Ground State Properties of a Tonks-Girardeau Gas in a Split Trap

TL;DR

This paper provides an exact analysis of a Tonks-Girardeau gas in a split harmonic trap, revealing how barrier strength and particle number influence ground state properties and coherence, with odd-even effects observed.

Contribution

It introduces an exact method to analyze the ground state and coherence properties of a Tonks-Girardeau gas in a split trap, highlighting odd-even particle number effects.

Findings

Ground state occupation diverges from expected $\

Momentum distribution shows distinct odd-even effects.

Interference fringe visibility varies with barrier height and particle number.

Abstract

We determine the exact many-body properties of a bosonic Tonks-Girardeau gas confined in a harmonic potential with a tunable $\delta$-function barrier at the trap center. This is done by calculating the reduced single particle density matrix, the pair-distribution function, and momentum distribution of the gas as a function of barrier strength and particle number. With increasing barrier height we find that the ground state occupation in a diagonal basis diverges from the $\sqrt N$ behavior that is expected for the case of a simple harmonic trap. In fact, the scaling of the occupation number depends on whether one has an even or odd number of particles. Since this quantity is a measure of the coherence of our sample we show how the odd-even effect manifests itself in both the momentum distribution of the Bose gas and interference fringe visibility during free temporal evolution.