This preprint contains the English translation of the paper "Relaxation dynamics of a quantum chain of harmonic oscillators", which was published by the author in 1980 in the Ukrainian Physical Journal. Comments describing its motivations, background ideas and results are presented as well. This paper addressed to the problem of approach to the thermal equilibrium in an isolated macroscopic quantum system, which was studied on the example of the quantum chain of weakly interacting harmonic oscillators. In the initial state, macroscopic energy was supplied to one oscillator (atom) in the chain. Subsequent evolution of the quantum state of the whole chain was determined due to the model integrability. The main subject of interest was the time evolution of the reduced density operators characterizing the quantum state of a particular atom. On the short time-scale, the energy perturbation expands along the chain with the velocity of the fastest phonon mode. On the long-time scale, the single-atom density operators display strong fluctuations around the canonical Gibbs distribution. These fluctuations are caused by the degeneracy of the energy level differences (presence of resonances) in the model of coupled harmonic oscillators. After lifting this degeneracy, fluctuations are suppressed providing, that the reduced density operator of each atom in the chain becomes very close to the Gibbs distribution at almost any time moment.
This preprint contains the English translation of the paper "Relaxation dynamics of a quantum chain of harmonic oscillators", which was published by the author in 1980 in the Ukrainian Physical Journal. Comments describing its motivations, background ideas and results are presented as well. This paper addressed to the problem of approach to the thermal equilibrium in an isolated macroscopic quantum system, which was studied on the example of the quantum chain of weakly interacting harmonic oscillators. In the initial state, macroscopic energy was supplied to one oscillator (atom) in the chain. Subsequent evolution of the quantum state of the whole chain was determined due to the model integrability. The main subject of interest was the time evolution of the reduced density operators characterizing the quantum state of a particular atom. On the short time-scale, the energy perturbation expands along the chain with the velocity of the fastest phonon mode. On the long-time scale, the single-atom density operators display strong fluctuations around the canonical Gibbs distribution. These fluctuations are caused by the degeneracy of the energy level differences (presence of resonances) in the model of coupled harmonic oscillators. After lifting this degeneracy, fluctuations are suppressed providing, that the reduced density operator of each atom in the chain becomes very close to the Gibbs distribution at almost any time moment.
We study the dissipative quantum harmonic oscillator with general nonthermal preparations of the harmonic oscillator bath. The focus is on equilibration of the oscillator in the long-time limit and the additional requirements for thermalization. Our study is based on the exact solution of the microscopic model obtained by means of operator equations of motion, which provides us with the time evolution of the central oscillator density matrix in terms of the propagating function. We find a hierarchy of conditions for thermalization, together with the relation of the asymptotic temperature to the energy distribution in the initial bath state. We discuss the presence and absence of equilibration for the example of an inhomogeneous chain of harmonic oscillators, and we illustrate the general findings about thermalization for the nonthermal environment that results from a quench.
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).
In the present note, we discuss a simple example of a macroscopic quantum many-body system in which the approach to thermal equilibrium from an arbitrary initial state in the microcanonical energy shell is proved without relying on any unproven assumptions. The model, which is equivalent to a free fermion chain, is designed to be a toy model for a weakly heat-conducting one-dimensional solid. We take a phenomenological point of view and perceive that the system is in thermal equilibrium when the measured coarse-grained energy distribution is uniform. The result on thermalization reported here is a variation (and an improvement) of our previous result on the irreversible expansion in a free fermion chain. As far as we know, this is the first concrete and rigorous realization of the philosophy on the foundation of equilibrium statistical mechanics proposed by von Neumann in 1929, and further developed recently by Goldstein, Lebowitz, Mastrodonato, Tumulka, and Zangh\`ıand the present author, namely, to characterize thermal equilibrium from a macroscopic viewpoint and to make use of the strong ETH to control the long-time dynamics. This note will be the most technical part of my longer article on thermalization, "What is thermal equilibrium and how do we get there?". I am making this document public at this stage since I have already announced (and will announce) the results at some of my talks.
In this Thesis I present the results of four research projects I carried out during my Ph.D. training at SISSA. The broad topic of my research was the mechanism of relaxation in isolated, integrable quantum systems. Roughly speaking, integrable quantum systems are one-dimensional lattice models or field theories that feature an extensively large number of conservation laws beyond energy and momentum conservation, which greatly influence their dynamical behaviour. Integrable systems have enjoyed a considerable interest in the last decade, mainly due to the development of ultracold atomic techniques. These techniques made it possible to implement experimentally a wealth of one-dimensional quantum models whose significance had been purely theoretical before. The theoretical interest in integrable systems stems in part from the fact that in many aspects they are exactly solvable. Exact analytical formulas are available for their spectrum and their equilibrium thermodynamic properties, creating an incentive to study analytically their non-equilibrium behaviour as well. Exact results for the dynamics of integrable systems may serve as a benchmark for confirming theories in statistical mechanics or testing approximative methods. To this day, computing analytically the entire time evolution of observables in a genuinely interacting integrable system is prohibitively difficult in general. In this Thesis, I concentrate on the description of the system at long times after it is brought out of equilibrium. Homogeneous, translationally invariant integrable systems are expected to relax to a stationary state, in which local observables are described by a generalization of the Gibbs ensemble that takes into consideration all the conservation laws beyond the energy. Two of the projects presented here concern this relaxed state. In Chapter 2, a method is presented for computing the Rényi entropies of the Generalized Gibbs Ensemble describing the relaxed state in the integrable Heisen- berg XXZ spin chain. Rényi entropies provide information about the relaxed state and its entanglement properties, going beyond the ordinary von Neumann thermo- dynamic entropy. The method presented there is very general; in principle it works in any integrable model and any translationally invariant thermodynamic macrostate. An alternative to the GGE in describing relaxed states after a quantum quench is the Quench Action approach. This approach is based on the explicit analytical knowledge of the overlaps between the initial state and the Hamiltonian eigenstates. In Chapter 3, this approach is generalized to a spin-1 integrable spin chain, the SU(3) invariant Lai–Sutherland model. The evolution of entanglement is then studied using the quasiparticle picture, which features ballistically propagating classical particle pairs carrying entanglement. If the requirement of translational invariance of the initial state is relaxed, various transport phenomena arise in integrable systems. The evolution of slowly varying, smoothly distributed states is described by a generalization of hydrodynamics, which takes into consideration the infinitely many local conservation laws. At the scale of non-diffusive, ballistic transport, the main feature of this generalized hydrodynamics (GHD) is the existence of continuity equations for the occupation number of each mode. Further two projects are presented in this Thesis which are concerned with GHD.I n Chapter 4, a numerical method is presented for computing the evolution of entanglement entropies within GHD. This method combines the quasiparticle picture of entanglement with a molecular dynamics simulation of GHD, the so-called fleagas. Numerical evidence is shown that this method reproduces previously known analytical results in GHD. Its versatility is demonstrated by computing another entanglement-related quantity, the mutual information, for which no analytical results are available. Finally, in Chapter 5, the GHD framework is generalized to a system of spin-1/2 fermions with repulsive contact interaction, the Yang–Gaudin model. It is shown that at low temperatures, the dynamics of the system is described by the superposition of two uncoupled, inhomogeneous conformal field theories. This fact can be interpreted as a separation of the spin and the charge degrees of freedom of the system.
A very fundamental problem in quantum statistical mechanics involves whether—and how—an isolated quantum system will reach thermal equilibrium after waiting a long time. In quantum systems that do thermalize, the long-time expectation value of any “reasonable” operator will match its predicted value in the canonical ensemble. The Eigenstate Thermalization Hypothesis (ETH) posits that this thermalization occurs at the level of each individual energy eigenstate; in fact, any single eigenstate in a microcanonical energy window will predict the expectation values of such operators exactly. In the first part of this dissertation, we identify, for a generic model system, precisely which operators satisfy ETH, as well as limits to the information contained in a single eigenstate. Remarkably, our results strongly suggest that a single eigenstate can contain information about energy densities—and therefore temperatures—far away from the energy density of the eigenstate.Additionally, we study the possible breakdown of quantum thermalization in a model of itinerant electrons on a one-dimensional chain, with both spin and charge degrees of freedom. This model exhibits peculiar properties in the entanglement entropy, the apparent scaling of which is modified from a “volume law” to an “area law” after performing a partial, site-wise measurement on the system. These properties and others suggest that this model realizes a new, non-thermal phase of matter, known as a Quantum Disentangled Liquid (QDL). The putative existence of this phase has striking implications for the foundations of quantum statistical mechanics.
Exactly solvable models provide an opportunity to study different aspects of reduced quantum dynamics in detail. We consider the reduced dynamics of a single spin in finite XX and XY spin 1/2 chains. First we introduce a general expression describing the evolution of the reduced density matrix. This expression proves to be tractable when the combined closed system (i.e. open system plus environment) is integrable. Then we focus on comparing decoherence and thermalization timescales in the XX chain. We find that for a single spin these timescales are comparable, in contrast to what should be expected for a macroscopic body. This indicates that the process of quantum relaxation of a system with few accessible states can not be separated in two distinct stages - decoherence and thermalization. Finally, we turn to finite-size effects in the time evolution of a single spin in the XY chain. We observe three consecutive stages of the evolution: regular evolution, partial revivals, irregular (apparently chaotic) evolution. The duration of the regular stage is proportional to the number of spins in the chain. We observe a "quiet and cold period" in the end of the regular stage, which breaks up abruptly at some threshold time.