Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates

TL;DR

This paper introduces new efficient numerical methods for accurately computing the ground states and dynamics of dipolar Bose-Einstein condensates by reformulating the governing equations to handle singular dipolar interactions.

Contribution

The paper proposes a novel mathematical reformulation and numerical algorithms that improve efficiency and accuracy in simulating dipolar BECs compared to existing methods.

Findings

Reformulated the GPE as a Gross-Pitaevskii-Poisson system

Proved existence, uniqueness, and nonexistence of ground states under various conditions

Demonstrated numerical efficiency and accuracy through extensive 3D simulations

Abstract

New efficient and accurate numerical methods are proposed to compute ground states and dynamics of dipolar Bose-Einstein condensates (BECs) described by a three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a dipolar interaction potential. Due to the high singularity in the dipolar interaction potential, it brings significant difficulties in mathematical analysis and numerical simulations of dipolar BECs. In this paper, by decoupling the two-body dipolar interaction potential into short-range (or local) and long-range interactions (or repulsive and attractive interactions), the GPE for dipolar BECs is reformulated as a Gross-Pitaevskii-Poisson type system. Based on this new mathematical formulation, we prove rigorously existence and uniqueness as well as nonexistence of the ground states, and discuss the existence of global weak solution and finite time blowup of the dynamics in different parameter regimes of dipolar BECs. In addition, a backward Euler sine pseudospectral method is presented for computing the ground states and a time-splitting sine pseudospectral method is proposed for computing the dynamics of dipolar BECs. Due to the adaption of new mathematical formulation, our new numerical methods avoid evaluating integrals with high singularity and thus they are more efficient and accurate than those numerical methods currently used in the literatures for solving the problem. Extensive numerical examples in 3D are reported to demonstrate the efficiency and accuracy of our new numerical methods for computing the ground states and dynamics of dipolar BECs.