An efficient compact splitting Fourier spectral method for computing the dynamics of rotating spin-orbit coupled spin-1 Bose-Einstein condensates

Abstract

This paper focuses on the dynamic simulation of spin-1 Bose-Einstein condensates (BECs) with rotation and spin-orbit coupling (SOC), and presents a high-order compact splitting Fourier spectral method with favorable numerical properties. The Hamiltonian is split into a linear part, which consists of the Laplace, rotation and SOC terms, and a nonlinear part that includes all the remaining terms. The wave function is well approximated by the Fourier spectral method and is numerically accessed with discrete Fast Fourier transform (FFT). For the linear subproblem, we rotate the wave function by a function-rotation mapping, which is realized easily with purely FFT achieving almost optimal efficiency. The rotation term vanishes, but the SOC term becomes time-dependent. Using a time-dependent matrix decomposition and the function-rotation mapping, we can integrate the linear subproblem exactly and explicitly. The nonlinear subproblem is integrated analytically in physical space. Such “compact” splitting involves only two operators and facilitates the design of high-order splitting schemes. Our method is spectrally accurate in space and high order in time. It is efficient, explicit, unconditionally stable and simple to implement. In addition, we derive some dynamical properties and carry out a systematic study, including accuracy and efficiency tests, dynamical property verification, the SOC effects and dynamics of quantized vortices.