Using very general and well established ideas of the statistical physics of macroscopic bodies, that is, of those composed of many degrees of freedom, we show how classical behavior of the center of mass motion arises from a fully quantum mechanical description of the dynamics of the whole body. We do not attempt to provide a rigorous proof of the latter statement, but rather, we show or, at least, indicate the hypotheses needed to obtain the purported result. Moreover, we neither attempt to deal with the "most general" physical situation and, instead, we concentrate on a stylized model of a small solid, yet macroscopic, that we shall call a "little stone". The main hypothesis is that a macroscopic body can be decomposed into several smaller pieces, still macroscopic, that become statistically independent due to the short-range interaction nature of their constituent atoms. The ensuing main result is that the quantum distributions of extensive variables of the body become sharply-peaked. The center of mass variables are of this type and hence their dynamics is essentially classical. We point out the crucial role played by the external potential, in which the motion occurs, as the macroscopic agent that executes the "measurement" process of the center of mass.
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
The question of how statistical physics can be seen to emerge from unitary quantum dynamics goes back to Schrödinger and von Neumann, however has been reaffirmed as a topic of importance due to recent advances in experimental capabilities, allowing the observation of the unitary evolution of many-particle systems over long periods of time. Understanding why such a system evolves to the specific thermal equilibrium state, effectively described by a relevant thermodynamic ensemble, has thus seen significantly more interest in recent years. Furthermore, observations of the dynamics of such systems open questions on the route to equilibrium, and subsequent fluctuations. In this thesis I will develop an approach to this problem, taking as a starting point the non-integrability of the system under study, which leads to a generic model in terms of random matrix theory (RMT), and more generally, chaotic wavefunctions. In the formulation of these methods, special attention is paid to the form of local observables of quantum systems, and an approach to their description is developed in terms of correlation functions of chaotic wavefunctions. From this approach a key conjecture in the understanding of thermalization, the eigenstate thermalization hypothesis (ETH), is derived in full, and the dynamical behaviour of such observables is obtained analytically. Further, I will show that emergent classical behaviour can be observed in the form of a fluctuation-dissipation theorem (FDT) of Brownian processes. This is exploited to develop an experimental proposal to measure the ‘complexity’ of a quantum device, by experimentally obtaining its effective Hilbert space dimension in terms of a fully connected system. Finally, quantum jump trajectories, describing stroboscopic projective measurements of local observables are studied, and shown to display classical Brownian dynamics in the form of a Markov process. Throughout this thesis, exact diagonalization calculations of realistic quantum spin systems are used to confirm analytical results.
In the previous companion paper, we proposed a subclass of wavefunctions to describe macroscopic solids that resolved and extended the theory quantum measurement and gave a more specific treatment of quasiparticles. Here we extend these notions to thermalization of solids and gases and to gas state hydrodynamics. This gives a modification of the thermodynamic limit to justify the canonical averages for "typical wavefunctions" without the use of ensembles. The energetic cost of vorticity is contrasted in the classical and ultracold gas limits. From this perspective, we then examine the applicability of thermo and hydro to ultracold gases and compare with the implications of pure state evolution. We illustrate how the proposed quantum limits on viscosity could be reinterpreted in terms of Schrödinger induced evolution of the one-body density function but some history dependent measurable properties should still persist.
Quantum thermalization describes how closed quantum systems can effectively reach thermal equilibrium, resolving the apparent incongruity between the reversibility of Schrödinger's equation and the second law of thermodynamics. Despite its ubiquity and conceptual significance, the precise conditions that give rise to quantum thermalization are still not well understood. After nearly a century of efforts, we have yet to find a complete mathematical proof that an effective statistical description naturally emerges the underlying quantum dynamics in generic settings. Here, we prove that quantum thermalization must occur in any qubit system with local interactions under three conditions: (i) high effective temperature, (ii) translation invariance, and (iii) no perfect resonances in the energy spectrum. Specifically, we show that a typical, low-complexity pure state drawn from any ensemble with large entropy and well-defined effective temperature becomes locally indistinguishable from a Gibbs state upon unitary evolution. In this setting, our rigorous results prove the widely anticipated notion that statistical physics should be understood as an emergent phenomenon, explicitly derived from the first principles of quantum mechanics.
Isolated, many-body quantum systems, evolving under their intrinsic dynamics, exhibit a multitude of exotic phenomena and raise foundational questions about statistical mechanics. A flurry of theoretical work has been devoted to understanding how these systems reach thermal equilibrium in the absence of coupling to an external bath and, when thermalization does not occur, investigating the emergent non-equilibrium phases of matter. With the advent of synthetic quantum systems, such as ultra-cold atoms in optical lattices or trapped ions, these questions are no longer academic and can be directly studied in the laboratory. This dissertation explores the non-equilibrium phenomena that stem from the interplay between interactions, disorder, symmetry, topology, and external driving. First, we study how strong disorder, leading to many-body localization, can arrest the heating of a Floquet system and stabilize symmetry-protected topological order that does not have a static analogue. We analyze its dynamical and entanglement properties, highlight its duality to a discrete time crystal, and propose an experimental implementation in a cold-atom setting.Quenched disorder and the many-body localized state are crucial ingredients in protecting macroscopic quantum coherence. We explore the stability of many-body localization in two and higher dimensions and analyze its robustness to rare regions of weak disorder.We then study a second example of non-thermal behavior, namely integrability. We show that a class of random spin models, realizable in systems of atoms coupled to an optical cavity, gives rise to a rich dynamical phase diagram, which includes regions of integrability, classical chaos, and of a novel integrable structure whose conservation laws are reminiscent of the integrals of motion found in a many-body localized phase.The third group of disordered, non-ergodic systems we consider, spin glasses, have fascinating connections to complexity theory and the hardness of constraint satisfaction. We define a statistical ensemble that interpolates between the classical and quantum limits of such a problem and show that there exists a sharp boundary separating satisfiable and unsatisfiable phases.