Mathematical and numerical study on the ground states of rotating spin-orbit coupled spin-1 Bose-Einstein condensates

TL;DR

This paper rigorously analyzes the ground states of rotating spin-orbit coupled spin-1 Bose-Einstein condensates using mathematical proofs and develops an efficient numerical method to verify properties and explore physical phenomena like vortices.

Contribution

It provides the first rigorous existence proof and analytical properties of ground states, along with a novel, efficient numerical algorithm for their computation.

Findings

Verified existence and properties of ground states

Achieved high accuracy and efficiency in numerical simulations

Discovered phenomena like giant vortices and U-shape vortex lines

Abstract

In this article, we study mathematically and numerically the ground states of three-component rotating spin-orbit coupled (SOC) spin-1 Bose-Einstein condensates modeled by the coupled Gross-Pitaevskii equations (CGPEs). Firstly, we rigorously prove existence result of the ground state and derive some analytical properties, including the virial identity and negativity of SOC energy. Secondly, we propose an efficient and accurate preconditioned nonlinear conjugate gradient (PCG) algorithm to compute the ground states. We truncate the whole space into a bounded rectangular domain and readily apply the Fourier spectral method to approximate the wave function. The PCG method is successfully adapted with appropriate modifications to the adaptive step size control strategy for the one-parameter energy minimization problem and to the choice of preconditioners, achieving great performance in terms of accuracy and efficiency. Lastly, we carry out extensive numerical experiments to verify the existence and property results of the ground states, confirm the spatial spectral accuracy by traversing the most commonly-used initial guesses for each component thanks to its great efficiency, which is also attributed to a utilization of cascadic multigrid and discrete Fast Fourier Transform (FFT). Moreover, we investigate the effects of local interaction, rotation and spin-orbit coupling and external trapping potential on the ground state, and unveil some interesting physical phenomena, such as giant vortex and U-shape vortex line.