Bose-Einstein Condensation in a Harmonic Potential

TL;DR

This paper explores Bose-Einstein condensation in harmonic traps, analyzing phase transitions, finite-size effects, and the differences between ideal and interacting gases across various dimensions.

Contribution

It provides a detailed analysis of BEC in harmonic potentials, including finite-size effects and the impact of interactions, extending understanding beyond the thermodynamic limit.

Findings

BEC occurs in harmonic traps for D≥2 in the thermodynamic limit.

Finite systems exhibit BEC below a pseudo-critical temperature without a true phase transition.

Momentum condensate fraction vanishes as 1/N^{1/2} in the thermodynamic limit.

Abstract

We examine several features of Bose-Einstein condensation (BEC) in an external harmonic potential well. In the thermodynamic limit, there is a phase transition to a spatial Bose-Einstein condensed state for dimension D greater than or equal to 2. The thermodynamic limit requires maintaining constant average density by weakening the potential while increasing the particle number N to infinity, while of course in real experiments the potential is fixed and N stays finite. For such finite ideal harmonic systems we show that a BEC still occurs, although without a true phase transition, below a certain ``pseudo-critical'' temperature, even for D=1. We study the momentum-space condensate fraction and find that it vanishes as 1/N^(1/2) in any number of dimensions in the thermodynamic limit. In D less than or equal to 2 the lack of a momentum condensation is in accord with the Hohenberg theorem, but must be reconciled with the existence of a spatial BEC in D=2. For finite systems we derive the N-dependence of the spatial and momentum condensate fractions and the transition temperatures, features that may be experimentally testable. We show that the N-dependence of the 2D ideal-gas transition temperature for a finite system cannot persist in the interacting case because it violates a theorem due to Chester, Penrose, and Onsager.