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arXiv:physics/9705006 · Physical Review A

Low-energy elementary excitations of a trapped Bose-condensed gas

Patrik Öhberg, E.L. Surkov, Ilkka Juhani Tittonen, Stig Stenholm, Martin Wilkens, Gora V. Shlyapnikov

Published Nov 1, 1997• 116 citations
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Abstract

We develop a method of finding analytical solutions of the Bogolyubov--de Gennes equations for the excitations of a Bose condensate in the Thomas-Fermi regime in harmonic traps of any asymmetry and introduce a classification of eigenstates. In the case of cylindrical symmetry we emphasize the presence of an accidental degeneracy in the excitation spectrum at certain values of the projection of orbital angular momentum on the symmetry axis and discuss possible consequences of the degeneracy in the context of new signatures of Bose-Einstein condensation.

DOI
10.1103/physreva.56.r3346

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We develop a method of finding analytical solutions of the Bogolyubov--de Gennes equations for the excitations of a Bose condensate in the Thomas-Fermi regime in harmonic traps of any asymmetry and introduce a classification of eigenstates. In the case of cylindrical symmetry we emphasize the presence of an accidental degeneracy in the excitation spectrum at certain values of the projection of orbital angular momentum on the symmetry axis and discuss possible consequences of the degeneracy in the context of new signatures of Bose-Einstein condensation.
We propose a method for finding analytical solutions of the Bogoliubov--de Gennes (BdG) equations for the low-lying collective excitations in a harmonically trapped Bose-Einstein condensate beyond the Thomas-Fermi limit. We first use a simple variational wave function for ground state to eliminate the divergence at the boundary layer of the condensate, which appears in the Thomas-Fermi approximation. We then solve the BdG equations analytically and obtain explicit and divergence-free expressions for the eigenvalues and eigenfunctions of the excitations for the traps with spherical and cylindrical symmetries. The solutions of the zero-energy mode of the BdG equations are also presented.
Bogoliubov–de Gennes equations (BdGEs) for collective excitations from a trapped Bose–Einstein condensate described by a spatially smooth ground-state wavefunction can be treated analytically. A new class of closed solutions for the BdGEs is obtained for the one-dimensional (1D) and 3D spherically harmonic traps. The solutions of zero-energy mode of the BdGEs are also provided. The eigenfunctions of the excitations consist of zero-energy mode, zero-quantum-number mode and entire excitation modes when the approximate ground state is a background Bose gas sea.
We consider noncentered vortices and their arrays in a cylindrically trapped Bose-Einstein condensate at zero temperature. We study the kinetic energy and the angular momentum per particle in the Thomas-Fermi regime and their dependence on the distance of the vortices from the center of the trap. Using a perturbative approach with respect to the velocity field of the vortices, we calculate, to first order, the frequency shift of the collective low-lying excitations due to the presence of an off-center vortex or a vortex array, and compare these results with predictions that would be obtained by the application of a simple sum-rule approach, previously found to be very successful for centered vortices. It turns out that the simple sum-rule approach fails for off-centered vortices.
We present a systematic classification of the elementary excitations of Bose-Einstein condensates in cylindrical traps in terms of their shapes. The classification generalizes the concept of families of excitations first identified by Hutchinson and Zaremba (1998 Phys. Rev. A 57 1280) by introducing a second classification number that allows all possible modes to be assigned to a family. We relate the energy ordering of the modes to their family classification, and provide a simple model which explains the relationship.
We investigate the low energy excitations of a dilute atomic Bose gas confined in a harmonic trap of frequency ${\ensuremath{\omega}}_{0}$ and interacting with repulsive forces. The dispersion law $\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\omega}}_{0}({2n}^{2}+2n\ensuremath{\ell}+3n+\ensuremath{\ell}{)}^{1/2}$ for the elementary excitations is obtained for large numbers of atoms in the trap, to be compared with the prediction $\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\omega}}_{0}(2n+\ensuremath{\ell})$ of the noninteracting harmonic oscillator model. Here nnn is the number of radial nodes and $\ensuremath{\ell}$ is the orbital angular momentum. The effects of the kinetic energy pressure are estimated using a sum rule approach. Results are also presented for deformed traps and attractive forces.