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 simpleharmonic 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 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

GPELab, a Matlab toolbox to solve Gross–Pitaevskii equations I: Computation of stationary solutions

“…For example, in [51], the authors propose a Fortran 90 code that solves the non rotating one-component GPE for a cubic nonlinearity and a quadratic potential by using the imaginary time method. In [52,53], the authors distribute codes developed around finite difference methods for solving the GPE with a radial or a spherical potential for a single-component, without rotation.…”

A Simple and Efficient Numerical Method for Computing the Dynamics of Rotating Bose--Einstein Condensates via Rotating Lagrangian Coordinates

“…Many numerical methods have been proposed to study the dynamics of nonrotating BECs without DDI, i.e., when Ω = 0 and η = 0 [5,8,12,15,21,38,45]. Among them, the time-splitting sine/Fourier pseudospectral method is one of the most successful methods.…”

Faraday and resonant waves in binary collisionally-inhomogeneous Bose–Einstein condensates

“…The numerical treatment of the GPE gradually developed into a research direction in its own right and now fast and accurate numerical algorithms exist for calculation of the ground and excited states of BECs using imaginary-time propagation [30], as well as explicit finite-difference scheme [31], time-splitting spectral methods [32], methods based on expansion of the condensate wave function in terms of the solutions of the harmonic oscillator which characterises the magnetic trap [33], symplectic shooting method [34], etc. A popular package of codes (available in Fortran and C [35][36][37][38][39][40] programming languages, including parallelised versions in MPI and CUDA) has proven to be particularly useful, as it provides the stationary states and the nonlinear dynamics of one-, two-and three-dimensional BECs.…”