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.
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.
This paper investigates the dynamics of spin-2 Bose-Einstein condensates (BECs) with rotation and spin-orbit coupling (SOC). In order to better simulate the dynamics, we present an efficient high-order compact splitting Fourier spectral method. This method splits the Hamiltonian 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 linear subproblem, the handling of rotation term and SOC term poses a major challenge. Using a function mapping based on rotation, we can integrate the linear subproblem exactly and explicitly. This mapping we propose not only helps eliminate the rotation term, but also prevents the SOC term from evolving into a time-dependent form. 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 vortex lattice.
We propose high-order compact splitting Fourier spectral methods to compute the dynamics of rotating spin-1 Bose–Einstein condensates (BECs) under an Ioffe-Pritchard magnetic field. We split the Hamiltonian into a linear part, which consists of the Laplace, rotation and Zeeman energy terms, and a nonlinear part, which includes all the remaining terms. This “compact” splitting, involving only two operators, significantly simplifies the construction of high-order schemes. For the linear subproblem, we factorize the linear operator as a product of simple sub-operators using the exact classical-quantum correspondence and semigroups decomposition, and such sub-operators are well approximated by Fourier spectral method and integrated exactly in phase space as usual. Importantly and surprisingly, the splitting coefficients, originally determined by a nonlinear equation, can actually be computed exactly by solving a linear system, ending up with explicit formulas. For the nonlinear subproblem, we derive an exact formula of the spin vector, which is not conserved in presence of the magnetic field, and transform it into a linear problem. Then we design explicit high-order schemes using Magnus integrators. Our scheme achieves spectral accuracy in space and high-order precision in time with near-optimal efficiency. It is explicit, easy to implement and unconditionally stable. In addition, we derive some properties of our numerical scheme and conduct a comprehensive investigation, including accuracy confirmation, efficiency test, property verification, interaction of quantized vortices and dynamics under honeycomb potential.
Abstract. We propose high-order compact splitting Fourier spectral methods to compute the dynamics of dipolar Bose–Einstein condensates (BECs) with arbitrary-angle rotation. The Hamiltonian is split into a linear part that consists of the Laplace and rotation and a nonlinear part that includes all the remaining terms. For the linear subproblem, we factorize the linear operator into a product of simple suboperators using the exact classical-quantum correspondence and semigroups decomposition, and these suboperators are integrated exactly in either physical or phase space. Importantly, we first derive explicit formulas for the splitting coefficients originally obtained by numerically solving a nonlinear system. The nonlinear part is solved exactly in physical space, and the convolution-type nonlocal potential with a singular kernel is computed via the Far-field Smooth Approximation (FSA) method with spectral accuracy. This “compact” splitting, involving only two operators, significantly simplifies the design of high-order schemes. Our method achieves high-order temporal precision and spectral spatial accuracy with near-optimal efficiency, and it is explicit, easy to implement and unconditionally stable. In addition, we derive some dynamical properties and conduct a comprehensive investigation, including accuracy confirmation, property verification, interaction of quantized vortices, and dynamics under honeycomb potential.
A time-splitting generalized-Laguerre–Fourier–Hermite pseudospectral method is proposed for computing the dynamics of rotating Bose–Einstein condensates (BECs) in two and three dimensions. The new numerical method is based on the following: (i) the use of a time-splitting technique for decoupling the nonlinearity; (ii) the adoption of polar coordinates in two dimensions and cylindrical coordinates in three dimensions such that the angular rotation term becomes constant coefficient; and (iii) the construction of eigenfunctions for the linear operator by properly scaling the generalized-Laguerre, Fourier, and Hermite functions. The new method enjoys the following properties: (i) it is explicit, time reversible, and time transverse invariant; (ii) it conserves the position density and is spectrally accurate in space and second-order or fourth-order accurate in time; and (iii) it solves the problem in the original whole space instead of in a truncated artificial computational domain. The method is also extended to solve the coupled Gross–Pitaevskii equations for the dynamics of rotating two-component and spin-1 BECs. Extensive numerical results for the dynamics of BECs are reported to demonstrate the accuracy and efficiency of the new method for rotating BECs.
In this article, we study mathematically and numerically the ground states of three-component rotating spin-orbit coupled (SOC) spin-1 Bose-Einstein condensates modeled by the coupled Gross-Pitaevskii equations (CGPEs). Firstly, we rigorously prove existence result of the ground state and derive some analytical properties, including the virial identity and negativity of SOC energy. Secondly, we propose an efficient and accurate preconditioned nonlinear conjugate gradient (PCG) algorithm to compute the ground states. We truncate the whole space into a bounded rectangular domain and readily apply the Fourier spectral method to approximate the wave function. The PCG method is successfully adapted with appropriate modifications to the adaptive step size control strategy for the one-parameter energy minimization problem and to the choice of preconditioners, achieving great performance in terms of accuracy and efficiency. Lastly, we carry out extensive numerical experiments to verify the existence and property results of the ground states, confirm the spatial spectral accuracy by traversing the most commonly-used initial guesses for each component thanks to its great efficiency, which is also attributed to a utilization of cascadic multigrid and discrete Fast Fourier Transform (FFT). Moreover, we investigate the effects of local interaction, rotation and spin-orbit coupling and external trapping potential on the ground state, and unveil some interesting physical phenomena, such as giant vortex and U-shape vortex line.
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.