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arXiv:1904.06259 · Physical review. E

Quench action and large deviations: Work statistics in the one-dimensional Bose gas

Gabriele Perfetto, Lorenzo Piroli, Andrea Gambassi

Published Sep 10, 2019• 33 citations
arXivHTMLPDF

Abstract

We study the statistics of large deviations of the intensive work done in an interaction quench of a one-dimensional Bose gas with a large number N of particles, system size L, and fixed density. We consider the case in which the system is initially prepared in the noninteracting ground state and a repulsive interaction is suddenly turned on. For large deviations of the work below its mean value, we show that the large-deviation principle holds by means of the quench action approach. Using the latter, we compute exactly the so-called rate function and study its properties analytically. In particular, we find that fluctuations close to the mean value of the work exhibit a marked non-Gaussian behavior, even though their probability is always exponentially suppressed below it as L increases. Deviations larger than the mean value exhibit an algebraic decay whose exponent cannot be determined directly by large-deviation theory. Exploiting the exact Bethe ansatz representation of the eigenstates of the Hamiltonian, we calculate this exponent for vanishing particle density. Our approach can be straightforwardly generalized to quantum quenches in other interacting integrable systems.

DOI
10.1103/physreve.100.032114

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We study the statistics of large deviations of the intensive work done in an interaction quench of a one-dimensional Bose gas with a large number N of particles, system size L, and fixed density. We consider the case in which the system is initially prepared in the noninteracting ground state and a repulsive interaction is suddenly turned on. For large deviations of the work below its mean value, we show that the large-deviation principle holds by means of the quench action approach. Using the latter, we compute exactly the so-called rate function and study its properties analytically. In particular, we find that fluctuations close to the mean value of the work exhibit a marked non-Gaussian behavior, even though their probability is always exponentially suppressed below it as L increases. Deviations larger than the mean value exhibit an algebraic decay whose exponent cannot be determined directly by large-deviation theory. Exploiting the exact Bethe ansatz representation of the eigenstates of the Hamiltonian, we calculate this exponent for vanishing particle density. Our approach can be straightforwardly generalized to quantum quenches in other interacting integrable systems.
Unless constrained by symmetry, measurement of an observable on an ensemble of identical quantum systems returns a distribution of values which are encoded in the full counting statistics. While the mean value of this distribution is important for determining certain properties of a system, the full distribution can also exhibit universal behavior. In this paper we study the full counting statistics of particle number in one-dimensional interacting Bose and Fermi gases which have been quenched far from equilibrium. In particular, we consider the time evolution of the Lieb-Liniger and Gaudin-Yang models quenched from a Bose-Einstein condensate initial state and calculate the full counting statistics of the particle number within a subsystem. We show that the scaled cumulants of the charge in the initial state and at long times are simply related and, in particular, the latter are independent of the model parameters. Using the quasiparticle picture we obtain the full time evolution of the cumulants and find that although their end points are fixed, the finite-time dynamics depends strongly on the model parameters. We go on to construct the scaled cumulant generating functions and from this determine the limiting charge probability distributions at long time which are shown to exhibit distinct, nontrivial, and non-Gaussian fluctuations and large deviations. Published by the American Physical Society 2024
We study quantum quenches to the one-dimensional Bose gas with attractive interactions in the case when the initial state is an ideal one-dimensional Bose condensate. We focus on properties of the stationary state reached at late times after the quench. This displays a finite density of multi-particle bound states, whose rapidity distribution is determined exactly by means of the quench action method. We discuss the relevance of the multi-particle bound states for the physical properties of the system, computing in particular the stationary value of the local pair correlation function g_2 g 2 .
We consider an impurity in a one-dimensional weakly interacting Bose gas and analytically calculate the density profile of the Bose gas. Within the mean-field approximation, by increasing the distance from the impurity, the Bose gas density saturates exponentially fast to its mean thermodynamic-limit value at distances beyond the healing length. The effect of quantum fluctuations drastically changes this behavior, leading to a power-law decay of the density deviation from the mean density. At distances longer than the healing length and shorter than a new length scale proportional to the impurity coupling strength, the power-law exponent is 2, while at the longest distances the corresponding exponent becomes 3. The latter crossover does not exist in two special cases. The first one is realized for infinitely strongly coupled impurity; then the density deviation always decays with the exponent 2. The second special case occurs when the new length scale is smaller than the healing length, i.e., at weak impurity coupling; then the density deviation always decays with the exponent 3. The obtained results are exact in the impurity coupling strength and account for the leading order in the interaction between the particles of the Bose gas.
This thesis is concerned with non-equilibrium phenomena in interacting quantum many-body systems. Specifically, we investigate the time-evolution and relaxation dynamics of Bose gases in a highly restricted geometry, which constrains the dynamics to one spatial dimension. This leads to a description of the system in terms of a simple model Hamiltonian which permits exact many-body quantum mechanical solutions due to its integrability. In the first part of this thesis, a computational method is developed to obtain experimentally relevant correlation functions in the framework of the coordinate Bethe ansatz. We employ this method to compute exact ground-state correlation functions of the Lieb-Liniger gas for up to seven particles covering the whole regime of repulsive interactions. We also investigate the dynamics of the system after an instantaneous change of the interaction strength. This quantum quench deposits large amounts of energy that cannot be dissipated due to the system being closed, and so the dynamics far from equilibrium are probed. We prepare the system in two different initial states, and quench to the same final interaction strength. The latter is determined in such a way that the added energy due to the quench is the same for both scenarios. Conventional statistical mechanics predicts the same relaxed state, but due to the integrability of the system all considered correlation functions of the relaxed states differ from each other and also from the thermal ones. We then investigate the dynamics and relaxed state for a quench from zero to repulsive interactions in more detail, focussing on the mechanism of relaxation and the involved time-scales. We find that local correlation functions relax on time-scales determined by the interaction strength, in contrast to non-local correlation functions, whose relaxation time-scale is proportional to the system size. Next, we employ the same methodology to study the one-dimensional Bose gas with attractive interactions. In this case many-body bound states are permissible solutions of the Lieb-Liniger model. We compare exact ground-state correlation functions of up to seven particles to their corresponding mean-field solution. The latter displays a quantum phase transition at a critical interaction strength, marking the transition from a uniform-density state to a localized bright soliton. Our exact results agree remarkably well with the corresponding mean-field solution past the critical point. We also investigate the dynamics following an interaction strength quench, starting again from the non-interacting ground state. Bound states strongly influence correlation functions for all post-quench interaction strengths, and local correlation functions are largely increased compared to their initial value. In the last part of this thesis, we investigate the behavior of the one-dimensional Bose gas under periodic driving of the interaction strength. To this end, we extend the coordinate Bethe-ansatz formalism by employing Floquet theory to obtain solutions for the full time- dependent Hamiltonian. This realizes an extension to non-integrable systems in a way that allows controlled breaking of integrability. We compare the dynamics of the system following an interaction quench to a fixed final interaction strength to that of a system with periodically modulated post-quench interaction strength. For fast driving, the system evolves according to the time-averaged Hamiltonian and correlation functions are nearly identical to those of the undriven system. The response of the system qualitatively changes at resonances of the time-averaged Hamiltonian, where we observe energy absorption and a marked change in correlation functions. However, the system does not absorb energy indefinitely.
We investigate the relaxation dynamics of the integrable Lieb-Liniger model of contact-interacting bosons in one dimension following a sudden quench of the collisional interaction strength. The system is initially prepared in its noninteracting ground state and the interaction strength is then abruptly switched to a positive value, corresponding to repulsive interactions between the bosons. We calculate equal-time correlation functions of the nonequilibrium Bose field for small systems of up to five particles via symbolic evaluation of coordinate Bethe-ansatz expressions for operator matrix elements between Lieb-Liniger eigenstates. We characterize the relaxation of the system by comparing the time-evolving correlation functions following the quench to the equilibrium correlations predicted by the diagonal ensemble and relate the behavior of these correlations to that of the quantum fidelity between the many-body wave function and the initial state of the system. Our results for the asymptotic scaling of local second-order correlations with increasing interaction strength agree with the predictions of recent generalized thermodynamic Bethe-ansatz calculations. By contrast, third-order correlations obtained within our approach exhibit a markedly different power-law dependence on the interaction strength as the Tonks-Girardeau limit of infinitely strong interactions is approached.