Momentum distribution of a freely expanding Lieb-Liniger gas

TL;DR

This paper investigates the free expansion dynamics of a few Lieb-Liniger bosons, analyzing how their momentum distribution evolves over time and establishing an analytical connection between initial states and asymptotic distributions.

Contribution

It provides a numerical and analytical study of the transient and long-term behavior of expanding Lieb-Liniger gases, highlighting the relation between initial states and asymptotic momentum distributions.

Findings

Momentum distribution asymptotically matches the real-space density shape.

Analytical formula derived using stationary phase approximation.

Numerical results agree with analytical predictions.

Abstract

We numerically study free expansion of a few Lieb-Liniger bosons, which are initially in the ground state of an infinitely deep hard-wall trap. Numerical calculation is carried out by employing a standard Fourier transform, as follows from the Fermi-Bose transformation for a time-dependent Lieb-Liniger gas. We study the evolution of the momentum distribution, the real-space single-particle density, and the occupancies of natural orbitals. Our numerical calculation allows us to explore the behavior of these observables in the transient regime of the expansion, where they are non-trivially affected by the particle interactions. We derive analytically (by using the stationary phase approximation) the formula which connects the asymptotic shape of the momentum distribution and the initial state. For sufficiently large times the momentum distribution coincides (up to a simple scaling transformation) with the shape of the real-space single-particle density (the expansion is asymptotically ballistic). Our analytical and numerical results are in good agreement.