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arXiv:quant-ph/0508186 · The European Physical Journal D

Atomic density of a harmonically trapped ideal gas near Bose-Einstein transition temperature

Rodolphe Hoppeler, José Viana‐Gomes, Denis Boiron

Published Aug 31, 2006• 7 citations
arXivHTMLPDF
DOI
10.1140/epjd/e2006-00197-8

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Abstract This chapter describes the main properties of the ideal Bose gas confined by a harmonic potential. It calculates the temperature dependence of the condensate density and identify the value of the critical temperature in terms of the oscillator frequency and the number of atoms. This identification provides an important temperature scale for experiments with ultracold atomic gases. The chapter also discusses the behaviour of the expanding cloud after the release of the trap and how the anisotropy of the gas evolves in time, reflecting the initial anisotropy of the momentum distribution. It finally discusses the possibility of getting Bose–Einstein condensation through adiabatic transformations, working with non-harmonic trapping potentials or with mixtures of two atomic species trapped by different harmonic potentials.
Evading the Mermin–Wagner–Hohenberg no-go theorem and revisiting with rigor the ideal Bose gas confined in a square box, we explore a discrete phase transition in two spatial dimensions. Through both analytic and numerical methods, we verify that thermodynamic instability emerges if the number of particles is sufficiently yet finitely large: specifically, N ⩾ 35131 . The instability implies that the isobar of the gas zigzags on the temperature–volume plane, featuring supercooling and superheating phenomena. The Bose–Einstein condensation can then persist from absolute zero to the superheating temperature. Without necessarily taking the large N limit, under a constant pressure condition, the condensation takes place discretely both in the momentum and in the position spaces. Our result is applicable to a harmonic trap. We assert that experimentally observed Bose–Einstein condensations of harmonically trapped atomic gases are a first-order phase transition that involves a discrete change of the density at the center of the trap.