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Physical Review A

Asymptotic formula for the condensate wave function of a trapped Bose gas

Dionisios Margetis

Published Apr 10, 2000• 8 citations
Publisher

Abstract

An analytical property is pointed out for the universal differential equation first derived by Dalfovo, Pitaevskii, and Stringari for the condensate wave function at the boundary of a trapped Bose gas. Specifically, the constant multiplying the Airy function of the solution asymptotically outside the trap is $\sqrt{2}.$ Accordingly, the Wentzel-Kramers-Brillouin approximation is determined in the case of a spherically symmetric harmonic potential. This calculation is related to Josephson-type currents flowing between well-separated traps.

DOI
10.1103/physreva.61.055601

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An analytical property is pointed out for the universal differential equation first derived by Dalfovo, Pitaevskii, and Stringari for the condensate wave function at the boundary of a trapped Bose gas. Specifically, the constant multiplying the Airy function of the solution asymptotically outside the trap is 2.\sqrt{2}.2​. Accordingly, the Wentzel-Kramers-Brillouin approximation is determined in the case of a spherically symmetric harmonic potential. This calculation is related to Josephson-type currents flowing between well-separated traps.
For a trapped, dilute atomic gas of short-range, repulsive interactions at extremely low temperatures, when Bose–Einstein condensation is nearly complete, some special forms of the time-dependent condensate wave function and the pair-excitation function, the latter being responsible for phonon creation, are investigated. Specifically, (i) a class of external potentials Ve(r,t) that allow for localized, shape-preserving solutions to the nonlinear Schrödinger equation for the condensate wave function, each recognized as a solitary wave moving along an arbitrary trajectory, is derived and analyzed in any number of space dimensions; and (ii) for any such external potential and condensate wave function, the nonlinear integro-differential equation for the pair-excitation function is shown to admit solutions of the same nature. Approximate analytical results are presented for a sufficiently slowly varying trapping potential. Numerical results are obtained for the condensate wave function when Ve is a time-independent, spherically symmetric harmonic potential.
Based on the Gross-Pitaevskii(G-P) energy functional and variational method, we present a new method to find the ground state wave function for Bose-condensed gas in a harmonic trap at zero temperature. With this method we are able to find the analytic expression of ground state wave function and explore the relevant quantities, such as energy, chemical potential and so on. These results agree well with previous ground state numerical solutions of the G-P equation given by Dalfovo et al. Under the large number of particles N limit for repulsive interaction, our results and Thomas-Fermi approximation model approach to reaching unanimity. This new method is simple compared to other methods used to solve numerically the G-P equation, and one can obtained the analytic and reliable results.