A basis-set based Fortran program to solve the Gross-Pitaevskii Equation for dilute Bose gases in harmonic and anharmonic traps

TL;DR

This paper presents a Fortran 90 program that solves the Gross-Pitaevskii equation for dilute Bose gases in harmonic and anharmonic traps using a basis set expansion, enabling analysis of condensate properties and entropy.

Contribution

The authors develop and validate a basis-set based Fortran program for solving the GPE in harmonic and anharmonic traps, including entropy analysis, which is a novel computational approach.

Findings

Excellent agreement with existing methods for harmonic traps.

Anharmonicity affects condensate properties significantly.

Shannon entropy increases monotonically with particle number.

Abstract

Inhomogeneous boson systems, such as the dilute gases of integral spin atoms in low-temperature magnetic traps, are believed to be well described by the Gross-Pitaevskii equation (GPE). GPE is a nonlinear Schroedinger equation which describes the order parameter of such systems at the mean field level. In the present work, we describe a Fortran 90 computer program developed by us, which solves the GPE using a basis set expansion technique. In this technique, the condensate wave function (order parameter) is expanded in terms of the solutions of the simple-harmonic oscillator (SHO) characterizing the atomic trap. Additionally, the same approach is also used to solve the problems in which the trap is weakly anharmonic, and the anharmonic potential can be expressed as a polynomial in the position operators x, y, and z. The resulting eigenvalue problem is solved iteratively using either the self-consistent-field (SCF) approach, or the imaginary time steepest-descent (SD) approach. Our results for harmonic traps are also compared with those published by other authors using different numerical approaches, and excellent agreement is obtained. GPE is also solved for a few anharmonic potentials, and the influence of anharmonicity on the condensate is discussed. Additionally, the notion of Shannon entropy for the condensate wave function is defined and studied as a function of the number of particles in the trap. It is demonstrated numerically that the entropy increases with the particle number in a monotonic way.