Free expansion of a Lieb-Liniger gas: Asymptotic form of the wave functions

TL;DR

This paper derives the asymptotic form of wave functions for a freely expanding Lieb-Liniger gas, showing they resemble Tonks-Girardeau states and decay in interaction energy as a universal power law.

Contribution

It provides a detailed derivation of the asymptotic wave functions using Fermi-Bose transformation and stationary phase approximation, highlighting differences from ground state properties.

Findings

Wave functions approach Tonks-Girardeau structure asymptotically

Interaction energy decays as t^{-3}

Asymptotic states depend on initial conditions

Abstract

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, $E_\mathrm{int}\propto t^{-3}$.