Recent progress in Cavity QED (CQED) has brought forth the possibility of wiring up individual CQED systems into lattices to investigate collective behavior and correlated phases of strongly coupled light and matter [1-9]. Such Lattice CQED systems operate on polaritonic quasi-particles that are hybrids of light and matter in a controllable proportion, combining long-range coherence of photons and strong interactions typically displayed by massive particles. In this talk, I will review recent work on proposed quantum simulation of Hubbard models with Lattice CQED systems, which are akin to the celebrated Bose-Hubbard model extensively studied with cold atomic gases in optical lattices and Josephson junction arrays. A crucial feature of Lattice CQED systems that I will focus on is their non-equilibrium nature. Unavoidable photon loss coupled with the ease of feeding in additional photons through continuous external driving renders such lattices open quantum systems. Under the influence of drive and dissipation, Lattice CQED systems typically display steady-state phases thereby providing a platform for realizing novel, non-equilibrium phases of light and matter that may be very dierent from those realized with condensed matter systems.
Recent progress in Cavity QED (CQED) has brought forth the possibility of wiring up individual CQED systems into lattices to investigate collective behavior and correlated phases of strongly coupled light and matter [1-9]. Such Lattice CQED systems operate on polaritonic quasi-particles that are hybrids of light and matter in a controllable proportion, combining long-range coherence of photons and strong interactions typically displayed by massive particles. In this talk, I will review recent work on proposed quantum simulation of Hubbard models with Lattice CQED systems, which are akin to the celebrated Bose-Hubbard model extensively studied with cold atomic gases in optical lattices and Josephson junction arrays. A crucial feature of Lattice CQED systems that I will focus on is their non-equilibrium nature. Unavoidable photon loss coupled with the ease of feeding in additional photons through continuous external driving renders such lattices open quantum systems. Under the influence of drive and dissipation, Lattice CQED systems typically display steady-state phases thereby providing a platform for realizing novel, non-equilibrium phases of light and matter that may be very dierent from those realized with condensed matter systems.
Collective quantum phenomena are fascinating, as they repeatedly challenge our comprehension of nature and its underlying mechanisms. The qualification ``quantum'' can be attributed to a generic many-body system whenever the interference effects related to the underlying wave nature of its elementary constituents can not be neglected anymore, and a naive classical description in terms of interacting billiard balls fails to catch its most essential features. This interference phenomenon called ``quantum degeneracy'' which occurs at weak temperatures, leads to spectacular collective behaviours such as the celebrated Bose-Einstein Condensation (BEC) phase transition, where a macroscopic fraction of a bosonic system of particles collapses below a critical temperature T_c on a single-particle state. Quantum coherence, when combined with inter-particle interactions, gives rise to highly non-classical frictionless hydrodynamic behaviours such as superfluidity (SF) and superconductivity (SC). \nEven more exotic quantum phases emerge in presence of important interactions as matter reaches a ``strongly correlated regime'' dominated by quantum fluctuations, where each particle is able to affect significantly the surrounding fluid: characteristic examples are the so-called Mott-Insulator (MI) quantum phase where particles are localized on a lattice due to a strong interaction-induced blockade, along with the Tonks-Girardeau (TG) gas where impenetrable bosons in one-dimension acquire effective fermionic statistics up to a unitary transformation, and the Fractional Quantum Hall (FQH) effect which occurs in presence of a gauge field, and features a special type of elementary excitation possessing a fractional charge and obeying to fractional statistics called `anyon'. \nThese quantum many-body effects were explored in a first place in systems well isolated from the external environment such as ultra-cold atomic gases or electrons in solid-state systems, within a physical context well described by ``equilibrium statistical mechanics''. Yet, over the last two decades a broad community has started investigating the possibility of stabilizing interacting quantum phases in novel nonlinear quantum optics architectures, where interacting photons have replaced their atomic and electronic counterpart. Thanks to their high level of controllability and flexibility, and the possibility of reaching the quantum degeneracy regime at exceptionally high temperatures, these platforms appear as extremely promising candidates for the ``quantum simulation'' of the most exotic many-body quantum problems: while the precursors experiments in semiconductor exciton-polariton already allow to reach the Bose-Einstein Condensation and superfluid regimes, novel platforms such as superconducting circuits, coupled cavity arrays or photons coupled to Rydberg EIT (Electromagnetically induced Transparency) atoms have entered the so-called `photon blockade' where photons behave as impenetrable particles, and open a encouraging pathway toward the future generation of strongly correlated phases with light. \nA specificity of quantum optics devices is their intrinsic ``non-equilibrium'' nature: the interplay between the practically unavoidable radiative and non-radiative losses and the external drive needed to replenish the photon gas leads the many-body system toward a steady-state presenting important non-thermal features. One one hand, an overwhelmingly large quantity of novel quantum phenomena is expected in the non-equilibrium framework, as breaking the thermal equilibrium condition releases severe constraints on the state of a quantum system and on the nature of its surrounding environment. On the other hand, we do not benefit yet of an understanding of non-equilibrium statistical mechanics comparable with its well-established equilibrium counterpart, which relies on strong historical foundations. Understanding how to tame (and possibly exploit) non-equilibrium effects in order to stabilize interesting quantum phases in a controlled manner often reveals a hard challenge. \nIn that prospect, an important conceptual issue in the non-equilibrium physics of strongly interacting photons regards the possibility of stabilizing ``incompressible quantum phases'' such as the Mott-Insulator or Fractional Quantum Hall states, and more generally to stabilize the ground-state of a given particle-number conserving Hamiltonian, in a physical context where dissipative losses can not be neglected. While being able to quantum simulate those emblematic strongly correlated quantum phases in this novel experimental context would strongly benefit to the quantum optics community, gaining such a kind of flexibility would also contribute to fill an important bridge between the equilibrium and the non-equilibrium statistical physics of open quantum systems, allowing to access in a controlled manner a whole new phenomenology at the interface between the two theories. \nIn this thesis I address those questions, which I reformulate in the following manner: \n \n-What are the conditions for the emergence of analogue equilibrium properties in open quantum systems in contact with a non-thermal environment ? \n \n-In particular, is it possible to stabilize strongly correlated quantum phases with a perfectly defined particle number in driven-dissipative photonic platforms, in spite of environment-induced losses and heating effects ? \n \nThe structure of the thesis is the following. \n \n \n[Chapter 1.] We give an overview of the physics of many-body photonic systems. As a first step we address the weakly interacting regime in the physical context of exciton-polaritons: after describing the microscopic aspects of typical experiments, we move to the discussion of non-equilibrium Bose-Einstein Condensation and the various mechanisms related to the emergence of thermal signatures at steady-state. The second part of this Chapter is dedicated to strongly interacting fluids. After drawing a quick overview of several experimental platforms presenting a good potential for the study of such physics in a near future, we discuss the relative performance of several schemes proposed in order to replenish the photonic population \n \n[Chapter 2.] We investigate the potential of a non-Markovian pump scheme with a narrow bandpass (Lorentzian shaped) emission spectrum for the generation of strongly correlated states of light in a Bose-Hubbard lattice. Our proposal can be implemented by mean of embedded inverted two-level emitters with a strong incoherent pumping toward the excited state. Our study confirms in a single cavity the possibility of stabilizing photonic Fock states in a single configuration, and strongly localized n=1 Mott-Insulator states in a lattice with n=1 density. We show that a relatively moderate hopping is responsible for a depletion of the Mott-state, which then moves toward a delocalized state reminiscent of the superfluid regime. Finally, we proceed to a mean-field analysis of the phase diagram, and unveil a Mott-to-Superfluid transition characterized by a spontaneous breaking of the U(1) symmetry and incommensurate density. The results of this Chapter are based on the following publications: \n- J. Lebreuilly, M. Wouters and I. Carusotto, ``Towards strongly correlated photons in arrays of dissipative nonlinear cavities under a frequency-dependent incoherent pumping'', C. R. Phys., 17(8), 836, 2016. \n- A. Biella, F. Storme, J. Lebreuilly, D. Rossini, R. Fazio, I. Carusotto and C. Ciuti, ``Phase diagram of incoherently driven strongly correlated photonic lattice'', Phys. Rev. A, 96, 023839, 2017. \n \n[Chapter 3.] In view of improving the performance of the scheme introduced in last chapter, and reproducing in particular the equilibrium zero temperature phenomenology in driven-dissipative photonic lattices, we develop a fully novel scheme based on the use of non-Markovian reservoirs with tailored broadband spectra which allows to mimick the effect of tunable chemical potential. Our proposal can be implemented by mean of a small number of emitters and absorbers and is accessible to current technologies. \n \nWe first analyse the case of a frequency-dependent emission with a square spectrum and confirm the possibility of stabilizing Mott insulator states with arbitrary integer density. Unlike the previous proposal the Mott state is robust against both losses and tunneling. A sharp transition toward a delocalized superfluid-like state can be induced by strong values of the tunneling or a change in the effective chemical potential. While an overall good agreement is found with the T=0 predictions, our analysis highlights small deviations from the equilibrium case in some parts of the parameters space, which are characterized by a non-vanishing entropy and the kinetic generation of doublon excitations. We finally consider an improved scheme involving additional frequency-dependent losses, and show in that case that the Hamiltonian ground-state is fully recovered for any choice of parameters. Our proposal, whose functionality relies on generic energy relaxation mechanisms and is not restricted to the Bose-Hubbard model, appears as a promising quantum simulator of zero temperature physics in photonic devices. The results of this Chapter are based on the following publication: \n- J. Lebreuilly, A. Biella, F. Storme, D. Rossini, R. Fazio, C. Ciuti and I. Carusotto, ``Stabilizing strongly correlated photon fluids with non-Markovian reservoirs'', Phys. Rev. A 96, 033828 (2017). \n \n[Chapter 4.] We adopt a broader perspective, and analyse the conditions for the emergence of analogous thermal properties in driven-dissipative quantum systems. We show that the impact of an equilibrated environment can be mimicked by several non-Markovian and non-equilibrated reservoirs. Chapter 2 already f
We review recent developments regarding the quantum dynamics and many-body physics with light, in superconducting circuits and Josephson analogues, by analogy with atomic physics. We start with quantum impurity models addressing dissipative and driven systems. Both theorists and experimentalists are making efforts towards the characterization of these non-equilibrium quantum systems. We show how Josephson junction systems can implement the equivalent of the Kondo effect with microwave photons. The Kondo effect can be characterized by a renormalized light frequency and a peak in the Rayleigh elastic transmission of a photon. We also address the physics of hybrid systems comprising mesoscopic quantum dot devices coupled with an electromagnetic resonator. Then, we discuss extensions to Quantum Electrodynamics (QED) Networks allowing one to engineer the Jaynes–Cummings lattice and Rabi lattice models through the presence of superconducting qubits in the cavities. This opens the door to novel many-body physics with light out of equilibrium, in relation with the Mott–superfluid transition observed with ultra-cold atoms in optical lattices. Then, we summarize recent theoretical predictions for realizing topological phases with light. Synthetic gauge fields and spin–orbit couplings have been successfully implemented in quantum materials and with ultra-cold atoms in optical lattices — using time-dependent Floquet perturbations periodic in time, for example — as well as in photonic lattice systems. Finally, we discuss the Josephson effect related to Bose–Hubbard models in ladder and two-dimensional geometries, producing phase coherence and Meissner currents. The Bose–Hubbard model is related to the Jaynes–Cummings lattice model in the large detuning limit between light and matter (the superconducting qubits). In the presence of synthetic gauge fields, we show that Meissner currents subsist in an insulating Mott phase.
Abstract The Jaynes‐Cummings model describes the coupling between photons and a single two‐level atom in a simplified representation of light‐matter interactions. In circuit QED, this model is implemented by combining microwave resonators and superconducting qubits on a microchip with unprecedented experimental control. Arranging qubits and resonators in the form of a lattice realizes a new kind of Hubbard model, the Jaynes‐Cummings‐Hubbard model, in which the elementary excitations are polariton quasi‐particles. Due to the genuine openness of photonic systems, circuit QED lattices offer the possibility to study the intricate interplay of collective behavior, strong correlations and non‐equilibrium physics. Thus, turning circuit QED into an architecture for quantum simulation, i.e., using a well‐controlled system to mimic the intricate quantum behavior of another system too daunting for a theorist to tackle head‐on, is an exciting idea which has served as theorists’ playground for a while and is now also starting to catch on in experiments. This review gives a summary of the most recent theoretical proposals and experimental efforts.
Systems of strongly interacting atoms and photons, which can be realized wiring up individual cavity QED systems into lattices, are perceived as a new platform for quantum simulation. While sharing important properties with other systems of interacting quantum particles, here we argue that the nature of light-matter interaction gives rise to unique features with no analogs in condensed matter or atomic physics setups. By discussing the physics of a lattice model of delocalized photons coupled locally with two-level systems through the elementary light-matter interaction described by the Rabi model, we argue that the inclusion of counterrotating terms, so far neglected, is crucial to stabilize finite-density quantum phases of correlated photons out of the vacuum, with no need for an artificially engineered chemical potential. We show that the competition between photon delocalization and Rabi nonlinearity drives the system across a novel Z(2) parity symmetry-breaking quantum criticality between two gapped phases that share similarities with the Dicke transition of quantum optics and the Ising critical point of quantum magnetism. We discuss the phase diagram as well as the low-energy excitation spectrum and present analytic estimates for critical quantities.
Recent advances in theoretical physics have increasingly focused on the dynamics of quantum systems with many interacting degrees of freedom, while ultracold atom platforms now enable the realization of diverse theoretical models with precise control over system parameters and advanced monitoring techniques. A prominent example is cavity quantum electrodynamics (CQED), where large atomic or molecular ensembles are coupled to high-finesse optical cavities, enabling strong interaction with confined electromagnetic modes and leading to phenomena such as the superradiant phase transition, in which atoms and photons form a coherent state. Most theoretical studies of CQED have concentrated on regimes of spatially uniform atom–cavity coupling, where collective behavior allows a description in terms of the classical dynamics of a few macroscopic variables. However, experimental capabilities now permit exploration beyond this limit, though theoretical progress is hindered by the exponential computational complexity of simulating large quantum systems, particularly when environmental coupling is included. This thesis addresses such challenges by developing efficient approaches, within the Keldysh formalism of non-equilibrium quantum field theory, to study systems with strongly disordered interactions coupled to external reservoirs. Two representative models are considered: a disordered spin–boson model describing light–matter interactions with quenched disorder, relevant to recent CQED experiments in confocal cavities, and a fermion–phonon system governed by the Yukawa–Sachdev–Ye–Kitaev (Yukawa-SYK) model. Both systems are quantum critical, exhibiting a continuum of low-energy modes above their stationary states, yet disorder produces qualitatively distinct dynamical behaviors: in the spin–boson system it induces slow, glassy relaxation, whereas in the Yukawa-SYK model it leads to rapid thermalization. These contrasting outcomes reveal the non-trivial role of disorder in quantum many-body systems and motivate further investigation into its effects on non-equilibrium quantum dynamics.
We review recent experimental and theoretical progress in realizing and simulating many-body phases of ultracold atoms in optical lattices, which gives access to analog quantum simulations of fundamental model Hamiltonians for strongly correlated condensed matter systems, such as the Hubbard model. After a general introduction to quantum gases in optical lattices, their preparation and cooling, and measurement techniques for relevant observables, we focus on several examples, where quantum simulations of this type have been performed successfully during the past years: Mott-insulator states, itinerant quantum magnetism, disorder-induced localization and its interplay with interactions, and topological quantum states in synthetic gauge fields.