Fortran programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap

TL;DR

This paper presents simple numerical Fortran algorithms for solving the time-dependent Gross-Pitaevskii equation in various trap geometries, providing both stationary and non-stationary solutions with verified results and accessible code implementations.

Contribution

It introduces new split-step Crank-Nicolson algorithms for anisotropic traps and provides comprehensive Fortran codes for diverse geometries of Bose-Einstein condensates.

Findings

Numerical results match previous studies for stationary states.

Non-stationary oscillation simulations demonstrate trap dynamics.

Codes are available in Fortran 77 and Fortran 90/95 for broad use.

Abstract

We develop simple numerical algorithms for both stationary and non-stationary solutions of the time-dependent Gross-Pitaevskii (GP) equation describing the properties of Bose-Einstein condensates at ultra low temperatures. In particular, we consider algorithms involving real and imaginary-time propagation based on a split-step Crank-Nicolson method. In a one-space-variable form of the GP equation we consider the one-dimensional linear, two-dimensional circularly symmetric, and the three-dimensional spherically-symmetric traps. In the two-space-variable form we consider the GP equation in two-dimensional anisotropic and three-dimensional axially-symmetric traps. The fully-anisotropic three-dimensional GP equation is also considered. Numerical results for the chemical potential and root-mean-square size of stationary states are reported using imaginary-time propagation programs for all the cases and compared with previously obtained results. Also presented are numerical results of non-stationary oscillation for different trap symmetries using real-time propagation programs. A set of convenient working codes developed in Fortran 77 are also provided for all these cases (twelve programs in all). In the case of two or three space variables, {Fortran 90/95 versions provide some simplification over the Fortran 77 programs}, and these programs are also included (six programs in all).