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An efficient compact splitting Fourier spectral methods for computing the dynamics of rotating spin-orbit coupled spin-2 Bose-Einstein condenstates

Xin Liu, Ziqing Xie, Yongjun Yuan, Yong Zhang, Xinyi Zhao

Abstract

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.

An efficient compact splitting Fourier spectral methods for computing the dynamics of rotating spin-orbit coupled spin-2 Bose-Einstein condenstates

Abstract

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.

Paper Structure

This paper contains 14 sections, 4 theorems, 66 equations, 8 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Dynamical properties
  3. Numerical methods
  4. Exact integrator for Laplace-Rotation-SOC subproblem
  5. Exact evaluation for nonlinear subproblem
  6. High-order compact splitting Fourier spectral method
  7. Stability
  8. Numerical results
  9. Accuracy confirmation.
  10. Efficiency test
  11. Dynamical properties verification.
  12. SOC effects.
  13. Dynamics of a vortex lattice.
  14. Conclusion

Key Result

Lemma 1

For CGPEs CGPEs with harmonic potential harmonic, the dynamics of angular momentum expectation is governed by the following ordinary differential equation (ODE) Thus the angular momentum expectation is conserved, i.e., when $\gamma_x = \gamma_y$ and $\gamma = 0$.

Figures (8)

  • Figure 1: Log-Log plots of timing results with the total grid $N_{tot}$ by TS2 and TS4 for both 2D (left) and 3D (right) cases in Example \ref{['efficiency']}.
  • Figure 2: Evolution of mass $\mathcal{N}(t)$ and energy $\mathcal{E}(t)$ (left) and magnetization $\mathcal{M}(t)$ (right) for Case i-Case iii in Example \ref{['properties1']}.
  • Figure 3: Evolution of $\frac{\mathcal{E}(t)}{\mathcal{E}(0)}-1$ (left) and $\mathcal{E}_{n+1}-\mathcal{E}_{n}$ (right) for Case i-Case iii in Example \ref{['properties1']}.
  • Figure 4: Evolution of angular momentum expectation $\langle L_z \rangle (t)$ (a) and condensate widths (b)-(d) for Case i-Case iii in Example \ref{['properties1']}.
  • Figure 5: Contour plots of the densities with $\gamma = 0, 0.7, 2$ (top to bottom) in Case i of Example \ref{['soc1']}.
  • ...and 3 more figures

Theorems & Definitions (18)

  • Lemma 1
  • proof 1
  • Lemma 2
  • proof 2
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3: Special cases
  • Remark 3.4: Extension to three-dimensional problem
  • Remark 3.5: Efficiency
  • Lemma 3: Stability
  • ...and 8 more