Variational study of a dilute Bose condensate in a harmonic trap

TL;DR

This paper develops a variational approach to approximate solutions of the Gross-Pitaevskii equation for a dilute Bose-Einstein condensate in a harmonic trap, covering both weakly and strongly interacting regimes.

Contribution

It introduces a two-parameter trial wave function that smoothly interpolates between ideal and strongly interacting condensates, providing a unified variational solution.

Findings

The variational method accurately estimates energy contributions across interaction strengths.

The approach simplifies analysis of condensates in isotropic and anisotropic harmonic traps.

Numerical values illustrate the transition from ideal to strongly interacting regimes.

Abstract

A two-parameter trial condensate wave function is used to find an approximate variational solution to the Gross-Pitaevskii equation for $N_0$ condensed bosons in an isotropic harmonic trap with oscillator length $d_0$ and interacting through a repulsive two-body scattering length $a>0$. The dimensionless parameter ${\cal N}_0 \equiv N_0a/d_0$ characterizes the effect of the interparticle interactions, with ${\cal N}_0 \ll 1$ for an ideal gas and ${\cal N}_0 \gg 1$ for a strongly interacting system (the Thomas-Fermi limit). The trial function interpolates smoothly between these two limits, and the three separate contributions (kinetic energy, trap potential energy, and two-body interaction energy) to the variational condensate energy and the condensate chemical potential are determined parametrically for any value of ${\cal N}_0$, along with illustrative numerical values. The straightforward generalization to an anisotropic harmonic trap is considered briefly.