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Journal of Computational Physics

Numerical methods for Bogoliubov-de Gennes excitations of Bose-Einstein condensates

Yali Gao, Yongyong Cai

Published Oct 25, 2019• 18 citations
Publisher
DOI
10.1016/j.jcp.2019.109058

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arXiv:0810.3088•Physical Review A•Apr 15, 2009•13 citations
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.
We construct quantum field theory in an analog curved spacetime in Bose-Einstein condensates based on the Bogoliubov--de Gennes equations by exactly relating quantum particles in curved spacetime with Bogoliubov quasiparticle excitations in Bose-Einstein condensates. Here, we derive a simple formula relating the two, which can be used to calculate the particle creation spectrum by solving the time-dependent Bogoliubov--de Gennes equations. Using our formulation, we numerically investigate particle creation in an analog expanding universe which can be expressed as Bogoliubov quasiparticles in an expanding Bose-Einstein condensate. We obtain its spectrum, which follows the thermal Maxwell-Boltzmann distribution, the temperature of which is experimentally attainable. Our derivation of the analogy is useful for general Bose-Einstein condensates and not limited to homogeneous ones, and our simulation is an example of particle creations by solving the Bogoliubov--de Gennes equation in an inhomogeneous condensate.