Collective oscillations of a confined Bose gas at finite temperature in the random-phase approximation

TL;DR

This paper develops a generalized random-phase approximation theory to analyze the collective oscillations of a finite-temperature Bose gas in a harmonic trap, highlighting the importance of density fluctuations and comparing with mean-field results.

Contribution

It introduces an extended RPA framework that includes dynamical coupling of density fluctuations, improving upon previous models for finite-temperature Bose gases.

Findings

The theory accurately obeys the generalized Kohn theorem for dipolar modes.

Normal and anomalous density fluctuations significantly affect monopolar excitations.

Mean-field theory suffices for quadrupolar modes, but not for monopolar modes.

Abstract

We present a theory for the linear dynamics of a weakly interacting Bose gas confined inside a harmonic trap at finite temperature. The theory treats the motions of the condensate and of the non-condensate on an equal footing within a generalized random-phase approximation, which ({\it i}) extends the second-order Beliaev-Popov approach by allowing for the dynamical coupling between fluctuations in the thermal cloud, and ({\it ii}) reduces to an earlier random-phase scheme when the anomalous density fluctuations are omitted. Numerical calculations of the low-lying spectra in the case of isotropic confinement show that the present theory obeys with high accuracy the generalized Kohn theorem for the dipolar excitations and demonstrate that combined normal and anomalous density fluctuations play an important role in the monopolar excitations of the condensate. Mean-field theory is instead found to yield accurate results for the quadrupolar modes of the condensate. Although the restriction to spherical confinement prevents quantitative comparisons with measured spectra, it appears that the non-mean field effects that we examine may be relevant to explain the features exhibited by the breathing mode as a function of temperature in the experiments carried out at JILA on a gas of $^{87}$Rb atoms.