Thermalization, the evolution of an interacting many-body system towards a thermal Gibbs ensemble after initialization in an arbitrary non-equilibrium state, is currently a phenomenon of great interest, both in theory and experiment. As the time evolution of a quantum system is unitary, the proposed mechanism of thermalization in quantum many-body systems corresponds to the so-called eigenstate thermalization hypothesis (ETH) and the typicality of eigenstates. Although this formally solves the contradiction of thermalizing but unitary dynamics in a closed quantum many-body system, it does neither make any statement on the dynamical process of thermalization itself nor in which way the coupling of the system to an environment can hinder or modify the relaxation dynamics. In this thesis, we address both the question whether or not a quantum system driven away from equilibrium is able to relax to a thermal state, which fulfills detailed balance, and if one can identify universal behavior in the non-equilibrium relaxation dynamics. As a first realization of driven quantum systems out of equilibrium, we investigate a system of Ising spins, interacting with the quantized radiation field in an optical cavity. For multiple cavity modes, this system forms a highly entangled and frustrated state with infinite correlation times, known as a quantum spin glass. In the presence of drive and dissipation, introduced by coupling the intra-cavity radiation field to the photon vacuum outside the cavity via lossy mirrors, the quantum glass state is modified in a universal manner. For frequencies below the photon loss rate, the dissipation takes over and the system shows the universal behavior of a dissipative spin glass, with a characteristic spectral density
mathcalA(omega)simsqrtomega. On the other hand, for frequencies above the loss rate, the system retains the universal behavior of a zero temperature, quantum spin glass. Remarkably, at the glass transition, the two subsystems of spins and photons thermalize to a joint effective temperature, even in the presence of photon loss. This thermalization is a consequence of the strong spin-photon interactions, which favor detailed balance in the system and detain photons from escaping the cavity. In the thermalized system, the features of the spin glass are mirrored onto the photon degrees of freedom, leading to an emergent photon glass phase. Exploiting the inherent photon loss of the cavity, we make predictions of possible measurements on the escaping photons, which contain detailed information of the state inside the cavity and allow for a precise, non-destructive measurement of the glass state. As a further set of non-equilibrium systems, we consider one-dimensional quantum fluids driven out of equilibrium, whose universal low energy theory is formed by the so-called Luttinger Liquid description, which, due to its large degree of universality, is of intense theoretical and experimental interest. A set of recent experiments in research groups in Vienna, Innsbruck and Munich have probed the non-equilibrium time-evolution of one-dimensional quantum fluids for different experimental realizations and are pushing into a time regime, where thermalization is expected. From a theoretical point of view, one-dimensional quantum fluids are particular interesting, as Luttinger Liquids are integrable and therefore, due to an infinite number of constants of motion, do not thermalize. The leading order correction to the quadratic theory is irrelevant in the sense of the renormalization group and does therefore not modify static correlation functions, however, it breaks integrability and will therefore, even if irrelevant, induce a completely different non-equilibrium dynamics as the quadratic Luttinger theory alone. In this thesis, we derive for the first time a kinetic equation for interacting Luttinger Liquids, which describes the time evolution of the excitation densities for arbitrary initial states. The resonant character of the interaction makes a straightforward derivation of the kinetic equation, using Fermi\'s golden rule, impossible and we have to develop non-perturbative techniques in the Keldysh framework. We derive a closed expression for the time evolution of the excitation densities in terms of self-energies and vertex corrections. Close to equilibrium, the kinetic equation describes the exponential decay of excitations, with a decay rate
sigmaR=mboxImSigmaR, determined by the self-energy at equilibrium. However, for long times
tau, it also reveals the presence of dynamical slow modes, which are the consequence of exactly energy conserving dynamics and lead to an algebraic decay
simtau−etasubD with
etaD=0.58. The presence of these dynamical slow modes is not contained in the equilibrium Matsubara formalism, while they emerge naturally in the non-equilibrium formalism developed in this thesis. In order to initialize a one-dimensional quantum fluid out of equilibrium, we consider an interaction quench in a model of interacting, dispersive fermions in Chap.~\\ref{sec:Quench}. In this scenario, the fermionic interaction is suddenly changed at time
t=0, such that for
t>0 the system is not in an eigenstate and therefore undergoes a non-trivial time evolution. For the quadratic theory, the stationary state in the limit
trightarrowinfty is a non-thermal, or prethermal, state, described by a generalized Gibbs ensemble (GGE). The GGE takes into account for the conservation of all integrals of motion, formed by the eigenmodes of the Hamiltonian. On the other hand, in the presence of non-linearities, the final state for
trightarrowinfty is a thermal state with a finite temperature
T>0. . The spatio-temporal, dynamical thermalization process can be decomposed into three regimes: A prequench regime on the largest distances, which is determined by the initial state, a prethermal plateau for intermediate distances, which is determined by the metastable fixed point of the quadratic theory and a thermal region on the shortest distances. The latter spreads sub-ballistically
simtalpha in space with
0<alpha<1 depending on the quench. Until complete thermalization (i.e. for times
t<infty), the thermal region contains more energy than the prethermal and prequench region, which is expressed in a larger temperature
Tt>Tinfty, decreasing towards its final value
Tinfty. As the system has achieved local detailed balance in the thermalized region, energy transport to the non-thermal region can only be performed by the macroscopic dynamical slow modes and the decay of the temperature
Tt−Tinftysimt−etasubD again witnesses the presence of these slow modes. The very slow spreading of thermalization is consistent with recent experiments performed in Vienna, which observe a metastable, prethermal state after a quench and only observe the onset of thermalization on much larger time scales. As an immediate indication of thermalization, we determine the time evolution of the fermionic momentum distribution after a quench from non-interacting to interacting fermions. For this quench scenario, the step in the Fermi distribution at the Fermi momentum
ksubF decays to zero algebraically in the absence of a non-linearity but as a stretched exponential (the exponent being proportional to the non-linearity) in the presence of a finite non-linearity. This can serve as a proof for the presence or absence of the non-linearity even on time-scales for which thermalization can not yet be observed. Finally, we consider a bosonic quantum fluid, which is driven away from equilibrium by permanent heating. The origin of the heating is atomic spontaneous emission of laser photons, which are used to create a coherent lattice potential in optical lattice experiments. This process preserves the system\'s
U(1)-invariance, i.e. conserves the global particle number, and the corresponding long-wavelength description is a heated, interacting Luttinger Liquid, for which phonon modes are continuously populated with a momentum dependent rate
partialtnqsimgamma∣q∣. In the dynamics, we identify a quasi-thermal regime for large momenta, featuring an increasing time-dependent effective temperature. In this regime, due to fast phonon-phonon scattering, detailed balance has been achieved and is expressed by a time-local, increasing temperature. The thermal region emerges locally and spreads in space sub-ballistically according to
xtsimt4/5. For larger distances, the system is described by an non-equilibrium phonon distribution
nqsim∣q∣, which leads to a new, non-equilibrium behavior of large distance observables. For instance, the phonon decay rate scales universally as
gammaqsim∣q∣5/3, with a new non-equilibrium exponent
eta=5/3, which differs from equilibrium. This new, universal behavior is guaranteed by the
U(1) invariant dynamics of the system and is insensitive to further subleading perturbations. The non-equilibrium long-distance behavior can be determined experimentally by measuring the static and dynamic structure factor, both of which clearly indicate the exponents for phonon decay,
eta=5/3 and for the spreading of thermalization
etaT=4/5. Remarkably, even in the presence of this strong external drive, the interactions and their aim to achieve detailed balance are strong enough to establish a locally emerging and spatially spreading thermal region. The physical setups in this thesis do not only reveal interesting and new dynamical features in the out-of-equilibrium time evolution of interacting systems, but they also strongly underline the high degree of universality of thermalization for the classes of models studied here. May it be a system of coupled spins and photons, where the photons are pulled away from a thermal state by Markovian photon d
Starting from a microscopic description of weak system-bath interactions, we derive from first principles a quantum master equation that does not rely on the well-known rotating wave approximation. This includes generic many-body systems, with Hamiltonians with vanishingly small energy spacings that forbid that approximation. The equation satisfies a general form of detailed balance, called KMS (Kubo-Martin-Schwinger), which ensures exact convergence to the many-body Gibbs state. Unlike the more common notion of GNS (Gelfand-Naimark-Segal) detailed balance, this notion is compatible with the absence of the rotating wave approximation. We show that the resulting Lindbladian dynamics not only reproduces the thermal equilibrium point up to a small renormalization of the system Hamiltonian, but it also approximates the true system evolution with an error that grows at most linearly in time, giving an exponential improvement upon previous estimates. This master equation has quasilocal jump operators, can be efficiently simulated on a quantum computer, and reduces to the usual Davies dynamics in the limit of a coarse-graining time much larger than the inverse of the smallest frequency difference. With it, we provide a rigorous model of many-body thermalization relevant to both open quantum systems and quantum algorithms.
Relaxation according to Fokker-Planck equations is a standard scenario in classical statistical mechanics. It is, however, not obvious how such a Fokker-Planck dynamics may emerge within a closed, finite quantum system. We present an analysis of a system comprising sixteen spins in which spatial inhomogeneities of the magnetization relax approximately in accord with a standard Fokker-Planck equation for a Brownian particle in a parabolic potential. While a single, specfic example is addressed in detail, arguments which indicate that the above findings may hold for a wider range of systems are given.
The question of thermalization of a closed quantum system is of central interest in nonequilibrium quantum many-body physics. Here we present one such study analyzing the dynamics of a closed coupled Majorana SYK system. We have a large- q SYK model prepared initially at equilibrium quenched by introducing a random hopping term, thus leading to nonequilibrium dynamics. We find that the final stationary state reaches thermal equilibrium with respect to the Green's functions and energy. Accordingly, the final state is characterized by calculating its final temperature and the thermalization rate. We provide a detailed review of analytical methods and derive the required Kadanoff-Baym equations, which are then solved using the algorithm developed in this work. Our results display rich thermalization dynamics in a closed quantum system in the thermodynamic limit.
The results presented in this PhD Thesis aim at understanding thermalization dynamics and, strictly speaking, pre-thermalization in both open and closed quantum spin chains driven out-of-equilibrium. In order to make analytical progress, we consider perturbations of the Quantum Ising Chain (QIC), one of the simplest integrable models, whose non-equilibrium dynamics has been studied in great details in the last few years. In Chapter 2 we introduce for the non-technical reader the main recent ideas behind quantum thermalization and relaxation in integrable and non-integrable quantum many body systems, while Chapter 3 contains an introduction to the current understanding of pre-thermalization. In Chapter 4 we start our study considering a QIC perturbed by a time dependent noise in the direction of the transverse magnetic field [6] [7]. We generalize the concept of pre-thermalization to a noisy quantum many body system driven away from equilibrium simultaneously by a quantum quench of the transverse field and by the noise, highlighting the mechanism behind pre-thermalization and the subsequent relaxation towards the thermal state in a wide class of physically relevant observables. Our main achievement is to show that pre-thermalization occurs quite generally up to time scales where the noise is dominated by the non-equilibrium dynamics induced by the quench of the transverse field, while for longer times pre-thermal plateaux disappear and the noise leaves different signatures in different observables. In Chapter 5 we consider the more challenging problem of pre-thermalization in a closed non-integrable quantum spin chain [8]. We show using a mapping of a weakly non-integrable spin chain into a hard core boson model, that pre-thermalization is successfully captured by a low-density description of the theory, which results in a integrable, bosonic model, that constitutes an excellent description for the out-of-equilibrium dynamics up to intermediate time scales. On the other hand, in order to investigate the late time dynamics we resort to a diagrammatic approach, which serves for a two-fold scope. First, we focus on the effect of the leading inelastic processes, realizing that they constitute only a sub-leading correction to the pre-thermal plateau, found within the effective low-energy description; and secondly, we show from a Quantum Boltzmann equation perspective how inelastic processes are the fundamental ingredient in order to have thermalization in the late time dynamics. The main technical tool used to derive the results in Chapter 4 and 5 is the Keldysh contour technique which is reviewed in the Appendix. This PhD Thesis is based on the following works: 1) J. Marino, A. Silva, Relaxation, pre-thermalization and diffusion in a noisy Quantum Ising Chain, Phys. Rev. B 86, 060408 (Rapid Communications) (2012). 2) M. Marcuzzi, J. Marino, A. Gambassi, A. Silva, Pre-thermalization in a non-integrable quantum spin chain after a quench, arXiv:1307.3738 (2013) (submitted to Phys. Rev. Lett.). 3) J. Marino, A. Silva, Non-Equilibrium Dynamics of a Noisy Quantum Ising Chain: statistics of the work and prethermalization after a sudden quench of the transverse field, arXiv:1309.7595 (2013) (submitted to Phys. Rev. B).
Understanding how closed quantum systems dynamically approach thermal equilibrium presents a major unresolved problem in statistical physics. Generically, non-integrable quantum systems are expected to thermalize as they comply with the Eigenstate Thermalization Hypothesis. However, in the presence of strong disorder, the dynamics can possibly slow down to a degree that systems fail to thermalize on experimentally accessible timescales, as in spin glasses or many-body localized systems. In general, particularly in long-range interacting quantum systems, the specific nature of the disorder necessary for the emergence of a prethermal, metastable state--distinctly separating the timescales of initial relaxation and subsequent slow thermalization--remains an open question. We study an ensemble of Heisenberg spins with a tunable distribution of random coupling strengths realized by a Rydberg quantum simulator. We observe a drastic change in the late-time magnetization when increasing disorder strength. The data is well described by models based on pairs of strongly interacting spins, which are treated as thermal for weak disorder and isolated for strong disorder. Our results indicate a crossover into a pair-localized prethermal regime in a closed quantum system of thousands of spins in the critical case where the exponent of the power law interaction matches the spatial dimension.