Ground state solution of Bose-Einstein condensate by directly minimizing the energy functional

TL;DR

This paper introduces a novel finite element numerical method to compute the ground state of Bose-Einstein condensates by directly minimizing the energy functional, applicable across different dimensions and symmetries.

Contribution

It presents a new finite element approach for directly minimizing the energy functional to find BEC ground states, including extensions to excited states and various symmetries.

Findings

Accurate ground state solutions for BEC in 1d, 2d, and 3d demonstrated.

Method effective for weak and strong interaction regimes.

Comparison with Thomas-Fermi approximation shows good agreement.

Abstract

In this paper, we propose a new numerical method to compute the ground state solution of trapped interacting Bose-Einstein condensation (BEC) at zero or very low temperature by directly minimizing the energy functional via finite element approximation. As preparatory steps we begin with the 3d Gross-Pitaevskii equation (GPE), scale it to get a three-parameter model and show how to reduce it to 2d and 1d GPEs. The ground state solution is formulated by minimizing the energy functional under a constraint, which is discretized by the finite element method. The finite element approximation for 1d, 2d with radial symmetry and 3d with spherical symmetry and cylindrical symmetry are presented in detail and approximate ground state solutions, which are used as initial guess in our practical numerical computation of the minimization problem, of the GPE in two extreme regimes: very weak interactions and strong repulsive interactions are provided. Numerical results in 1d, 2d with radial symmetry and 3d with spherical symmetry and cylindrical symmetry for atoms ranging up to millions in the condensation are reported to demonstrate the novel numerical method. Furthermore, comparisons between the ground state solutions and their Thomas-Fermi approximations are also reported. Extension of the numerical method to compute the excited states of GPE is also presented.