Computing Ground States of Spin-1 Bose-Einstein Condensates by the Normalized Gradient Flow

TL;DR

This paper introduces a novel numerical method using normalized gradient flow and a specialized discretization scheme to efficiently compute the ground states of spin-1 Bose-Einstein condensates, extending existing methods for single-component BEC.

Contribution

The paper develops a new normalized gradient flow approach with a unique projection condition for spin-1 BEC, along with an efficient BFSP discretization scheme, enabling accurate ground state computations.

Findings

Method successfully computes ground states for various interactions and potentials.

Numerical results demonstrate high efficiency and accuracy.

Extension of single-component BEC methods to spin-1 BEC achieved.

Abstract

In this paper, we propose an efficient and accurate numerical method for computing the ground state of spin-1 Bose-Einstein condensates (BEC) by using the normalized gradient flow or imaginary time method. The key idea is to find a third projection or normalization condition based on the relation between the chemical potentials so that the three projection parameters used in the projection step of the normalized gradient flow are uniquely determined by this condition as well as the other two physical conditions given by the conservation of total mass and total magnetization. This allows us to successfully extend the most popular and powerful normalized gradient flow or imaginary time method for computing the ground state of single component BEC to compute the ground state of spin-1 BEC. An efficient and accurate discretization scheme, the backward-forward Euler sine-pseudospectral method (BFSP), is proposed to discretize the normalized gradient flow. Extensive numerical results on ground states of spin-1 BEC with ferromagnetic/antiferromagnetic interaction and harmonic/optical lattice potential in one/three dimensions are reported to demonstrate the efficiency of our new numerical method.