An efficient and accurate numerical method for computing the ground states of three-dimensional rotating dipolar Bose-Einstein condensates under strongly anisotropic trap

Abstract

In this article, we propose an efficient and spectrally accurate numerical method to compute the ground states of three-dimensional (3D) rotating dipolar Bose-Einstein condensates (BEC) under strongly anisotropic trapping potentials.The kernel singularity, convolution non-locality and density anisotropy together complicate the dipolar potential evaluation. The fast rotation mechanism not only induces a complicated energy landscape with many local minima, but also creates a large number of vortices in the condensates. Such factors collectively make the ground state computation challenging in terms of convergence, accuracy and efficiency, especially for 3D anisotropic systems. Coupled with Fourier spectral discretization, we proposed a preconditioned conjugate gradient method (PCG) by integrating the anisotropic truncated kernel method (ATKM) for the dipolar potential evaluation. An adaptive step size control strategy is designed and ATKM allows for a spectral accuracy without introducing any extra anisotropy-dependent memory requirement or computational time. Our algorithm is spectrally accurate, highly efficient and memory-economic. Extensive numerical results are presented to confirm the accuracy and efficiency, together with applications to study impacts of the model parameters on critical rotational frequency, energies and chemical potential. Furthermore, these simulations reveal additional novel ground state patterns, such as bent vortices.

Figures (11)

• Figure 1: Schematic diagrams of cigar- and pancake-shape domain $\mathcal{D}_{L\bm{\xi}}$ with $\bm{\xi}=(\frac{1}{10},\frac{1}{10},1)^T$ (left) and $\bm{\xi}=(1,1,\frac{1}{10})^T$ (right).
• Figure 2: Contour plots of $|\phi_g(x, y, z=0)|^2$ and $|\phi_g^{\mathbf{h}}(x, y, z=0)|^2$ with $N=256^3$ for $\Omega=0.85$ of Case I in Example \ref{['TimeAccuracy']}.
• Figure 3: Computational time for Case I (left) and Case II (right) in Example \ref{['TimeAccuracy']}.
• Figure 4: Isosurface plots of $|\phi_g(\mathbf{x})|^2=3\times10^{-4}$ with different $N$ in Example \ref{['Shape_grids']}.
• Figure 5: Isosurfaces of $|\phi_g(\mathbf{x})|^2=3\times10^{-4}$ for $\Omega=0.41$ (top row) and $\Omega=0.49$ (bottom row) in Case I-IV (from left to right) of Example \ref{['DDInoDDI']}.

Theorems & Definitions (13)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Example 3.1
• Example 3.2
• Example 3.3
• Example 3.4
• Example 3.5
• Example 3.6