A comparison between microscopic methods for finite temperature Bose gases

TL;DR

This paper compares various theoretical methods for analyzing finite temperature properties of weakly interacting one-dimensional Bose gases, highlighting their validity ranges and proposing a practical way to extract condensate information.

Contribution

It evaluates the accuracy of stochastic and perturbative theories for 1D Bose gases at finite temperatures and introduces a simple method to determine the condensate from correlation functions.

Findings

ncB approach loses validity at low temperatures due to thermal fluctuations.

sGPe results show large number fluctuations in small systems.

Proposed method effectively extracts the condensate from correlation functions.

Abstract

We analyze the equilibrium properties of a weakly interacting, trapped quasi-one-dimensional Bose gas at finite temperatures and compare different theoretical approaches. We focus in particular on two stochastic theories: a number-conserving Bogoliubov (ncB) approach and a stochastic Gross-Pitaevskii equation (sGPe) that have been extensively used in numerical simulations. Equilibrium properties like density profiles, correlation functions, and the condensate statistics are compared to predictions based upon a number of alternative theories. We find that due to thermal phase fluctuations, and the corresponding condensate depletion, the ncB approach loses its validity at relatively low temperatures. This can be attributed to the change in the Bogoliubov spectrum, as the condensate gets thermally depleted, and to large fluctuations beyond perturbation theory. Although the two stochastic theories are built on different thermodynamic ensembles (ncB: canonical, sGPe: grand-canonical), they yield the correct condensate statistics in a large BEC (strong enough particle interactions). For smaller systems, the sGPe results are prone to anomalously large number fluctuations, well-known for the grand-canonical, ideal Bose gas. Based on the comparison of the above theories to the modified Popov approach, we propose a simple procedure for approximately extracting the Penrose-Onsager condensate from first- and second-order correlation functions that is computationally convenient. This also clarifies the link between condensate and quasi-condensate in the Popov theory of low-dimensional systems.