Approximating Steady States in Equilibrium and Nonequilibrium Condensates

TL;DR

This paper develops a generalized WKB method to approximate steady states of the Gross-Pitaevskii equation, providing accurate density profiles for Bose gases in traps, including nonequilibrium conditions, surpassing the Thomas-Fermi approximation.

Contribution

It introduces a generalized divergence-free WKB approach for the GP equation, yielding explicit, uniformly valid condensate density approximations in multiple dimensions, applicable to equilibrium and nonequilibrium systems.

Findings

Accurate approximation of condensate density including healing effects.

Excellent agreement between asymptotic and numerical solutions.

Applicable to both equilibrium and steady-state nonequilibrium Bose gases.

Abstract

We obtain approximations for the time-independent Gross-Pitaevskii (GP) and complex GP equation in two and three spatial dimensions by generalizing the divergence-free WKB method. The results include an explicit expression of a uniformly valid approximation for the condensate density of an ultracold Bose gas confined in a harmonic trap that extends into the classically forbidden region. This provides an accurate approximation of the condensate density that includes healing effects at leading order that are missing in the widely adopted Thomas-Fermi approximation. The results presented herein allow us to formulate useful approximations to a range of experimental systems including the equilibrium properties of a finite temperature Bose gas and the steady-state properties of a 2D nonequilibrium condensate. Comparisons between our asymptotic and numerical results for the conservative and forced-dissipative forms of the GP equations as applied to these systems show excellent agreement between the two sets of solutions thereby illustrating the accuracy of these approximations.