While the mechanisms governing the equilibrium properties of quantum gases are today relatively well understood, their non-equilibrium counterparts remain largely unknown. In this thesis, we theoretically explore the non-equilibrium physics of low-dimensional, interacting Bose gases in two representative configurations. In the first one, we consider an isolated Bose superfluid in two dimensions following a quench of the coupling constant. Using a quantum hydrodynamic description combined with a Keldysh field formalism, we derive quantum kinetic equations for the low-energy phononic excitations of the system. This enables us to describe the system's evolution from its short-time (prethermal) coherent dynamics to its long-time thermalization. In particular, we derive and discuss the Landau-Beliaev damping rates both in the standard perturbative approach and within a self-consistent extension of the theory. The results are also discussed in the light of recent experiments. Our second study explores a one-dimensional, periodically-driven quantum system, a Bose gas subjected to periodically-kicked interactions. In the limit of infinitely short kicks, this system was previously shown to exhibit a fast exponential spreading of the wave function. By revisiting this problem for kicks of arbitrary duration, we demonstrate that, in this case, the spreading is not exponential but sub-diffusive at long time. Our findings indicate that the breakdown of exponential spreading occurs at very short times even for extremely short kicks, suggesting that sub-diffusion is more the rule than the exception in that system.
While the mechanisms governing the equilibrium properties of quantum gases are today relatively well understood, their non-equilibrium counterparts remain largely unknown. In this thesis, we theoretically explore the non-equilibrium physics of low-dimensional, interacting Bose gases in two representative configurations. In the first one, we consider an isolated Bose superfluid in two dimensions following a quench of the coupling constant. Using a quantum hydrodynamic description combined with a Keldysh field formalism, we derive quantum kinetic equations for the low-energy phononic excitations of the system. This enables us to describe the system's evolution from its short-time (prethermal) coherent dynamics to its long-time thermalization. In particular, we derive and discuss the Landau-Beliaev damping rates both in the standard perturbative approach and within a self-consistent extension of the theory. The results are also discussed in the light of recent experiments. Our second study explores a one-dimensional, periodically-driven quantum system, a Bose gas subjected to periodically-kicked interactions. In the limit of infinitely short kicks, this system was previously shown to exhibit a fast exponential spreading of the wave function. By revisiting this problem for kicks of arbitrary duration, we demonstrate that, in this case, the spreading is not exponential but sub-diffusive at long time. Our findings indicate that the breakdown of exponential spreading occurs at very short times even for extremely short kicks, suggesting that sub-diffusion is more the rule than the exception in that system.
We study theoretically the nonequilibrium dynamics of a two-dimensional (2D) uniform Bose superfluid following a quantum quench, from its short-time (prethermal) coherent dynamics to its long-time thermalization. Using a quantum hydrodynamic description combined with a Keldysh field formalism, we derive quantum kinetic equations for the low-energy phononic excitations of the system and characterize both their normal and anomalous momentum distributions. We apply this formalism to the interaction quench of a 2D Bose gas and study the ensuing dynamics of its quantum structure factor and coherence function, both recently measured experimentally. Our results indicate that, in two dimensions, a description in terms of independent quasiparticles becomes quickly inaccurate and should be systematically questioned when dealing with nonequilibrium scenarios.
This thesis manuscript is devoted to different aspects of the non-equilibrium dynamics of a moving two-dimensional ultra-cold Bose gas, under the simultaneous action of a spatially disordered potential and interactions between particles. In the absence of interactions, the momentum distribution of the gas is the sum of two well-known components: a classical, diffusive background made of particles multiply scattered by the disorder, and a coherent backscattering interference peak (CBS peak), consequence of the weak localization phenomenon. In the presence of weak interactions, mean field simulations show that these two contributions evolve in time. This evolution is described by two coupled kinetic equations, which we derive using a microscopic quantum transport theory. These predict the destruction of the CBS peak by an effective dephasing mechanism, and the thermalization of the diffusive background. As the interaction strength increases, the effects of disorder become negligible, and the dynamics of the gas resembles that of a superfluid. In this regime, the gas quickly enters a so-called "pre-thermal" state: the spatial correlations g_1 decrease algebraically and propagate along a light cone, in accordance with theoretical predictions. Finally, at long times, our simulations describe the global evolution of the system towards a thermal equilibrium state, characterized by a Kosterlitz-Thouless transition separating a normal fluid state from a superfluid state. Some properties of this phase transition, resulting from the non-equilibrium evolution in the presence of disorder, are discussed.
Dynamique hors d’équilibre et thermalisation des gaz de Bose isolés et forcés en basse dimension Si les mécanismes régissant les propriétés d'équilibre des gaz quantiques sont aujourd'hui relativement bien compris, leurs équivalents hors équilibre restent largement inconnus. Dans cette thèse, nous explorons théoriquement la physique hors équilibre des gaz de Bose en interaction à basse dimension dans deux situations représentatives. Dans la première, nous considérons un superfluide de Bose isolé en dimension deux suite à un changement brutal de la constante de couplage. En utilisant une description hydrodynamique quantique combinée à un formalisme de champ de Keldysh, nous obtenons des équations cinétiques quantiques pour les excitations phononiques de basse énergie du système. Cette méthode nous permet de décrire l'évolution du système depuis sa dynamique cohérente aux temps courts (régime préthermal) jusqu'à sa thermalisation aux temps longs. En particulier, nous calculons et discutons les taux d'amortissement Landau-Beliaev à la fois dans une approche perturbative standard et dans son extension auto-cohérente. Les résultats sont également discutés au regard d'expériences récentes. Notre deuxième étude porte sur un système quantique unidimensionnel forcé, un gaz de Bose soumis à des interactions périodiques. Dans la limite de kicks infiniment courts, il a été montré récemment que ce système présentait un étalement exponentiellement rapide de la fonction d'onde. En revisitant ce problème pour des kicks de durée arbitraire, nous montrons que, dans ce cas, l'étalement n'est pas exponentiel mais sous-diffusif aux temps longs. Nos résultats indiquent que le passage du comportement exponentiel à sous-diffusif se produit à des temps très courts, même pour des kicks extrêmement brefs, ce qui suggère que la sous-diffusion est le mécanisme générique de transport dans ce type de système.
The study of out-of-equilibrium dynamics in ultra-cold gases is a new and exciting field, driven largely by the recent experimental advances in controlling and imaging cold clouds. The experimental and theoretical work thus far has been somewhat exploratory and largely numerical in nature, as the very paradigms for thinking about these systems are not well established. In this thesis I consider several different scenarios of ultra-cold bosonic and fermionic gases driven out of equilibrium and study their properties. In Chapter 1, I provide an overview of the phenomenology of ultra-cold gases, highlighting the timescales governing these systems and how the experimentalist can tune them. I discuss how cold gases can be cooled and trapped and discuss the basic physics behind optical lattices. I also discuss experimental probes of these gases, in particular the new high resolution imaging techniques developed recently at Chicago, Munich and Harvard. In Chapter 2, I discuss an early experiment (circa. 2008) which observed long lived spin dynamics in a thermal spin-1/2 Fermi gas. This experiment is an nice illustration of interesting physics resulting from the separation of timescales between spin and collisional dynamics. In my opinion, it is an excellent example of why cold gases are naturally suited to studying non-equilibrium dynamics. I simulate the experiment numerically using a collisionless Boltzmann equation and explain the observed spin dynamics both qualitatively and quantitively. In Chapter 3, I continue the discussion of spin waves in thermal gases by extending previous works on spin-1/2 gases to spin-1 Bose gases. In contrast to Chapter 2, the bulk of the work in this Chapter is analytic in nature. In particular, I find a spin wave instability in the thermal spin-1 Bose gas, which is the high temperature analog of the polar to ferromagnetic transition in a spin-1 Bose Einstein condensate. In Chapter 3, I turn my attention to bosonic systems and briefly review the the Bogoliubov mean-field theory. I calculate the momentum distribution and density-density correlation function of an interacting Bose gas within the Bogoliubov framework. Then I consider bosons in an optical lattice, and introduce the Bose Hubbard model. I calculate the mean-field phase diagram of the Bose Hubbard model and then consider fluctuations about the mean field, and derive the excitation spectrum of the lattice gas in the superfluid and insulating regimes. In Chapter 4, I ask what we learn by studying the dynamics of correlation functions following a sudden change in the interactions in a superfluid. Using the Bogoliubov theory developed in Chapter 3, I will show how the underlying excitation spectrum influences the long and short time behavior of the correlation functions. By considering a lattice dispersion, I study the analogous problem in a weak optical lattice and discuss how the lattice dispersion leads to additional features in the correlation functions. I will also discuss the timescale governing the revival of the condensate fraction in a quantum depleted gas. In Chapter 5, I derive equations of motion governing the dynamics of one and two body correlation functions in the single-band Bose Hubbard model, applicable to bosons in deep lattices. I then consider a simple quench from a Mott insulating initial state to a weakly interacting final state and produce analytic expressions describing the dynamics of correlations following such a quench. I discuss the timescale for the development of long range order following such a quench. I study the problem of chapter 4 using an equations of motion approach. This approach complements the Bogoliubov approach of Chapter 4. First, I derive exact expressions for a quench to a non-interacting state. I then consider how interactions redistribute quasi-momentum to first order in perturbation theory in different dimensions. In Chapter 6, I calculate the relevant timescales for local and global dynamics in trapped lattice Bose gases, a work done in collaboration with Dr. Kaden R. A Hazzard. Using a time-dependent Gutzwiller mean-field theory, I show that the timescale for local equilibration in these systems is fast in experimental terms. I then show that due to the spatial inhomogeneities inherent to cold gases, achieving global equilibrium can be quite complicated, sometimes taking longer than the lifetime of the experiment, an issue of practical importance to current day experiments. I continue this discussion in Chapter 7 which is a collaborative work with experimentalists David McKay and Prof. Brian DeMarco from the University of Toronto and the University of Illinois, Urbana Champaign. Using experimental and numerical methods, we show that the rapid timescales for local dynamics in interacting systems invalidates a frequently used cold atom technique for mapping out the momentum distribution of atoms in an optical lattice.
The experimental control and observation of quantum many-body systems has become a reality with the advent of ultracold quantum matter. The high level of isolation of these experiments, together with the development of novel measurement techniques, has revived a fundamental debate concerning the thermal equilibration of isolated quantum systems, commonly named "quantum thermalization". In this thesis we make use of a quantum-gas microscope to explore the thermalizing dynamics of highly isolated systems of ultracold bosonic atoms in optical lattices. The ability to prepare and control quantum systems made up of hundreds of atoms makes it possible to explore regimes that represent a challenge for classical numeric simulations. A major part of this dissertation deals with Bose-Hubbard systems in the presence of quenched disorder. We begin by studying the microscopic properties of its phases near equilibrium, where by tuning the strength of the disorder, observe features consistent with the emergence of a so-called Bose-glass phase. We then continue by preparing states far from equilibrium and exploring their quantum many-body dynamics. In particular, we observe signatures of the phenomenon of "many-body localization", which implies a breakdown of quantum thermalization. In addition, we study whether localized systems can be thermalized via the coupling to a bath of few degrees of freedom, i.e. a quantum bath. We do so by preparing a mixture of two interacting atomic species, where one acts as the bath and the other as the localized system. We do observe delocalizing dynamics for large enough baths, though in regimes of weak coupling localization can survive for extremely long times. The second main topic of this thesis is the thermalization of periodically driven many-body systems, so-called Floquet thermalization. In these systems, the absence of energy conservation eventually brings any initial state into a featureless infinite-temperature one. However, for sufficiently high frequencies this thermalization process can take arbitrarily long times, which can enable the engineering of exotic long-lived prethermal states. We use the high isolation of our system, together with the high sensitivity of quantum-gas microscopy, to measure the heating rates for a range of driving frequencies and interaction regimes. Our results show a dramatic suppression of the heating as the frequency of the drive is increased, which is consistent with theoretical expectations.
Abstract We explore the quench dynamics of a two-dimensional, weakly interacting disordered Bose gas for various relative strengths of interactions and disorder. This allows us to identify two well distinct out-of-equilibrium regimes. When interactions are smaller than the disorder, the gas experiences multiple scattering and exhibits a short-range spatial coherence. At short time this coherence is only smoothly affected by interactions, via a diffusion process of the particles' energies. When interactions are larger than the disorder, scattering ceases and the gas behaves more and more like a fluid, ultimately like a superfluid at low energy. In the superfluid regime, the gas exhibits a long-range algebraic coherence, characteristic of a pre-thermal regime in disorder.