# Ground states and their characterization of spin-$F$ Bose-Einstein condensates

## TL;DR

This paper introduces a novel optimization-based numerical method for computing ground states of spin-F Bose-Einstein condensates, including complex spin-3 cases, using manifold optimization and multigrid acceleration.

## Contribution

It develops the first applicable algorithm for arbitrary spin BECs, employing Fourier pseudospectral discretization, manifold optimization, and multigrid techniques.

## Key findings

- Efficient computation of ground states for spin-1, spin-2, and spin-3 BECs.
- Demonstration of interesting physical phenomena in various dimensions.
- Validation of the method's effectiveness through extensive numerical experiments.

## Abstract

The computation of the ground states of spin-$F$ Bose-Einstein condensates (BECs) can be formulated as an energy minimization problem with two quadratic constraints. We discretize the energy functional and constraints using the Fourier pseudospectral schemes and view the discretized problem as an optimization problem on manifold. Three different types of retractions to the manifold are designed. They enable us to apply various optimization methods on manifold to solve the problem. Specifically, an adaptive regularized Newton method is used together with a cascadic multigrid technique to accelerate the convergence. According to our limited knowledege, our method is the first applicable algorithm for BECs with an arbitrary integer spin, including the complicated spin-3 BECs. Extensive numerical results on ground states of spin-1, spin-2 and spin-3 BECs with diverse interaction and optical lattice potential in one/two/three dimensions are reported to show the efficiency of our method and to demonstrate some interesting physical phenomena.