Collective oscillations of a trapped quantum gas in low dimensions

TL;DR

This paper analyzes collective oscillations of trapped quantum gases across different dimensions and temperatures, deriving a unified hydrodynamic equation and providing explicit mode solutions for Bose and Fermi gases.

Contribution

It introduces a comprehensive hydrodynamic framework applicable to quantum gases in various confinement regimes, including finite temperature effects and different quantum statistics.

Findings

Derived a closed hydrodynamic equation for velocity fields in quantum gases.

Obtained explicit analytic solutions for collective modes in multiple geometries.

Validated results against existing analytical predictions in limiting cases.

Abstract

We present a comprehensive study of the discretized modes of an atomic gas in different conditions of confinement. Starting from the equations of hydrodynamics we derive a closed equation for the velocity field, depending on the adiabatic and isothermal compressibilities and applicable to different dimensions and quantum statistics. At zero temperature the equation reproduces the irrotational behavior of superfluid hydrodynamics. It is also applicable above the critical temperature in the collisional regime, where the appearence of rotational components in the velocity field is caused by the external potential. In the presence of harmonic trapping, a general class of analytic solutions is obtained for systems exhibiting a polytropic equation of state, characterized by a power law isoentropic dependence of the pressure on the density. Explicit results for the compressional modes are derived for both Bose and Fermi gases in the pancake, cigar as well as in the deep 2D and 1D regimes. Our results agree with the analytical predictions available in the literature in some limiting cases. They are particularly relevant in 1D configurations, where the study of the collective frequencies could provide a unique test of the achievement of the collisional regime at finite temperature.