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Multiscale Modeling and Simulation

An Efficient Fourier Spectral Eigensolver for Computing the Bogoliubov-de Gennes Excitations of Spin-1 Bose-Einstein Condensates

Yu Li, Z L Li, Manting Xie, Yong Zhang

Published Jul 8, 2026• 0 citations
Publisher

Abstract

Abstract. In this paper, we propose a spectrally accurate solver for computing the Bogoliubov-de Gennes (BdG) excitations of spin-1 Bose-Einstein condensates (BECs), which are governed by the BdG equations, around the mean-field ground state. The BdG equations are essentially a constrained eigenvalue/eigenfunction system. First, we investigate their analytical properties, including exact eigenpairs, generalized nullspace, and bi-orthogonality of eigenspaces. Second, by combining the standard Fourier spectral method for spatial discretization and a stable Gram-Schmidt bi-orthogonal algorithm, we develop a subspace iterative solver for such a large-scale dense eigenvalue problem, and it proves to be numerically stable, efficient, and accurate. Our solver is matrix-free, and the operator-function evaluation is accelerated by a discrete fast Fourier transform (FFT) with almost optimal efficiency. Therefore, it is memory-friendly and efficient for large-scale problems. Furthermore, we give a rigorous and detailed numerical analysis of the stability and spectral convergence. Finally, we present extensive numerical results to illustrate the spectral accuracy and efficiency, and we investigate the excitation spectrum and Bogoliubov amplitudes around the ground state in 1–3 spatial dimensions.

DOI
10.1137/25m1775853

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Abstract. In this paper, we propose a spectrally accurate solver for computing the Bogoliubov-de Gennes (BdG) excitations of spin-1 Bose-Einstein condensates (BECs), which are governed by the BdG equations, around the mean-field ground state. The BdG equations are essentially a constrained eigenvalue/eigenfunction system. First, we investigate their analytical properties, including exact eigenpairs, generalized nullspace, and bi-orthogonality of eigenspaces. Second, by combining the standard Fourier spectral method for spatial discretization and a stable Gram-Schmidt bi-orthogonal algorithm, we develop a subspace iterative solver for such a large-scale dense eigenvalue problem, and it proves to be numerically stable, efficient, and accurate. Our solver is matrix-free, and the operator-function evaluation is accelerated by a discrete fast Fourier transform (FFT) with almost optimal efficiency. Therefore, it is memory-friendly and efficient for large-scale problems. Furthermore, we give a rigorous and detailed numerical analysis of the stability and spectral convergence. Finally, we present extensive numerical results to illustrate the spectral accuracy and efficiency, and we investigate the excitation spectrum and Bogoliubov amplitudes around the ground state in 1–3 spatial dimensions.
In this paper, we propose an efficient Fourier spectral algorithm for an eigenvalue problem, that is, the Bogoliubov-de Gennes (BdG) equation arsing from spin-1 Bose-Einstein condensates (BEC) to describe the elementary/collective excitations around the mean-field ground state. The BdG equation is essentially a constrained eigenvalue/eigenfunction system. Firstly, we investigate its analytical properties, including exact eigenpairs, generalized nullspace, and bi-orthogonality of eigenspaces. Secondly, by combining the standard Fourier spectral method for spatial discretization and a stable Gram-Schmidt bi-orthogonal algorithm, we develop a subspace iterative solver for such a large-scale dense eigenvalue problem, and it proves to be numerically stable, efficient, and accurate. Our solver is matrix-free and the operator-function evaluation is accelerated by discrete Fast Fourier Transform (FFT) with almost optimal efficiency. Therefore, it is memory-friendly and efficient for large-scale problems. Furthermore, we give a rigorous and detailed numerical analysis on the stability and spectral convergence. Finally, we present extensive numerical results to illustrate the spectral accuracy and efficiency, and investigate the excitation spectrum and Bogoliubov amplitudes around the ground state in 1-3 spatial dimensions.
In this paper, we propose an efficient and robust numerical method to study the elementary excitation of dipolar Bose--Einstein condensates, which is governed by the Bogoliubov--de Gennes equations (BdGEs) with nonlocal dipole-dipole interaction, around the mean field ground state. Analytical properties of the BdGEs are investigated, which could serve as benchmarks for the numerical methods. To evaluate the nonlocal interactions accurately and efficiently, we propose a new simple Fourier spectral convolution (SFSC) method. Then, integrating SFSC with the standard Fourier spectral method for spatial discretization and implicitly restarted Arnoldi methods (IRAM) for the eigenvalue problem, we derive an efficient and spectrally accurate method, named the SFSC-IRAM, for the BdGEs. Ample numerical tests are provided to illustrate the accuracy and efficiency. Finally, we apply the new method to study systematically the excitation spectrum and Bogoliubov amplitudes around the ground state with different parameters in different spatial dimensions.
An efficient spectrally accurate multigrid method for the Bogoliubov-de Gennes excitations of the quasi-2D dipolar Bose-Einstein condensates is proposed. The wave functions/eigenmodes are spatially discretised by the Fourier spectral method. The convolution-type nonlocal potentials are computed in O(Nlog(N))\mathscr{O}(N log(N))O(Nlog(N)) operations with a spectral accuracy by the kernel truncation method. In addition, the influence of the model parameters on the eigenvalue distribution is studied and for various dipole orientations and an anisotropic external potential the phase diagrams of the eigenmodes are presented. Examples verify the spectral accuracy of the method.
We propose efficient and accurate numerical methods for computing the ground state and dynamics of the dipolar Bose-Einstein condensates utilising a newly developed dipole-dipole interaction (DDI) solver that is implemented with the non-uniform fast Fourier transform (NUFFT) algorithm. We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a DDI term and present the corresponding two-dimensional (2D) model under a strongly anisotropic confining potential. Different from existing methods, the NUFFT based DDI solver removes the singularity by adopting the spherical/polar coordinates in the Fourier space in 3D/2D, respectively, thus it can achieve spectral accuracy in space and simultaneously maintain high efficiency by making full use of FFT and NUFFT whenever it is necessary and/or needed. Then, we incorporate this solver into existing successful methods for computing the ground state and dynamics of GPE with a DDI for dipolar BEC. Extensive numerical comparisons with existing methods are carried out for computing the DDI, ground states and dynamics of the dipolar BEC. Numerical results show that our new methods outperform existing methods in terms of both accuracy and efficiency.
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.
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.