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arXiv:2603.05194

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

Qinglin Tang, Hanquan Wang, Shaobo Zhang, Yong Zhang

Published Mar 5, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2603.05194

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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.
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The Bose-Einstein condensate is a phase of matter that arises when cooling gases of bosons to extremely low temperatures. When studying these condensates one may use the Gross-Pitaevskii equation, which is a non-linear variant of the Schrödinger equation. An interesting phenomenon that arises when rotating a Bose-Einstein condensate is the appearance of vortices. We implement a semi-implicit Euler scheme using spectral methods proposed in [1] to numerically calculate the ground state of a rotating Bose-Einstein condensate. We start with implementing a simpler iterative fixed-point method to solve the Euler scheme but show that this method fails to converge for large rotations. Because of this we implement multiple Krylov subspace solvers that in fact do converge for large rotations and show that the Preconditioned Conjugate Gradient method has better performance than the BiConjugate Gradient Stabilized method. After the implementation we briefly look at the performance of the method and improve it with simple tricks that do not compromise the accuracy or robustness and which reduce the computation time slightly. Lastly we look at the formation of vortices in 2-dimensional and 3-dimensional Bose-Einstein condensates. We show that the number of vortices increases exponentially for increasing angular velocity in 2D until the condensate breaks apart, but in 3D we ultimately find that the required computation time and RAM storage is too large to be able to analyze the vortices in a similar way on our personal computers.
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.