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

Paramjeet Banger, Pardeep Kaur, Sandeep Gautam

Published Jan 1, 2022• 10 citations
Publisher

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.

DOI
10.1142/s0129183122500462

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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.
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.
In this work, a high-order numerical method with mass and energy conservation obtained by the operator-compensation (OC) technique is proposed for the coupled Gross–Pitaevskii equations (CGPEs) modelling the spin–orbit-coupled Bose–Einstein condensates. The OC technique is utilized to seek high-order approximations to the first- and the second-order spatial derivatives in the CGPEs, after which a time-dependent semi-discrete system is obtained. Then we obtain a fully discrete scheme by applying the Crank–Nicolson method to discretize this system in time. We theoretically show that the semi-discrete system and the fully discrete scheme conserve both mass and energy on the discrete level. In addition, the stability analysis is discussed for the proposed method. Numerical experiments are given to test accuracy and verify conservative properties. The dynamics of the two-dimensional CGPEs and the effects of the parameters in the CGPEs are simulated.
In this paper, an efficient time-splitting Fourier spectral method is proposed to simulate the dynamics of spin-orbit coupled spin-1 Bose-Einstein condensates (SOC spin-1 BECs). We split the Hamiltonian into a linear part, which consists of the Laplace and SOC terms, and a nonlinear part that includes all the remaining terms. The linear subproblem is integrated analytically in phase space by solving an ordinary differential system of constant coefficient matrix. While, for the nonlinear subproblem, it is proved the coefficient matrix is actually time-independent in physical space, therefore, the nonlinear subproblem can be integrated exactly. Based on such two-step splitting, we construct high-order schemes to simulate the dynamics. 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 for SOC spin-1 BECs. Extensive numerical results are presented to confirm the accuracy and efficiency, illustrate the dynamical properties at discrete level, and show interesting physics of SOC spin-1 BECs, such as the SOC effects and different wave patterns.
An effective spin-orbit coupling can be generated in a cold atom system by engineering atom-light interactions. In this Letter we study spin-1/2 and spin-1 Bose-Einstein condensates with Rashba spin-orbit coupling, and find that the condensate wave function will develop nontrivial structures. From numerical simulation we have identified two different phases. In one phase the ground state is a single plane wave, and often we find the system splits into domains and an array of vortices plays the role of a domain wall. In this phase, time-reversal symmetry is broken. In the other phase the condensate wave function is a standing wave, and it forms a spin stripe. The transition between them is driven by interactions between bosons. We also provide an analytical understanding of these results and determine the transition point between the two phases.
In this paper, we investigate the dynamic properties of spin–orbit coupling spin-1 ferromagnetic Bose–Einstein condensates with different trap geometry. Our results are obtained in terms of the three-component Gross–Pitaevskii equation of mean-field theory. Two kinds of trap potential are discussed: isotropic and anisotropic. It is shown that the spin-exchange dynamics are greatly influenced by trap geometry. For the latter with weak spin–orbit coupling strength, we find that the three-component oscillation accelerates and some small difference emerges between component m=1 and m=-1. With the increase in spin–orbit coupling strength, the three components reach almost the same population. In addition, the kinetic energy of the system changes within a small scope for strong spin–orbit coupling, as opposed to a constant value in an isotropic trap. The density distributions display that the stripe phase appears with the increase in spin–orbit coupling strength. The method of generating stripe structure is different from the ground state of ferromagnetic condensate. For isotropic trap, the spatial separation of top and bottom spin–orbit condensates in component m=1 and m=-1 occurs at weak spin–orbit coupling, and square lattice appears at strong spin–orbit coupling.
We develop a symmetry classification scheme to find ground states of pseudo-spin-12\frac{1}{2}21​, spin-1, and spin-2 spin-orbit-coupled spinor Bose-Einstein condensates and show that as the SO(2) symmetry of simultaneous spin and space rotations is broken into discrete cyclic groups, various types of lattice structures emerge in the absence of a lattice potential. Examples include two different kagome lattices for pseudo-spin-12\frac{1}{2}21​ condensates and a nematic vortex lattice in which uniaxial and biaxial spin textures align alternatively for spin-2 condensates. For the pseudo-spin-12\frac{1}{2}21​ system, although mean-field states always break time-reversal symmetry, there exists a time-reversal invariant many-body ground state, which is fragmented and expected to be observed in a microcondensate.