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

Abstract

We develop time-splitting finite-difference methods, using an implicit Backward–Euler and a semi-implicit Crank–Nicolson discretization schemes, to study the spin-orbit-coupled (SO-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 a time implicit Backward–Euler or a semi-implicit Crank–Nicolson method. We explicitly develop the methods 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 SO-coupled spinor condensates. We confirm the emergence of different phases in SO-coupled pseudospin-1/2, spin-1 and spin-2 condensates with coherent coupling.