Abstract.
We have studied the atomic density of a cloud confined in an isotropic harmonic trap at the vicinity of the Bose-Einstein transition temperature. We show that, for a non-interacting gas and near this temperature, the ground-state density has the same order of magnitude as the excited states density at the centre of the trap. This holds in a range of temperatures where the ground-state population is negligible compared to the total atom number. We compare the exact calculations, available in a harmonic trap, to semi-classical approximations. We show that these latter should include the ground-state contribution to be accurate.
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In reference pathria, the transition temperature was indeed Tsc (see later in the text). The transition temperature defined by the maximum of the second derivative of the condensate fraction has been calculated for atom number in the range 103–108; the relative deviation is below ∼ 10-3 on the transition temperature and ∼ 10-2 on the condensate peak density fraction
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After submission of this article, we have been aware of a different type of semi-classical approximations which does not give rise to divergences. See V.I. Yukalov, Phys. Rev. A 72, 033608 (2005)
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Hoppeler, R., Viana Gomes, J. & Boiron, D. Atomic density of a harmonically trapped ideal gas near Bose-Einstein transition temperature. Eur. Phys. J. D 41, 157–162 (2007). https://doi.org/10.1140/epjd/e2006-00197-8
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DOI: https://doi.org/10.1140/epjd/e2006-00197-8