Mathematical models and numerical methods for spinor Bose-Einstein condensates

TL;DR

This paper reviews mathematical models and numerical methods for analyzing ground states and dynamics of spinor Bose-Einstein condensates across various spin systems, emphasizing existence, structure, and computational techniques.

Contribution

It provides a comprehensive review of models, theories, and numerical methods for spinor BECs, including new insights into ground state properties and extensions to higher spin systems.

Findings

Analysis of ground state existence and uniqueness under different parameters

Development of efficient numerical methods for simulating dynamics

Extension of models to higher spin and dipolar systems

Abstract

In this paper, we systematically review mathematical models, theories and numerical methods for ground states and dynamics of spinor Bose-Einstein condensates (BECs) based on the coupled Gross-Pitaevskii equations (GPEs). We start with a pseudo spin-1/2 BEC system with/without an internal atomic Josephson junction and spin-orbit coupling including (i) existence and uniqueness as well as non-existence of ground states under different parameter regimes, (ii) ground state structures under different limiting parameter regimes, (iii) dynamical properties, and (iv) efficient and accurate numerical methods for computing ground states and dynamics. Then we extend these results to spin-1 BEC and spin-2 BEC. Finally, extensions to dipolar spinor systems and/or general spin-F (F>=3) BEC are discussed.