The approach to thermal equilibrium, or thermalization, in isolated quantum systems is among the most fundamental problems in statistical physics. Recent theoretical studies have revealed that thermalization in isolated quantum systems has several remarkable features, which emerge from quantum entanglement and are quite distinct from those in classical systems. Experimentally, well isolated and highly controllable ultracold quantum gases offer an ideal system to study the nonequilibrium dynamics in isolated quantum systems, triggering intensive recent theoretical endeavors on this fundamental subject. Besides thermalization, many isolated quantum systems show intriguing behavior in relaxation processes, especially prethermalization. Prethermalization occurs when there is a clear separation in relevant time scales and has several different physical origins depending on individual systems. In this review, we overview theoretical approaches to the problems of thermalization and prethermalization.

Through an exact method, we numerically solve the time evolution of the density profile for an initially localized state in the one-dimensional bosons with repulsive short-range interactions. We show that a localized state with a density notch is constructed by superposing one-hole excitations. The initial density profile overlaps the plot of the squared amplitude of a dark soliton in the weak coupling regime. We observe the localized state collapsing into a flat profile in equilibrium for a large number of particles such as N=1000. The relaxation time increases as the coupling constant decreases, which suggests the existence of off-diagonal long-range order. We show a recurrence phenomenon for a small number of particles such as N=20.

We present a series of quantum states that are characterized by dark solitons of the nonlinear Schrödinger equation (i.e. the Gross-Pitaevskii equation) for the one-dimensional Bose gas interacting through the repulsive delta-function potentials. The classical solutions satisfy the periodic boundary conditions and we simply call them classical dark solitons. Through exact solutions we show corresponding aspects between the states and the solitons in the weak coupling case: the quantum and classical density profiles completely overlap with each other not only at an initial time but also at later times over a long period of time, and they move together with the same speed in time; the matrix element of the bosonic field operator between the quantum states has exactly the same profiles of the square amplitude and the phase as the classical complex scalar field of a classical dark soliton not only at the initial time but also at later times, and the corresponding profiles move together for a long period of time. We suggest that the corresponding properties hold rigorously in the weak coupling limit. Furthermore, we argue that the lifetime of the dark soliton-like density profile in the quantum state becomes infinitely long as the coupling constant approaches zero, by comparing it with the quantum speed limit time. Thus, we call the quantum states quantum dark soliton states.

A well-isolated system often shows relaxation to a quasi-stationary state before reaching thermal equilibrium. Such a prethermalization [1] has attracted considerable interest recently in association with closely related fundamental problems of relaxation and thermalization of isolated quantum systems [2][3][4][5]. Motivated by the recent experiment in ultracold atoms [2], we study the dynamics of a one-dimensional Bose gas which is split into two subsystems, and find that individual subsystems relax to Gibbs states, yet the entire system does not due to quantum entanglement. In view of recent experimental realization on a small well-defined number of ultracold atoms [6], our prediction based on exact few-body calculations is amenable to experimental test.Relaxation and thermalization have been studied over a wide range of fields [7][8][9][10][11][12][13][14]. Especially, isolated quantum systems have recently been realized in cold atom experiments [15,16], giving an enormous impetus to researchers to explore such fundamental issues. In the case of an integrable 1D Bose gas, thermalization does not take place due to the existence of as many conserved quantities as subsystem's degrees of freedom and the system stays at a quasi-stationary state [2,15]. It is conjectured that the prethermalized state can be described by a generalized Gibbs ensemble [11, 17] (see, however, Ref [18,19]). In an actual system, there are nonnegligible integrability-breaking perturbations which cause the eventual thermalization of the system. Thus, non-thermal steady states found in an idealized situation should be regarded as having a finite lifetime in a more realistic setup, so we call them "prethermalized states". Prethermalization has been experimentally observed in a 1D Bose gas which is coherently split into two parts, left and right [2], with the same phase. Because the subsystems are spatially separated, such an initial correlation can persist non-locally. Can such a nonlocal correlation or quantum entanglement affect prethermalization of the entire system? We will answer this question in the affirmative. The memory of those initial nonlocal correlations persists throughout time evolution, which is expected to significantly influence prethermalization. We emphasize that the mechanism of entanglement prethermalization is general as we will see later, although we focus on the integrable 1D Bose gas in this paper. We will also propose a different experimental system which is expected to exhibit entanglement prethermalization.Here, we study prethermalization in a few-body system, motivated by experimental realizations of isolated quantum systems with a few but well-defined number of particles [6]. Theoretical analysis of the experiment [2] has been done by using the Tomonaga-Luttinger theory [2,20,21], but it cannot be applied to a few-body system. The difficulty of theoretically studying a few-body system is that we need to exactly solve the Schrödinger equation for an arbitrarily long time. To overcome this difficulty, we adopt t...

Dark soliton solutions in the one-dimensional classical nonlinear Schrödinger equation has been considered to be related to the yrast states corresponding to the type-II excitations in the Lieb-Liniger model. However, the relation is nontrivial and remains unclear because a dark soliton localized in space breaks the translation symmetry, while yrast states are translationally invariant. In this work, we construct a symmetry-broken quantum soliton state and investigate the relation to the yrast states. By interpreting a quantum dark soliton as a Bose-Einstein condensation to the wave function of a classical dark soliton, we find that the quantum soliton state has a large weight only on the yrast states, which is analytically proved in the free-boson limit and numerically verified in the weak-coupling regime. By extending these results, we derive a parameter-free expression of a quantum soliton state that is written as a superposition of yrast states with Gaussian weights. The density profile of this quantum soliton state excellently agrees to that of the classical dark soliton. The dynamics of a quantum dark soliton is also studied, and it turns out that the density profile of a dark soliton decays, but the decay time increases as the inverse of the coupling constant in the weak-coupling limit.

We propose a methodology to construct excited states with a fixed angular momentum, namely, "yrast excited states" of finite-size one-dimensional bosonic systems with periodic boundary conditions. The excitation energies such as the first yrast excited energy are calculated through the system-size asymptotic expansion and expressed analytically by dressed energy. Interestingly, they are grouped into sets of almost degenerate energy levels. The low-lying excitation spectrum near the yrast state is consistent with the U(1) conformal field theories if the total angular momentum is given by an integral multiple of particle number; i.e., if the system is supercurrent.

Recurrence time is evaluated for some initial quantum states in the one-dimensional Bose gas with repulsive short-range interactions. In the relatively strong and weak coupling cases some different types of initial states show almost complete recurrence and the estimates of recurrence time are proportional to some powers of the system size at least in some range of the system size. They are much longer than in the case of free particles such as 100 times. In the free-bosonic and freefermionic regimes we evaluate the recurrence time rigorously, which is proportional to the square of the system size. The estimate of recurrence time is given by the order of ten milliseconds in the corresponding experimental systems of cold atoms trapped in one dimension of ten micrometers in length. It is much shorter than the estimate in a generic quantum many-body system, which may be as long as the age of the universe.

We study quantum double dark-solitons, which give pairs of notches in the density profiles, by constructing corresponding quantum states in the Lieb–Liniger model for the one-dimensional Bose gas. Here, we expect that the Gross–Pitaevskii (GP) equation should play a central role in the long distance mean-field behavior of the 1D Bose gas. We first introduce novel quantum states of a single dark soliton with a nonzero winding number. We show them by exactly evaluating not only the density profile but also the profiles of the square amplitude and phase of the matrix element of the field operator between the N-particle and (N−1)-particle states. For elliptic double dark-solitons, the density and phase profiles of the corresponding states almost perfectly agree with those of the classical solutions, respectively, in the weak coupling regime. We then show that the scheme of the mean-field product state is quite effective for the quantum states of double dark solitons. Assigning the ideal Gaussian weights to a sum of the excited states with two particle-hole excitations, we obtain double dark-solitons of distinct narrow notches with different depths. We suggest that the mean-field product state should be well approximated by the ideal Gaussian weighted sum of the low excited states with a pair of particle-hole excitations. The results of double dark-solitons should be fundamental and useful for constructing quantum multiple dark-solitons.