This thesis is devoted to developing of an universal description of strongly interacting mesoscopic systems out of equilibrium. The nature of universalities we are looking for is twofold. Firstly, our goal is to find novel and simple dependences of physical observables on the experimental parameters. Secondly, we are looking for an universal theoretical approach that can be applied to a class of seemingly different strongly interacting physical problems. To achieve this generality, we rely on the bosonization technique, which allows one to account for interactions exactly. However, it is not trivial to apply this technique to out-of-equilibrium systems, because an injection of non-equilibrium electrons is naturally described at the fermion level, whereas the interactions are very elegantly incorporated within the bosonic framework. In this thesis, we discuss an approach that treats out-of-equilibrium corrections as perturbations. This allows us to stay within the framework of bosonization and to study non-equilibrium systems.
This thesis is devoted to developing of an universal description of strongly interacting mesoscopic systems out of equilibrium. The nature of universalities we are looking for is twofold. Firstly, our goal is to find novel and simple dependences of physical observables on the experimental parameters. Secondly, we are looking for an universal theoretical approach that can be applied to a class of seemingly different strongly interacting physical problems. To achieve this generality, we rely on the bosonization technique, which allows one to account for interactions exactly. However, it is not trivial to apply this technique to out-of-equilibrium systems, because an injection of non-equilibrium electrons is naturally described at the fermion level, whereas the interactions are very elegantly incorporated within the bosonic framework. In this thesis, we discuss an approach that treats out-of-equilibrium corrections as perturbations. This allows us to stay within the framework of bosonization and to study non-equilibrium systems.
This Ph.D. thesis concerns a very active area of current research in theoretical physics, which is non-equilibrium statistical mechanics. More specifically, the focus will be on the study of the non-equilibrium dynamics and fluctuations in isolated many body systems driven out of equilibrium. We shall consider primarily quantum systems, albeit many of the theoretical techniques that will be used can be applied to both classical and quantum statistical systems. The novel contribution of this work is two-fold. First, we have developed new analytical techniques to study exactly the large-deviation statistics of physical quantities relevant for the description of systems driven out of equilibrium, like the work done in a quantum quench or the time-integrated current characterizing transport of matter, energy or charge etc, discussed in Part I of the thesis. Second, we have singled out a class of experimentally relevant nondisordered quantum systems showing a dramatically slow approach to thermal equilibrium and we have provided quantitative predictions for this behavior, which is discussed in Part II.
The study of non-equilibrium behaviors of quantum matter is one of the research directions at the frontiers of condensed matter physics and, in general, of theoretical and experimental physics. Understanding the mechanisms which prevent many quantum particles from reaching thermal equilibrium can ultimately shed light on the foundations of statistical physics and provide insight into the anomalous long-time collective behaviors observed in experiments. In this thesis, we present new theoretical developments in this direction. In the first part of the thesis we discuss the non-equilibrium dynamics of the two-dimensional quantum Ising model, the prototypical model of magnetism. In particular, we investigate the dynamics of interfaces separating different spin domains. In some regimes, we can provide an exact description of their unitary time evolution, but we also discuss the robustness of our predictions away from the exactly-solvable limit. Moreover, we address the same problem in the presence of disorder and provide a hydrodynamic interpretation of our findings. In the second part of the thesis, we investigate the localization properties of disordered quantum systems. In particular, we develop a renormalization-group picture of the Anderson localization transition in generic geometries. This provides a new framework to understand the phase diagram of such systems and their critical behavior beyond the usual finite-size scaling analysis. We also provide further insights on this topic both by studying resonances and their connection with observables and by employing unsupervised non-parametric learning techniques to probe disordered systems in an experimentally accessible form. The results presented in this thesis constitute a step forward in understanding the non-equilibrium phases and regimes of quantum matter and they pave the way to interesting future developments.
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.
Understanding the non-equilibrium quantum dynamics of many-body systems is one of the most challenging problems in modern theoretical physics. While numerous approximate and exact solutions exist for systems in equilibrium, examples of non-equilibrium dynamics of many-body systems that allow reliable theoretical analysis are few and far between. In this paper, we discuss a broad class of time-dependent interacting systems subject to external linear and parabolic potentials, for which the many-body Schrödinger equation can be solved using a scaling transformation. We demonstrate that scaling solutions exist for both local and non-local interactions, and derive appropriate self-consistency equations. We apply this approach to several specific experimentally relevant examples of interacting bosons in one and two dimensions. As an intriguing result, we find that weakly and strongly interacting Bose gases expanding from a parabolic trap can exhibit very similar dynamics.
This thesis presents a series of theoretical studies of ultra cold atomic systems which model and propose experiments, and develop new computational techniques in order to elucidate aspects of many-body physics and non-equilibrium dynamics. In the first two studies I model the dynamics of nonlinear solitonic excitations in ultracold fermionic superfluids: the first simulates recent experiments and supports the hypothesis that the solitons generated in those experiments are unstable to the formation of vortex rings; the second demonstrates how population imbalance between up and down spin fermions can be used to prevent this instability. In the next study I discuss a method for generating and probing topologically protected edge states using periodically driven optical lattices potentials. Next I use a perturbative approach to study the spectral density of fermions with strong attractive interactions in the normal phase. After that I develop a novel cluster expansion technique to model the dynamics of interacting fermions in a disordered optical lattice. Finally I apply a Ginzurg-Landau theory to model experimental studies of superfluid 3He embedded in nematically ordered aerogel, finding evidence for a new phase of matter --the ``polar phase"-- which is not seen in bulk 3He.
This thesis deals with the study of transport properties of integer and fractional QH edge states and it is based on the work I performed during my Ph.D. studies. The focus of this thesis is on Luttinger liquids far from equilibrium and their relaxation dynamics. Since Boltzmann, a fundamental aspect of statistical mechanics has been the understanding of the emergence of an equilibrium state. Interactions play a crucial role in the thermalization process that drives a system through states described by the Gibbs equilibrium ensemble. Therefore, it seems counterintuitive that a strongly interacting system, such as the Luttinger liquid, should not present any relaxation dynamics. This peculiar fact is due to the integrability of the Luttinger model, i.e. the existence of an infinite number of conserved quantities that precludes the equilibration process. However, in the past few years it has become clear that integrable systems can present some kind of relaxation, even though not towards the Gibbs equilibrium ensemble. Remarkably, the necessity of correctly taking into account some particular non-equilibrium configurations, also revealed the necessity of modifying bosonization, a technique widely used to study strongly interacting systems in one dimension. In this work we focus on three different cases: • Relaxation of high energy electrons injected in a ν = 1/3 chiral Luttinger liquid and in a standard Luttinger liquid. • Heating and the emergence of effective temperatures in a Quantum Hall system at fractional filling fraction ν = 2/3 partitioned by a Quantum Point Contact. • Effect of relaxation on shot-noise measurement of the quasi-particle charge in a ν = 2 QH state.