A robust and efficient numerical method to compute the dynamics of the rotating two-component dipolar Bose-Einstein condensates

TL;DR

This paper introduces a robust, spectrally accurate numerical method for simulating the complex dynamics of rotating two-component dipolar Bose-Einstein condensates, effectively handling nonlocal interactions and dynamic phenomena.

Contribution

It develops a new numerical approach combining rotating Lagrangian coordinates, a time-splitting Fourier pseudospectral method, and a Gaussian-sum solver for efficient, accurate simulation of dipolar BEC dynamics.

Findings

Validated spectral accuracy in space and second-order accuracy in time.

Simulated and analyzed vortex lattice dynamics and collapse phenomena.

Confirmed conservation of physical quantities like mass and energy.

Abstract

In this paper, we propose a robust and efficient numerical method to compute the dynamics of the rotating two-component dipolar Bose-Einstein condensates (BEC). Using the rotating Lagrangian coordinates transform \cite{BMTZ2013}, we reformulate the original coupled Gross-Pitaevskii equations (CGPE) into new equations where the rotating term vanishes and the potential becomes time-dependent. A time-splitting Fourier pseudospectral method is proposed to simulate the new equations where the nonlocal Dipole-Dipole Interactions (DDI) are computed by a newly-developed Gaussian-sum (GauSum) solver \cite{EMZ2015} which helps achieve spectral accuracy in space within $O(N\log N)$ operations ($N$ is the total number of grid points). The new method is spectrally accurate in space and second order accurate in time, and the accuracies are confirmed numerically. Dynamical properties of some physical quantities, including the total mass, energy, center of mass and angular momentum expectation, are presented and confirmed numerically. Interesting dynamics phenomena that are peculiar to the rotating two-component dipolar BECs, such as dynamics of center of mass, quantized vortex lattices dynamics and the collapse dynamics of 3D cases, are presented.