Semi-Implicit finite-difference methods to study the spin-orbit and coherently coupled spinor Bose-Einstein condensates

TL;DR

This paper introduces semi-implicit finite-difference numerical methods for studying spin-orbit coupled spinor Bose-Einstein condensates, demonstrating their accuracy and ability to reveal phase phenomena across different spin states.

Contribution

The paper develops and validates semi-implicit finite-difference schemes for simulating spinor BECs with spin-orbit coupling, extending to multiple spin cases and confirming phase emergence.

Findings

Methods agree well with spectral solutions

Different phases observed in various spin condensates

Effective for quasi-1D and quasi-2D traps

Abstract

We develop time-splitting finite difference methods, using implicit Backward-Euler and semi-implicit Crank-Nicolson discretization schemes, to study the spin-orbit coupled spinor Bose Einstein condensates with coherent coupling in quasi-one and quasi-two-dimensional traps. The split equations involving kinetic energy and spin-orbit coupling operators are solved using either time-implicit Backward-Euler or semi-implicit Crank-Nicolson methods. We explicitly develop the method for pseudospin-1/2, spin-1, and spin-2 condensates. The results for ground states obtained with time-splitting Backward-Euler and Crank-Nicolson methods are in excellent agreement with time-splitting Fourier spectral method which is one of the popular methods to solve the mean-field models for spin-orbit coupled spinor condensates. We confirm the emergence of different phases in spin-orbit coupled pseudospin-1/2, spin-1, and spin-2 condensates with coherent coupling.