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Physical Review A

Nonclassical spatial and momentum distributions in a Bose-condensed gas

Lev I. Plimak, Daniel F. Walls

Published Jul 1, 1996• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.54.652

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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.
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.
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.
For a non-self-interacting Bose gas with a fixed, large number of particles confined to a trap, as the ground-state occupation becomes macroscopic, the condensate number fluctuations remain microscopic. However, this is the only significant aspect in which the grand canonical description differs from canonical or microcanonical in the thermodynamic limit. General arguments and estimates including some of the vanishingly small quantities are compared to explicit, fixed-number calculations for 102{10}^{2}102--106{10}^{6}106 particles. \textcopyright{} 1996 The American Physical Society.
The properties and stability of a trapped Bose-Einstein condensate are strongly influenced by attractive interactions between the particles. We describe the spatial distribution, stability, and collisional loss rates for a weakly interacting gas in the mean-field limit. We show how the condensate contracts and becomes unstable as the number of condensate atoms increases. We further show how the number of atoms is limited by the collisional loss rates associated with the contraction of the condensate; this loss is in addition to the particle ejection decay indentified by Kagan et al. \textcopyright{} 1996 The American Physical Society.
Present experiments with Bose condensed gases can be largely described by a semi-ideal two-gas model. In this model, the condensate is influenced only by the mean-field repulsion among condensed atoms, while the thermal cloud is considered an ideal gas confined by an effective potential that consists of the external trap and the mean-field repulsion by the condensate. This simple, intuitive model provides explicit analytical expressions for the density distributions of the condensate and the thermal component. It describes the reduction of the condensate fraction relative to that of an ideal gas as a consequence of the positive chemical potential due to interactions in the condensate.
The anisotropic density distribution of Bose particles after expansion from a magnetic trap has been taken as an important signature of the formation of Bose-Einstein condensate. In this letter we show that even for a non-condensed trapped gas one can get an anisotropic distribution due to anisotropic collision rates along radial and axial directions. We demonstrate that the mean energies along those directions, different from the energy in the trap, have opposite evolution tendencies with the expansion. This difference is particularly important for the case where a large number of bosons is trapped. We also show that the velocity distribution of the non-condensed gas in the trap, assumed initially isotropic and Gaussian, becomes anisotropic and with linewidth dependent on the expansion direction.