Fortran and C programs for the time-dependent dipolar Gross-Pitaevskii equation in an anisotropic trap

TL;DR

This paper develops and provides numerical algorithms and source codes in Fortran and C for solving the time-dependent dipolar Gross-Pitaevskii equation in various dimensions, enabling detailed study of dipolar Bose-Einstein condensates.

Contribution

It introduces comprehensive numerical algorithms and publicly available codes for solving the full 3D and reduced 1D/2D dipolar GP equations, including static and dynamic cases.

Findings

Numerical results for energy, chemical potential, and density of dipolar BECs.

Comparison with other methods and approximations validates the algorithms.

Provision of source codes facilitates further research in dipolar BECs.

Abstract

Many of the static and dynamic properties of an atomic Bose-Einstein condensate (BEC) are usually studied by solving the mean-field Gross-Pitaevskii (GP) equation, which is a nonlinear partial differential equation for short-range atomic interaction. More recently, BEC of atoms with long-range dipolar atomic interaction are used in theoretical and experimental studies. For dipolar atomic interaction, the GP equation is a partial integro-differential equation, requiring complex algorithm for its numerical solution. Here we present numerical algorithms for both stationary and non-stationary solutions of the full three-dimensional (3D) GP equation for a dipolar BEC, including the contact interaction. We also consider the simplified one- (1D) and two-dimensional (2D) GP equations satisfied by cigar- and disk-shaped dipolar BECs. We employ the split-step Crank-Nicolson method with real- and imaginary-time propagations, respectively, for the numerical solution of the GP equation for dynamic and static properties of a dipolar BEC. The atoms are considered to be polarized along the z axis and we consider ten different cases, e.g., stationary and non-stationary solutions of the GP equation for a dipolar BEC in 1D (along x and z axes), 2D (in x-y and x-z planes), and 3D, and we provide working codes in Fortran 90/95 and C for these ten cases (twenty programs in all). We present numerical results for energy, chemical potential, root-mean-square sizes and density of the dipolar BECs and, where available, compare them with results of other authors and of variational and Thomas-Fermi approximations.