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arXiv:cond-mat/0302213 · Physical Review Letters

Dynamics of Lieb-Liniger Gases

Marvin D. Girardeau

Published Jul 24, 2003• 22 citations
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Abstract

It is proved that the Lieb-Liniger cusp condition implementing the delta function interaction in one-dimensional Bose gases is dynamically conserved under phase imprinting by pulses of arbitrary spatial form, and the subsequent many-body dynamics in the thermodynamic limit is expressed approximately in terms of solutions of the time-dependent single-particle Schrödinger equation for a set of time-dependent orbitals evolving from an initial Lieb-Liniger-Fermi sea. As an illustrative application, a generation of gray solitons in a Lieb-Liniger gas on a ring by a phase-imprinting pulse is studied.

DOI
10.1103/physrevlett.91.040401

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It is proved that the Lieb-Liniger cusp condition implementing the delta function interaction in one-dimensional Bose gases is dynamically conserved under phase imprinting by pulses of arbitrary spatial form, and the subsequent many-body dynamics in the thermodynamic limit is expressed approximately in terms of solutions of the time-dependent single-particle Schrödinger equation for a set of time-dependent orbitals evolving from an initial Lieb-Liniger-Fermi sea. As an illustrative application, a generation of gray solitons in a Lieb-Liniger gas on a ring by a phase-imprinting pulse is studied.
Exact solutions of the Schrödinger equation describing a freely expanding Lieb-Liniger gas of delta-interacting bosons in one spatial dimension are constructed. We demonstrate that for any interaction strength the system enters a strongly correlated regime during such expansion. The asymptotic form of the wave function is shown to have the form characteristic for "impenetrable-core" bosons. Exact solutions are obtained by transforming a fully antisymmetric (fermionic) time-dependent wave function that obeys the Schrödinger equation for a free gas. This transformation employs a differential Fermi-Bose mapping operator that depends on the strength of the interaction and the number of particles.
Abstract The coordinate Bethe ansatz can be extended to a model, the Lieb–Liniger model, of a one-dimensional gas of Bosons interacting with repulsive δ-function potentials. It has attracted attention due to its relevance for experimental developments in the fields of ultracold gases and optical lattices. This chapter provides an exposition of the related classical nonlinear Schrödinger equation, followed by its generalization to the quantum model. It explores a limiting case, the Tonks-Girardeau gas. The δ-function potentials supply a kind of boundary condition on the wave functions allowing us to analyze the eigenfunctions of the Bethe ansatz, which are examined on the infinite line and for periodic boundary conditions. The latter leads to the Bethe ansatz equations. The solution of these equations is achieved in the thermodynamic limit for the ground state and for low-lying excited states.
We derive a 1/c1/c1/c expansion for the single-particle density matrix of a strongly interacting time-dependent one-dimensional Bose gas, described by the Lieb-Liniger model (ccc denotes the strength of the interaction). The formalism is derived by expanding Gaudin's Fermi-Bose mapping operator up to 1/c1/c1/c terms. We derive an efficient numerical algorithm for calculating the density matrix for time-dependent states in the strong coupling limit, which evolve from a family of initial conditions in the absence of an external potential. We have applied the formalism to study contraction dynamics of a localized wave packet upon which a parabolic phase is imprinted initially.
The asymptotic form of the wave functions describing a freely expanding Lieb-Liniger gas is derived by using a Fermi-Bose transformation for time-dependent states, and the stationary phase approximation. We find that asymptotically the wave functions approach the Tonks-Girardeau (TG) structure as they vanish when any two of the particle coordinates coincide. We point out that the properties of these asymptotic states can significantly differ from the properties of a TG gas in a ground state of an external potential. The dependence of the asymptotic wave function on the initial state is discussed. The analysis encompasses a large class of initial conditions, including the ground states of a Lieb-Liniger gas in physically realistic external potentials. It is also demonstrated that the interaction energy asymptotically decays as a universal power law with time, ${E}_{\mathrm{int}}\ensuremath{\propto}{t}^{\ensuremath{-}3}$.
We study how dark solitons, i.e. solutions of one-dimensional single-particle nonlinear time-dependent Schrödinger equation, emerge from eigenstates of a linear many-body model of contact interacting bosons moving on a ring, the Lieb-Liniger model. This long-standing problem was addressed by various groups, which presented different, seemingly unrelated, procedures to reveal the solitonic waves directly from the many-body model. Here, we propose a unification of these results using a simple Ansatz for the many-body eigenstate of the Lieb-Liniger model, which gives us access to systems of hundreds of atoms. In this approach, mean-field solitons emerge in a single-particle density through repeated measurements of particle positions in the Ansatz state. The post-measurement state turns out to be a wave packet of yrast states of the reduced system.
We obtain exact results on interaction quenches in the 1D Bose gas described by the integrable Lieb-Liniger model. We show that in the long time limit integrability leads to significant deviations from the predictions of the grand canonical ensemble and a description within the generalized Gibbs ensemble (GGE) is needed. For a non-interacting initial state and arbitrary final interactions, we find that the presence of infinitely many conserved charges generates a non-analytic behavior in the equilibrated density of quasimomenta. This manifests itself in a dynamically generated Friedel-like oscillation of the non-local correlation functions with interaction dependent oscillation momenta. We also exactly evaluate local correlations and the generalized chemical potentials within GGE.