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

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 Schrödinger 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. Iterations can be initiated using either the simple-harmonic-oscillator ground state solution, or the Thomas–Fermi (TF) solution. It is found that for condensates containing up to a few hundred atoms, both approaches lead to rapid convergence. However, in the strong interaction limit of condensates containing thousands of atoms, it is the SD approach coupled with the TF starting orbitals, which leads to quick convergence. 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. Program summary Title of program:bose.x Catalogue identifier:ADWZ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADWZ_v1_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format:tar.gz Computers:PC's/Linux, Sun Ultra 10/Solaris, HP Alpha/Tru64, IBM/AIX Programming language used:mostly Fortran 90 Number of bytes in distributed program, including test data, etc.:27 313 Number of lines in distributed program, including test data, etc.:28 015 Card punching code:ASCII Nature of physical problem:It is widely believed that the static properties of dilute Bose condensates, as obtained in atomic traps, can be described to a fairly good accuracy by the time-independent Gross–Pitaevskii equation. This program presents an efficient approach of solving this equation. Method of solution:The solutions of the Gross–Pitaevskii equation corresponding to the condensates in atomic traps are expanded as linear combinations of simple-harmonic oscillator eigenfunctions. Thus, the Gross–Pitaevskii equation which is a second-order nonlinear differential equation, is transformed into a matrix eigenvalue problem. Thereby, its solutions are obtained in a self-consistent manner, using methods of computational linear algebra. Unusual features of the program:None