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arXiv:quant-ph/9807036 · Physical Review A

Momentum distribution of confined bosons: Temperature dependence

Jacques Tempere, F. Brosens, L. F. Lemmens, J. T. Devreese

Published Oct 1, 1998• 6 citations
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Abstract

The momentum distribution function of a parabolically confined gas of bosons with harmonic interparticle interactions is derived. In the Bose-Einstein condensation region, this momentum distribution substantially deviates from a Maxwell-Boltzmann distribution. It is argued that the determination of the temperature of the boson gas from the Bose-Einstein momentum distribution function is more appropriate than the currently used fitting to the high-momentum tail of the Maxwell-Boltzmann distribution.

DOI
10.1103/physreva.58.3180

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Kluwer Academic Publishers eBooks•Dec 10, 2005•1 citation
The momentum distribution function of a parabolically confined gas of bosons with harmonic interparticle interactions is derived. In the Bose-Einstein condensation region, this momentum distribution substantially deviates from a Maxwell-Boltzmann distribution. It is argued that the determination of the temperature of the boson gas from the Bose-Einstein momentum distribution function is more appropriate than the currently used fitting to the high-momentum tail of the Maxwell-Boltzmann distribution.
Based on the Hellman-Feynman theorem, it is shown that the average square radius of a cloud of interacting bosons in a parabolic well can be derived from their free energy. As an application, the temperature dependence of the moment of inertia of noninteracting bosons in a parabolic trap is determined as a function of the number of bosons. Well below the critical condensation temperature, the Bose-Einstein statistics are found to substantially reduce the moment of inertia of this system, as compared to a gas of ``distinguishable'' particles in a parabolic well.
The spatial and momentum distributions of the uncondensed cloud associated with a confined Bose-condensed gas are found to have a universal form which differs from the classical Maxwell-Boltzmann distribution. This universality is a direct consequence of the condensation which stabilizes the density of the cloud on the threshold of condensation. The deviation of the distributions from the Maxwell-Boltzmann distribution is a macroscopic signature of the quantum statistics. \textcopyright{} 1996 The American Physical Society.
We present a new method to identify the critical point for the Bose-Einstein condensation (BEC) of a trapped Bose gas. We calculate the momentum distribution of an interacting Bose gas near the critical temperature, and find that it deviates significantly from the Gaussian profile as the temperature approaches the critical point. More importantly, the standard deviation between the calculated momentum spectrum and the Gaussian profile at the same temperature shows a turning point at the critical point, which can be used to determine the critical temperature. These predictions are also confirmed by our BEC experiment for magnetically trapped 87 Rb gases.
Because of experimental interest in the effects of particle interactions in Bose condensates, such as a dilute atomic vapour of 87Rb, we have worked out the momentum distribution in an N-Boson many-body assembly at T = 0, for which both confinement and interbosonic interactions are modelled as purely harmonic. The half-width of the Gaussian momentum distribution is displayed as a function of N and of the strength and sign of the harmonic interactions. The bosonic kinetic energy is finally treated.
A peculiarity of trapped atomic gases is that Bose-Einstein condensation (BEC) shows up not only in momentum space, where it is usually discussed in traditional textbooks, but also in coordinate space. This is the consequence of the inhomogeneity of these systems, which makes it possible to separate the condensate from the thermal component also in coordinate space. Actually, most of measurements in trapped gases have been so far limited to the study of density profiles and of the effects of BEC in coordinate space. The possibility of a direct measurement of the momentum distribution, as emerged from recent experiments based on two-photon Bragg scattering, is highly appealing. In fact, despite the inhomogeneity of the gas, BEC shows up more deeply in momentum than in coordinate space. In current experiments on harmonically-confined Bose gases, the sizes of the condensate R and of the thermal cloud R_T are in fact typically comparable.