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 manybody 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 BoseEinstein Condensation (BEC) phase transition, where a macroscopic fraction of a bosonic system of particles collapses below a critical temperature T_c on a singleparticle state. Quantum coherence, when combined with interparticle interactions, gives rise to highly nonclassical frictionless hydrodynamic behaviours such as superfluidity (SF) and superconductivity (SC). Even 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 socalled MottInsulator (MI) quantum phase where particles are localized on a lattice due to a strong interactioninduced blockade, along with the TonksGirardeau (TG) gas where impenetrable bosons in onedimension 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'. These quantum manybody effects were explored in a first place in systems well isolated from the external environment such as ultracold atomic gases or electrons in solidstate 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 manybody quantum problems: while the precursors experiments in semiconductor excitonpolariton already allow to reach the BoseEinstein 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 socalled `photon blockade' where photons behave as impenetrable particles, and open a encouraging pathway toward the future generation of strongly correlated phases with light. A specificity of quantum optics devices is their intrinsic ``nonequilibrium'' nature: the interplay between the practically unavoidable radiative and nonradiative losses and the external drive needed to replenish the photon gas leads the manybody system toward a steadystate presenting important nonthermal features. One one hand, an overwhelmingly large quantity of novel quantum phenomena is expected in the nonequilibrium 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 nonequilibrium statistical mechanics comparable with its wellestablished equilibrium counterpart, which relies on strong historical foundations. Understanding how to tame (and possibly exploit) nonequilibrium effects in order to stabilize interesting quantum phases in a controlled manner often reveals a hard challenge. In that prospect, an important conceptual issue in the nonequilibrium physics of strongly interacting photons regards the possibility of stabilizing ``incompressible quantum phases'' such as the MottInsulator or Fractional Quantum Hall states, and more generally to stabilize the groundstate of a given particlenumber 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 nonequilibrium statistical physics of open quantum systems, allowing to access in a controlled manner a whole new phenomenology at the interface between the two theories. In this thesis I address those questions, which I reformulate in the following manner: What are the conditions for the emergence of analogue equilibrium properties in open quantum systems in contact with a nonthermal environment ? In particular, is it possible to stabilize strongly correlated quantum phases with a perfectly defined particle number in drivendissipative photonic platforms, in spite of environmentinduced losses and heating effects ? The structure of the thesis is the following. [Chapter 1.] We give an overview of the physics of manybody photonic systems. As a first step we address the weakly interacting regime in the physical context of excitonpolaritons: after describing the microscopic aspects of typical experiments, we move to the discussion of nonequilibrium BoseEinstein Condensation and the various mechanisms related to the emergence of thermal signatures at steadystate. 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 [Chapter 2.] We investigate the potential of a nonMarkovian pump scheme with a narrow bandpass (Lorentzian shaped) emission spectrum for the generation of strongly correlated states of light in a BoseHubbard lattice. Our proposal can be implemented by mean of embedded inverted twolevel 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 MottInsulator states in a lattice with n=1 density. We show that a relatively moderate hopping is responsible for a depletion of the Mottstate, which then moves toward a delocalized state reminiscent of the superfluid regime. Finally, we proceed to a meanfield analysis of the phase diagram, and unveil a MotttoSuperfluid 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:  J. Lebreuilly, M. Wouters and I. Carusotto, ``Towards strongly correlated photons in arrays of dissipative nonlinear cavities under a frequencydependent incoherent pumping'', C. R. Phys., 17(8), 836, 2016.  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. [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 drivendissipative photonic lattices, we develop a fully novel scheme based on the use of nonMarkovian 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. We first analyse the case of a frequencydependent 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 superfluidlike 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 nonvanishing entropy and the kinetic generation of doublon excitations. We finally consider an improved scheme involving additional frequencydependent losses, and show in that case that the Hamiltonian groundstate is fully recovered for any choice of parameters. Our proposal, whose functionality relies on generic energy relaxation mechanisms and is not restricted to the BoseHubbard 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:  J. Lebreuilly, A. Biella, F. Storme, D. Rossini, R. Fazio, C. Ciuti and I. Carusotto, ``Stabilizing strongly correlated photon fluids with nonMarkovian reservoirs'', Phys. Rev. A 96, 033828 (2017). [Chapter 4.] We adopt a broader perspective, and analyse the conditions for the emergence of analogous thermal properties in drivendissipative quantum systems. We show that the impact of an equilibrated environment can be mimicked by several nonMarkovian and nonequilibrated reservoirs. Chapter 2 already features a preliminary result in that direction, showing that in presence of a broad reservoir spectral density a given quantum system will evolve toward a Gibbs ensemble with an artificial chemical potential and temperature. In this chapter we develop a broader analysis focusing as a counterpart part on the exactly solvable model of a weakly interacting Bose Gas in the \acs{BEC} regime. Our formalism based on a quantum Langevin model, allows in particular to access both static and dynamical properties: remarkably, we demonstrate not only the presence of an equilibrium static signature, but also the validity of the fluctuationdissipation theorem. While our results apply only for lowenergy excitations for an arbitrary choice of reservoir spectral densities, we predict that a fine tuned choices of reservoirs mimicking the socalled Kennard Stepanov condition will lead to a full apparent equilibration. Such effect that we call ``pseudothermalization'' implies that under very specific conditions, an open quantum system can present all the properties of an equilibrated one in spite of the presence of an highly non equilibrated environment. The results of this Chapter are based on the following paper:  J. Lebreuilly, A. Chiocchetta and I. Carusotto, ``Pseudothermalization in drivendissipative nonMarkovian open quantum systems'', arXiv:1710.09602 (submitted for publication).
Strongly correlated quantum fluids and effective thermalization in nonMarkovian drivendissipative photonic systems / Lebreuilly, José Rafael Eric.  (2017), pp. 1134.
Strongly correlated quantum fluids and effective thermalization in nonMarkovian drivendissipative photonic systems
Lebreuilly, José Rafael Eric
20170101
Abstract
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 manybody 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 BoseEinstein Condensation (BEC) phase transition, where a macroscopic fraction of a bosonic system of particles collapses below a critical temperature T_c on a singleparticle state. Quantum coherence, when combined with interparticle interactions, gives rise to highly nonclassical frictionless hydrodynamic behaviours such as superfluidity (SF) and superconductivity (SC). Even 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 socalled MottInsulator (MI) quantum phase where particles are localized on a lattice due to a strong interactioninduced blockade, along with the TonksGirardeau (TG) gas where impenetrable bosons in onedimension 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'. These quantum manybody effects were explored in a first place in systems well isolated from the external environment such as ultracold atomic gases or electrons in solidstate 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 manybody quantum problems: while the precursors experiments in semiconductor excitonpolariton already allow to reach the BoseEinstein 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 socalled `photon blockade' where photons behave as impenetrable particles, and open a encouraging pathway toward the future generation of strongly correlated phases with light. A specificity of quantum optics devices is their intrinsic ``nonequilibrium'' nature: the interplay between the practically unavoidable radiative and nonradiative losses and the external drive needed to replenish the photon gas leads the manybody system toward a steadystate presenting important nonthermal features. One one hand, an overwhelmingly large quantity of novel quantum phenomena is expected in the nonequilibrium 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 nonequilibrium statistical mechanics comparable with its wellestablished equilibrium counterpart, which relies on strong historical foundations. Understanding how to tame (and possibly exploit) nonequilibrium effects in order to stabilize interesting quantum phases in a controlled manner often reveals a hard challenge. In that prospect, an important conceptual issue in the nonequilibrium physics of strongly interacting photons regards the possibility of stabilizing ``incompressible quantum phases'' such as the MottInsulator or Fractional Quantum Hall states, and more generally to stabilize the groundstate of a given particlenumber 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 nonequilibrium statistical physics of open quantum systems, allowing to access in a controlled manner a whole new phenomenology at the interface between the two theories. In this thesis I address those questions, which I reformulate in the following manner: What are the conditions for the emergence of analogue equilibrium properties in open quantum systems in contact with a nonthermal environment ? In particular, is it possible to stabilize strongly correlated quantum phases with a perfectly defined particle number in drivendissipative photonic platforms, in spite of environmentinduced losses and heating effects ? The structure of the thesis is the following. [Chapter 1.] We give an overview of the physics of manybody photonic systems. As a first step we address the weakly interacting regime in the physical context of excitonpolaritons: after describing the microscopic aspects of typical experiments, we move to the discussion of nonequilibrium BoseEinstein Condensation and the various mechanisms related to the emergence of thermal signatures at steadystate. 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 [Chapter 2.] We investigate the potential of a nonMarkovian pump scheme with a narrow bandpass (Lorentzian shaped) emission spectrum for the generation of strongly correlated states of light in a BoseHubbard lattice. Our proposal can be implemented by mean of embedded inverted twolevel 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 MottInsulator states in a lattice with n=1 density. We show that a relatively moderate hopping is responsible for a depletion of the Mottstate, which then moves toward a delocalized state reminiscent of the superfluid regime. Finally, we proceed to a meanfield analysis of the phase diagram, and unveil a MotttoSuperfluid 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:  J. Lebreuilly, M. Wouters and I. Carusotto, ``Towards strongly correlated photons in arrays of dissipative nonlinear cavities under a frequencydependent incoherent pumping'', C. R. Phys., 17(8), 836, 2016.  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. [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 drivendissipative photonic lattices, we develop a fully novel scheme based on the use of nonMarkovian 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. We first analyse the case of a frequencydependent 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 superfluidlike 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 nonvanishing entropy and the kinetic generation of doublon excitations. We finally consider an improved scheme involving additional frequencydependent losses, and show in that case that the Hamiltonian groundstate is fully recovered for any choice of parameters. Our proposal, whose functionality relies on generic energy relaxation mechanisms and is not restricted to the BoseHubbard 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:  J. Lebreuilly, A. Biella, F. Storme, D. Rossini, R. Fazio, C. Ciuti and I. Carusotto, ``Stabilizing strongly correlated photon fluids with nonMarkovian reservoirs'', Phys. Rev. A 96, 033828 (2017). [Chapter 4.] We adopt a broader perspective, and analyse the conditions for the emergence of analogous thermal properties in drivendissipative quantum systems. We show that the impact of an equilibrated environment can be mimicked by several nonMarkovian and nonequilibrated reservoirs. Chapter 2 already features a preliminary result in that direction, showing that in presence of a broad reservoir spectral density a given quantum system will evolve toward a Gibbs ensemble with an artificial chemical potential and temperature. In this chapter we develop a broader analysis focusing as a counterpart part on the exactly solvable model of a weakly interacting Bose Gas in the \acs{BEC} regime. Our formalism based on a quantum Langevin model, allows in particular to access both static and dynamical properties: remarkably, we demonstrate not only the presence of an equilibrium static signature, but also the validity of the fluctuationdissipation theorem. While our results apply only for lowenergy excitations for an arbitrary choice of reservoir spectral densities, we predict that a fine tuned choices of reservoirs mimicking the socalled Kennard Stepanov condition will lead to a full apparent equilibration. Such effect that we call ``pseudothermalization'' implies that under very specific conditions, an open quantum system can present all the properties of an equilibrated one in spite of the presence of an highly non equilibrated environment. The results of this Chapter are based on the following paper:  J. Lebreuilly, A. Chiocchetta and I. Carusotto, ``Pseudothermalization in drivendissipative nonMarkovian open quantum systems'', arXiv:1710.09602 (submitted for publication).File  Dimensione  Formato  

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