Unitarity-Limited Elastic Collision Rate in a Harmonically-Trapped Fermi Gas

TL;DR

This paper derives a universal formula for the elastic collision rate in a harmonically-trapped Fermi gas at unitarity, revealing temperature-dependent behavior and providing estimates relevant for experimental conditions.

Contribution

It presents a new analytical expression for the collision rate at unitarity, incorporating temperature dependence and offering insights for experimental Fermi gas systems.

Findings

Peak collision rate at T/T_F ≈ 0.4

Collision rate scales as (T/T_F)^2 at low temperatures

Collision rate scales as (T_F/T)^2 at high temperatures

Abstract

We derive the elastic collision rate for a harmonically-trapped Fermi gas in the extreme unitarity limit where the s-wave scattering cross section is $\sigma (k) =4\pi/k^2$, with $\hbar k$ the relative momentum. The collision rate is given in the form $\Gamma=\gamma I(T/T_F)$--the product of a universal collision rate $\gamma =k_B T_F/(6 \pi \hbar)$ and a dimensionless function of the ratio of the temperature $T$ to the Fermi temperature $T_F$. We find $I$ has a peak value of $\simeq4.6$ at $T/T_F\simeq0.4$, $I\simeq 82 (T/T_F)^2$ for $T/T_F\leq 0.15$, and $I\simeq 2(T_F/T)^2$ for $T/T_F>1.5$. We estimate the collision rate for recent experiments on a strongly-interacting degenerate Fermi gas of atoms.