Density profile of the weakly interacting Bose gas confined in a potential well: Nonzero temperature

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Abstract
A theory is developed for the density profile of a weakly interacting Bose gas confined within an arbitrary potential well at nonzero temperature. This problem is of interest in connection with magnetically confined spin-aligned atomic hydrogen (H↓). The approach taken is conceptually simpler and of comparable accuracy as compared to previous approaches. It is based on the statement of constancy of the chemical potential in equilibrium. This in turn involves the internal chemical potential, a functional of the density, which we treat in a local-density approximation. For a given temperature we develop a form for the local internal chemical potential for a wide range of density using known low-density, intermediate-density (i.e., near the Bose-condensation density), and high-density analytic forms, with appropriate smooth interpolants connecting these three forms. Density profiles are evaluated within the Bogoliubov approximation for the local internal chemical potential for parameters appropriate to magnetically confined H↓ for both symmetric linear and quadratic one-dimensionally varying wells. Comparison is made to previous work. The nonanalytic region of the density profile near the Bose-condensation density is analyzed. We also calculate density profiles including the next-highest-order interaction correction (i.e., beyond Bogoliubov) in the internal chemical potential. For conditions relevant to hypothetical H↓ at moderate densities this leads to a small but discernable effect on the density profile.