phys. Rev. A 95, 012128 (2017).[16] E. M. Kessler, G. Giedke, A. Imamoglu, S. F. Yelin, M. D. Lukin, and J. I. Cirac, Phys. Rev. A

86, 012116 (2012).

Linked cluster expansions for dissipative quantum systems

D. Rossini,1 J. Jin,2 A. Biella,3 O. Viyuela,4 and R. Fazio5

1Scuola Normale Superiore and Istituto Nanoscienze-CNR, Pisa, Italy2School of Physics and Optoelectronic Engineering,Dalian University of Technology, Dalian, China

3Materiaux et Phenomenes Quantiques, Universite Paris Diderot, Paris, France4Departamento de Fısica Teorica I, Universidad Complutense, Madrid, Spain

5International Centre for Theoretical Physics, Trieste, Italy

The recent developments of quantum technologies have drastically enhanced our capabilityto access the many-body properties of driven-dissipative systems at the nanoscale, such as forcoupled QED cavities or for superconducting circuits [1, 2]. The investigation of nonequilibriumproperties of driven-dissipative systems has thus entered the quantum world (see e.g. Ref. [3]).Preliminary theoretical studies beyond the crude mean-field approximation have put forwardan intriguing scenario, which seems to substantially di↵er from that of closed-system situa-tions. Renormalization-group calculations using the Keldysh formalism have conjectured thatthe critical behavior of quantum many-body systems may be modified both by the presence ofthe external environment and by nonequilibrium e↵ects [4]. Important qualitative di↵erenceswith respect to the equilibrium properties have been put forward even by means of complemetarynumerical approaches. With cluster mean-field techniques it has been possible to highlight thedramatic impact of short-range correlations on the steady-state dissipative phase diagram [5];the corner-space renormalization method [6] enabled to access finite-size scaling properties ofthe entanglement and one critical exponent of a dissipative phase transition [7].

In this work we provide an alternative numerical approach, which is based on the general-ization of numerical linked cluster expansions for quantum lattice models [8, 9] in a dissipativeframework. Clever resummation techniques can be used to significantly accelerate the conver-gence of the series expansions, in such a way that it is possible to obtain properties at thethermodynamic limit up to a given order, by numerically treating very small clusters of arbi-trary size. Combining this strategy with a Pade-approximant analysis of the obtained results,it is possible to compute the critical exponents associated to a given singularity of the thermo-dynamic properties in proximity of the phase-transition points.

[1] A. Tomadin and R. Fazio, J. Opt. Soc. Am. B 27, A130 (2010).[2] I. Carusotto and C. Ciuti, Rev. Mod. Phys. 85, 299 (2013).[3] A. A. Houck, H. E. Tureci, and J. Koch, Nat. Phys. 8, 292 (2012).[4] L. M. Sieberer, S. D. Huber, E. Altman, and S. Diehl, Phys. Rev. Lett. 110, 195301 (2013).[5] J. Jin, A. Biella, O. Viyuela, L. Mazza, J. Keeling, R. Fazio, and D. Rossini, Phys. Rev. X 6, 031011

(2016).[6] S. Finazzi, A. Le Boite, F. Storme, A. Baksic, and C. Ciuti, Phys. Rev. Lett. 115, 080604 (2015).[7] R. Rota, F. Storme, N. Bartolo, R. Fazio, and C. Ciuti, arXiv:1609.02848[8] M. Rigol, T. Bryant, and R. R. P. Singh, Phys. Rev. Lett. 97, 187202 (2006).[9] H. C. Oitmaa and J. W. Zhang, Series expansion methods for strongly interacting lattice models,

Cambridge University Press, Cambridge, UK (2006).

Critical behavior of 2D dissipative spin lattices

R. Rota,1 F. Storme,1 N. Bartolo,1 R. Fazio,2, 3 and C. Ciuti1

1Laboratoire Materiaux et Phenomenes Quantiques,

Universite Paris Diderot - Paris 7, Paris, France

2ICTP, Trieste, Italy and NEST, Scuola Normale Superiore

3Istituto Nanoscienze-CNR, Pisa, Italy

The study of dissipative phase transitions is an emerging topic of research for non-equilibriumquantum manybody systems, which can be realized in artificial platforms using Rydberg atoms,semiconductor microstructures or superconducting circuits. Recently, unconventional magneticphase transitions have been predicted in spin lattices described by a dissipative Heisenbergmodel with anisotropic spin-spin coupling and incoherent spin relaxation: in particular, thepredictions have been based on single-site [1] and cluster mean-field [2] theory. A crucial problemis to explore the physical properties beyond mean-field.

By applying the corner-space renormalization method [3], we have explored the critical be-havior of such class of spin systems [4]. We have been able to investigate the finite-size scalingand to calculate the critical exponent of the magnetic linear susceptibility. We show that theVon Neumann entropy increases across the critical point, revealing a strongly mixed character ofthe ferromagnetic phase. At the same time, the quantum Fisher information, an entanglementwitness, exhibits a critical behavior at the transition point, showing that quantum correlationsplay a crucial role. Our results suggest that dissipative phase transition can share properties ofboth thermal and quantum phase transitions.

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Inset: maximum value �max

av

of the susceptibility as a function of the size L of the lattice.

[1] T. E. Lee, S. Gopalakrishnan, and M. D. Lukin, Phys. Rev. Lett. 110, 257204 (2013).[2] J. Jin, A. Biella, O. Viyuela, L. Mazza, J. Keeling, R. Fazio, and D. Rossini, Phys. Rev. X 6, 031011

(2016)[3] S. Finazzi, A. Le Boite, F. Storme, A. Baksic and C. Ciuti, Phys. Rev. Lett. 115, 080604 (2015).[4] R. Rota, F. Storme, N. Bartolo, R. Fazio and C. Ciuti, Phys. Rev. B 95, 134431 (2017).

Dissipation-Induced Superradiance in a Non-Markovian Open Dicke Model

Marco Schiro1 and Orazio Scarlatella1

1Institut de Physique Theorique, Universite Paris Saclay,

CNRS, CEA, F-91191 Gif-sur-Yvette, France

We consider the Dicke model, describing an ensemble of N quantum spins interacting witha cavity field, and study how the coupling to a non-Markovian environment with power-lawspectrum changes the physics of superradiant phase transition. Quite remarkably we find thatdissipation can induce, rather than suppress, the ordered phase, a result which is in strikingcontrast with both thermal and Markovian quantum baths. We interpret this dissipation-induced superradiance as a genuine dissipative quantum phase transition that exists even atfinite N due to the coupling with the bath modes and whose nature and critical propertiesstrongly depend on the spectral features of the non-Markovian environment [1].

[1] Scarlatella, O. and Schiro, M., arXiv:1611.09378

Observation of the thermal de Broglie wavelength in a two-dimensionalphotonic quantum gas

T. Damm, D. Dung, F. Vewinger, J. Schmitt, and M. Weitz1

1Institut for Applied Physics, University of Bonn, Wegelerstrae 8, 53115 Bonn

Tailoring the dispersion of particles is the background of many advances of modern physics,with the linear (i.e. light-like) dispersion of electrons in graphene or in topological materialsconstituting important examples. In reverse, photons and polaritons confined in microcavitiescan become two-dimensional systems with a quadratic (i.e. matter-like) dispersion, allowing forBose-Einstein condensation when thermalization is achieved [1, 2].

N/Nc ≈ 1.76

10 µm

x (µm)10-10 0

x (µm)10-10 0

y (µm)

10

0

-10

10 µm

N/Nc ≈ 10-4

Position-resolved interference of the photon gas

emitted from the microcavity in the thermal

(left) and condensed (right) phase regime. While

below the critical photon number Nc the spatial

correlations are short-range and determined by

the thermal de Broglie wavelength, the onset of

condensation is accompanied by the emergence

of long-range order.

In this talk, I will present an exper-imental determination of the thermal deBroglie wavelength of a two-dimensionalphoton gas independent from quantumstatistics, using position-resolved interfer-ometric measurements of the emission of adye-filled optical microcavity [3]. Informa-tion about the thermal de Broglie wave-length is encoded in the transverse spa-tial correlations of the photon gas, whichyield �th ' 1.48(1) µm, in good agreementwith theory. We observe the expected de-pendence with temperature T , which gen-uinely verifies the thermal character. Asthe central phase-space density of the pho-ton gas is increased beyond the criticalvalue of ⇡2/3, a Bose-Einstein condensateof photons is realized in the ground stateof the harmonic trapping potential inducedby the bispherical microcavity. The onset

of Bose-Einstein condensation agrees with the concept that quantum statistical e↵ects emergewhen the thermal de Broglie wave packets overlap, upon which we observe long-range ordereventually exceeding the condensate diameter of 12.7 µm.

For the future, spatial first-order coherence measurements are expected to reveal possiblelong-lived phase singularities from vortices in thermo-optically or Kerr nonlinearity inducedphoton superfluids [4]. Other than atomic condensates, optical quantum gases can be subjectto grand canonical statistics [5], which could give rise to novel vortex dynamics. Finally, oursetup might be a tool to study critical scaling at the phase transition and universality in opticalquantum gases.

[1] H. Deng, H. Haug, and Y. Yamamoto, Rev. Mod. Phys. 82, 1489 (2010).[2] J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, Nature 468, 545 (2010) .[3] T. Damm, D. Dung, F. Vewinger, J. Schmitt, and M. Weitz, in preparation (2016).[4] M. Strinati, and C. Conti, Phys. Rev. A. 90, 043853 (2014).[5] J. Schmitt, T. Damm, D. Dung, F. Vewinger, J. Klaers, and M. Weitz, Phys. Rev. Lett. 112, 030401

(2014).

Topological pumping and non-equilibrium string-like order in

driven-dissipative photonic lattices

Jirawat Tangpanitanon,1 Victor M. Bastidas,1 Sarah Al-Assam,2

Pedram Roushan,3 Dieter Jaksch,2, 1 and Dimitris G. Angelakis1, 4

1Centre for Quantum Technologies, National Universityof Singapore, 3 Science Drive 2, Singapore 117543

2Clarendon Laboratory, University of Oxford,Parks Road, Oxford OX1 3PU, United Kingdom

3Google Inc., Santa Barbara, California 93117, USA4School of Electrical and Computer Engineering,

Technical University of Crete, Chania, Crete, Greece, 73100

In the first part, we show how to implement topological or Thouless pumping of interactingphotons in one dimensional nonlinear resonator arrays, by simply modulating the frequency ofthe resonators periodically in space and time. The interplay between interactions and the adia-batic modulations enables robust transport of Fock states with few photons per site. We analyzethe transport mechanism via an e↵ective analytic model and study its topological properties andits protection to noise [1]. In the second part, we present a study on the extended Bose-Hubbardmodel under external time-dependent driving that preserves the global Z2 symmetry. We anal-yse the possibility for generating an out of equilibrium phase exhibiting topological string-likeorder characterising Haldane Insulator phases. We provide an analytic model discussing therole of the drive in generating the exotic phase and back up our results with extensive tensornetwork based numerics [2]. In both cases, we provide the details of a possible implementationin existing circuit QED architectures.

[1] J. Tangpanitanon, V. M. Bastidas, S. Al-Assam, P. Roushan, D. Jaksch, and D. G. Angelakis, PRL,117, 21, 213603, 2016

[2] J. Tangpanitanon, S. Clark, S. Al-Assam, R. Fazio, D. Jaksch, and D. G. Angelakis, Out of equilib-rium string-like order phases in driven dissipative many-body photonics, in preparation.

FIG. 1. (a) Quantized transport of photonic Fock states in 1D nonlinear lattices due to topologicalpumping. The system is prepared initially at |...030030...i and the particles are shifted by one latticepoint corresponding to topological transport with Chern number 1 (b) A scheme for observing Haldaneinsulator like phases in driven dissipative nonlinear lattices under time dependent driving. Under theright type of drive many-body states exhibiting non-zero string-like order can emerge in this far out ofequilibrium setting.

Electrically tunable artificial gauge potential for magnetopolaritons

Hyang-Tag Lim,1 Emre Togan,1 and Atac Imamoglu1

1Institute of Quantum Electronics, ETH Zurich, CH-8093 Zurich, Switzerland.

Neutral particles subject to artificial gauge potentials can behave as charged particles inmagnetic fields. This fascinating premise has led to demonstrations of one-way waveguides [1],topologically protected edge states [2, 3] in photonic systems and Landau levels for photons innon-planar cavities [4]. In ultracold neutral atoms artificial gauge potentials have allowed theemulation of matter under strong magnetic fields leading to realization of Harper-Hofstadter [5]and Haldane models [6]. Here we show that application of perpendicular electric and magneticfields e↵ects an artificial gauge potential for two-dimensional microcavity exciton polaritons. Forverification, we perform interferometric measurements to determine the magnitude of the gaugepotential induced phase accumulated during coherent polariton transport. Since the gauge po-tential originates from the magnetoelectric Stark e↵ect [7], it can be realized for photons stronglycoupled to excitations in any polarizable medium. Together with strong polariton-polariton in-teractions [8–10] and engineered polariton lattices [11], artificial gauge potentials could play akey role in investigation of non-equilibrium dynamics of strongly correlated photons [12].

[1] Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, Nature 461, 772 (2009).[2] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev,

and A. Szameit, Nature 496, 196 (2013).[3] M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, Nature Photonics 7, 1001 (2013).[4] N. Schine, A. Ryou, A. Gromov, A. Sommer, and J. Simon, Nature 534, 671 (2016).[5] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Physical Review

Letters 111, 185301 (2013).[6] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Nature

515, 237 (2014).[7] C. Kallin and B. I. Halperin, Physical Review B 30, 5655 (1984).[8] A. Amo, J. Lefrere, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. Houdre, E. Giacobino, and

A. Bramati, Nature Physics 5, 805 (2009).[9] L. Ferrier, S. Pigeon, E. Wertz, M. Bamba, P. Senellart, I. Sagnes, A. Lematre, C. Ciuti, and

J. Bloch, Applied Physics Letters 97, 031105 (2010).[10] L. Ferrier, E. Wertz, R. Johne, D. D. Solnyshkov, P. Senellart, I. Sagnes, A. Lemaıtre, G. Malpuech,

and J. Bloch, Physical Review Letters 106, 126401 (2011).[11] T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin,

A. Lemaıtre, J. Bloch, and A. Amo, Physical Review Letters 112, 116402 (2014).[12] I. Carusotto and C. Ciuti, Reviews of Modern Physics 85, 299 (2013).

Topological pumps and synthetic dimensions

O. Zilberberg,1 Y. Lahini,2 Z. Ringel,3 M. Lohse,4, 5 C. Schweizer,4, 5

M. Aidelsburger,4, 5 I. Bloch,4, 5 Tomoki Ozawa,6 Hannah M.

Price,6 Nathan Goldman,7 Iacopo Carusotto,6 and K. Kraus†

1Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland.2Harvard John A. Paulson School of Engineering and Applied Sciences,

Harvard University, Cambridge, Massachusetts 02138, USA3Theoretical Physics, Oxford University, 1,

Keble Road, Oxford OX1 3NP, United Kingdom4Fakultat fur Physik, Ludwig-Maximilians-Universitat,

Schellingstrasse 4, 80799 Munchen, Germany5Max-Planck-Institut fur Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany

6INO-CNR BEC Center and Dipartimento di Fisica,Universita di Trento, I-38123 Povo, Italy

7CENOLI, Faculte des Sciences, Universite Libre de Bruxelles (U.L.B.), B-1050 Brussels, Belgium

Topological phases of matter have fascinated researchers since over thirty years. Indeed, thisyear’s Nobel prize joins the two Nobel awards for the quantum Hall e↵ects in commending thisunique field. In my talk, I will start with introducing the quantum Hall e↵ect and demonstratehow it is related to topological pumps [1, 2]. I will, then, present our realizations of topologicalpumps using two completely di↵erent bosonic systems, namely, using coupled photonic waveg-uide arrays [3, 4] and with trapped atoms in optical superlattices [5]. Last, I will present howthe notion of two-dimensional topological pumps naturally leads to a path for realizing the 4Dquantum Hall e↵ect in the lab using synthetic dimensions [6–8].

[1] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).[2] D. J. Thouless, Phys. Rev. B 27, 6083 (1983).[3] Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and OZ, Phys. Rev. Lett. 109, 106402 (2012).[4] M. Verbin, OZ, Y. Lahini, Y. E. Kraus, and Y. Silberberg, Phys. Rev. B 91, 064201 (2015).[5] M. Lohse, C. Schweizer, OZ, M. Aidelsburger, and I. Bloch, Nature Phys. 12, 350 (2016).[6] Y. E. Kraus, Z. Ringel, and OZ, Phys. Rev. Lett. 111, 226401 (2013).[7] H. M. Price, OZ, T. Ozawa, I. Carusotto, and N. Goldman, Phys. Rev. Lett. 115, 195303 (2015).[8] T. Ozawa, H. M. Price, N. Goldman, OZ, and I. Carusotto, Phys. Rev. A 93, 043827 (2016).† Deceased 9 November 2016.