Expansion of harmonically trapped interacting particles and time dependence of the contact

TL;DR

This paper analytically investigates the expansion dynamics of various interacting quantum gases released from harmonic traps, focusing on the evolution of the contact parameter and its relation to observed density and momentum distribution tails.

Contribution

It provides exact analytical results on the time evolution of the contact parameter during expansion for different quantum systems, linking theoretical predictions to experimental observations.

Findings

The contact parameter diminishes over time during expansion.

Analytic expressions describe the density and momentum distribution evolution.

The study connects the contact's behavior to observed density tails in experiments.

Abstract

We study the expansion of an interacting atomic system at zero temperature, following its release from an isotropic three-dimensional harmonic trap and calculate the time dependence of its density and momentum distribution, with special focus on the behavior of the contact parameter. We consider different quantum systems, including the unitary Fermi gas of infinite scattering length, the weakly interacting Bose gas, and two interacting particles with highly asymmetric mass imbalance. In all cases analytic results can be obtained, which show that the initial value of the contact, fixing the $1/k^4$ tail of the momentum distribution, disappears for large expansion times. Our results raise the problem of understanding the recent experiment of Chang \textit{et al.} [Phys. Rev. Lett. \textbf{117}, 235303 (2016)] carried out on a weakly interacting Bose gas of metastable $^4$He atoms, where a $1/r^4$ tail in the density distribution was observed after a large expansion time, implying the existence of the $1/k^4$ tail in the asymptotic momentum distribution.