Computing the Bogoliubov-de Gennes excitations of two-component Bose-Einstein condensates

TL;DR

This paper introduces a highly efficient, spectrally accurate numerical method for computing excitations in two-component Bose-Einstein condensates by solving the Bogoliubov-de Gennes equation, utilizing Fourier spectral discretization and a structure-preserving iterative eigenvalue solver.

Contribution

The paper develops a matrix-free, Fourier spectral-based iterative method for large-scale non-Hermitian eigenproblems in BEC excitations, improving accuracy and efficiency over existing techniques.

Findings

Method achieves near-optimal complexity with FFT acceleration.

Spectral accuracy demonstrated in 1D, 2D, and 3D applications.

Outperforms traditional methods in accuracy and computational efficiency.

Abstract

In this paper, we present an efficient and spectrally accurate numerical method to compute elementary/collective excitations in two-component Bose-Einstein condensates (BEC), around their mean-field ground state, by solving the associated Bogoliubov-de Gennes (BdG) equation. The BdG equation is essentially an eigenvalue problem for a non-Hermitian differential operator with an eigenfunction normalization constraint. Firstly, we investigate its analytical properties, including the exact eigenpairs, generalized nullspace structure and bi-orthogonality of eigenspaces. Subsequently, by combining the Fourier spectral method for spatial discretization and a stable modified Gram-Schmidt bi-orthogonal algorithm, we propose a structure-preserving iterative method for the resulting large-scale dense non-Hermitian discrete eigenvalue problem. Our method is matrix-free, and the matrix-vector multiplication (or the operator-function evaluation) is implemented with a near-optimal complexity ${\mathcal O}(N_{\rm t}\log(N_{\rm t}))$, where $N_{\rm t}$ is the total number of grid points, thanks to the utilization of the discrete Fast Fourier Transform (FFT). Therefore, it is memory-friendly, spectrally accurate, and highly efficient. Finally, we carry out a comprehensive numerical investigation to showcase its superiority in terms of accuracy and efficiency, alongside some applications to compute the excitation spectrum and Bogoliubov amplitudes in one, two, and three-dimensional problems.