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Mathematical and numerical study on the ground states of rotating spin-orbit coupled spin-1 Bose-Einstein condensates

Jing Wang, Wei Yang, Yongjun Yuan, Yong Zhang

Abstract

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.

Mathematical and numerical study on the ground states of rotating spin-orbit coupled spin-1 Bose-Einstein condensates

Abstract

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.
Paper Structure (11 sections, 3 theorems, 83 equations, 8 figures, 2 tables, 3 algorithms)

This paper contains 11 sections, 3 theorems, 83 equations, 8 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Existence and properties of the ground states
  3. Existence of the ground states
  4. Properties of the ground states
  5. Numerical method
  6. Preconditioned nonlinear conjugate gradient method
  7. Spatial discretization by Fourier spectral method
  8. The multigrid PCG method
  9. Stopping criteria and Preconditioners
  10. Numerical results
  11. Conclusion

Key Result

Theorem 1

Suppose Then there exists a ground state of minimizer if $(1)~ d=2$ $(2)~ d=3$

Figures (8)

  • Figure 1: Residual error and the energy error against the number of iterations for PCG and PGF for 2D BEC in Example \ref{['perf-comp']}.
  • Figure 2: Contour plots of the densities for Case 1, 2, 3 (from top to bottom, $c_0$=100,200,600) in Example \ref{['Effect-local']} to investigate the effect of spin-independent interaction.
  • Figure 3: Contour plots of the densities for Case 4, 5, 6 (from top to bottom, $c_1$=-20,10,30) in Example \ref{['Effect-local']} to investigate the effect of spin-exchange interaction.
  • Figure 4: Contour plots of the densities for Case 1 (from top to bottom, $\Omega = 0, 0.45, 0.90$) in Example \ref{['Effect-Rot']} in harmonic potential to investigate the effect of rotation.
  • Figure 5: Contour plots of the densities for Case 2 (from top to bottom, $\Omega = 3.0, 6.0, 14.0$) in Example \ref{['Effect-Rot']} in harmonic-plus-quartic potential to investigate the effect of rotation.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Theorem 2: Virial identity
  • proof
  • Theorem 3: Negativity of the SOC energy
  • proof
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • ...and 3 more