Fermi-Bose Transformation for the Time-Dependent Lieb-Liniger Gas

Abstract

Exact solutions of the Schrödinger equation describing a freely expanding Lieb-Liniger (LL) gas of delta-interacting bosons in one spatial dimension are constructed. The many-body wave function is obtained by transforming a fully antisymmetric (fermionic) time-dependent wave function which obeys the Schrödinger equation for a free gas. This transformation employs a differential Fermi-Bose mapping operator which depends on the strength of the interaction and the number of particles.HD-THEP-07-26 Nonequilibrium phenomena in quantum many-body systems are among the most fundamental and intriguing phenomena in physics. One-dimensional (1D) interacting Bose gases provide a unique opportunity to study such phenomena. In some cases, the models describing these systems [1,2,3] allow to determine exact timedependent solutions of the Schrödinger equation [4,5] providing insight beyond various approximations, which is particularly important in strongly correlated regimes. These 1D systems are experimentally realized with atoms tightly confined in effectively 1D waveguides [6,7,8], where nonequilibrium dynamics is considerably affected by the kinematic restrictions of the geometry [8], while quantum effects are enhanced [9,10,11]. Today, experiments have the possibility to explore 1D Bose gases for various interaction strengths, from the Lieb-Liniger (LL) gas with finite coupling [6,8] up to the so-called TonksGirardeau (TG) regime of "impenetrable-core" bosons [7,8]. However, most theoretical studies of the exact time-dependence address the TG regime (see, e.g., Refs. [4,12,13,14,15,16,17,18]). In this limit, the complex many-body problem is considerably simplified due to the Fermi-Bose mapping property [4] where dynamics follows a set of uncoupled single-particle (SP) Schrödinger equations [4]. It is therefore desirable to employ an efficient method for calculating the time-evolution of a LL gas with finite interaction strength.In 1963, Lieb and Liniger [1] presented, on the basis of the Bethe ansatz, a solution for a homogeneous Bose gas with (repulsive) δ-function interactions, for arbitrary interaction strength c; periodic boundary conditions were imposed. This system was analyzed by McGuire on an infinite line with attractive interactions [3]. The renewed interest in 1D Bose gases stimulated recent studies of static LL wave functions [19,20,21] including a LL gas in box confinement [21]. Besides the wave functions, the correlations of a LL system with finite coupling [22,23,24,25,26,27,28,29,30,31] provide a link to many observables and were analyzed by using various techniques, including the inverse scattering method [24,25,30,31]

Exact quantum decay of an interacting many-particle system: the Calogero–Sutherland model

“…Its derivation required (i) the scaling dynamics of the exact timedependent coherent states, (ii) the use of the Bijl-Jastrow form of the initial state, and (iii) the identification of the leading term at long times. The scaling dynamics holds exactly for the CS gas and simplifies the ensuing analysis by contrast to other many-body systems such as, e.g., the 1D Bose gas, where only recently moderate progress accounting for its dynamics has been reported [52][53][54][55]. Comparing the first leading terms in a longtime asymptotic expansion, it is found that the power-law (15) sets in when the time of evolution satisfies…”

“…In systems lacking self-similar dynamics, the role of interactions can be disentangled from that of the exclusion statistics and further studies will be illuminating. A prominent example is the one-dimensional Bose gas with contact interactions, where recent advances in describing its dynamics have been reported [52][53][54][55].…”

Analytic results for a quantum quench from free to hard-core one-dimensional bosons

“…This quench has been studied in the past [29,53,56], but resisted any analytical computation. Apart from the direct interest, our results will also be a benchmark for the numerically exact methods based on integrability [27,[32][33][34][35].…”

“…Many numerical investigations seem to confirm this scenario [10,[16][17][18][19][20][21][22][23][24][25][26], but due to their intrinsic limitations (such as finite size and finite time effects) exact analytic calculations are playing a central role. However, while solving the nonequilibrium dynamics of nonintegrable models is clearly impossible, even the analytic study of integrable interacting systems in the thermodynamic limit (TDL) is still beyond our present capabilities, despite several attempts in this direction [27][28][29][30][31][32][33][34][35][36]. For these reasons, analytic calculations have concentrated on two main aspects.…”

A coordinate Bethe ansatz approach to the calculation of equilibrium and nonequilibrium correlations of the one-dimensional Bose gas

“…Other calculations make explicit use of the integrability of the system. These are based on various Bethe ansatz approaches, and include utilizing Fermi-Bose mapping [73,74] and strong coupling expansions of the coordinate Bethe ansatz wave function [75][76][77], combining the algebraic Bethe ansatz with other numerical methods [78][79][80], and using the Yudson contour-integral representation for infinite-length systems [81,82]. Recently, it was conjectured that the dynamics following an interaction strength quench are captured by a thermodynamic Bethe ansatz saddle point state and excitations around it-the so-called quench action approach [83][84][85][86][87][88].…”