Bose-Einstein condensation in a harmonic potential

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 dimensionD≥2. The thermodynamic limit requires maintaining constant average density by weakening the potential while increasing the particle numberN to infinity, while of course in real experiments the potential is fixed andN 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 forD=1. We study the momentum-space condensate fraction and find that it vanishes as${1 \mathord{\left/ {\vphantom {1 {\sqrt N }}} \right. \kern-\nulldelimiterspace} {\sqrt N }}$ in any number of dimensions in the thermodynamic limit. InD≤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 inD=2. For finite systems we derive theN-dependence of the spatial and momentum condensate fractions and the transition temperatures, features that may be experimentally testable. We show that theN-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.