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Journal of Physics B Atomic Molecular and Optical Physics

Monopole and quadrupole shape oscillations of a Bose-Einstein condensate

M. O. C. Pires, E. J. V. de Passos

Published Sep 14, 2000• 9 citations
Publisher

Abstract

We use a variational ansatz to derive a closed dynamics involving only the monopole and quadrupole shape degrees of freedom of the condensate, which is a generalization of the work of Perez-Garcia and collaborators. Taking the limit of small oscillations around the equilibrium configuration, we calculate the excitation energies of the low-lying even-parity states of the condensate. In this limit, the diagonal and non-diagonal quadrupole degrees of freedom are decoupled. For the diagonal elements, our results agree exactly with those of Perez-Garcia and collaborators. For the non-diagonal elements we obtain analytical expressions for the excitation energies which are valid for any trap geometry and any particle number and that agree, in the Thomas-Fermi limit and for axially symmetric traps, with Stringari's results. For negative scattering lengths we show that the collapsing state is the breathing mode, independent of the trap geometry.

DOI
10.1088/0953-4075/33/19/308

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We use a variational ansatz to derive a closed dynamics involving only the monopole and quadrupole shape degrees of freedom of the condensate, which is a generalization of the work of Perez-Garcia and collaborators. Taking the limit of small oscillations around the equilibrium configuration, we calculate the excitation energies of the low-lying even-parity states of the condensate. In this limit, the diagonal and non-diagonal quadrupole degrees of freedom are decoupled. For the diagonal elements, our results agree exactly with those of Perez-Garcia and collaborators. For the non-diagonal elements we obtain analytical expressions for the excitation energies which are valid for any trap geometry and any particle number and that agree, in the Thomas-Fermi limit and for axially symmetric traps, with Stringari's results. For negative scattering lengths we show that the collapsing state is the breathing mode, independent of the trap geometry.
We solve the time-dependent Gross-Pitaevskii equation by a variational ansatz to calculate the excitation spectrum of a Bose-Einstein condensate in a trap. The trial wave function is a Gaussian which allows an essentially analytical treatment of the problem. Our results reproduce numerical calculations over the whole range from small to large particle numbers, and agree exactly with the Stringari results in the strong interaction limit. Excellent agreement is obtained with the recent JILA experiment and predictions for the negative scattering length case are also made.
The oscillation frequencies of collective excitations of a trapped Bose-Einstein condensate, when calculated in the mean-field approximation and in the Thomas-Fermi limit, are independent of the scattering length aaa. We calculate the leading corrections to the frequencies from quantum fluctuations around the mean field. The semiclassical correction is proportional to N1/5a6/5{N}^{1/5}{a}^{6/5}N1/5a6/5, where NNN is the number of atoms in the condensate. The correction is positive semidefinite and is zero for surface modes whose eigenfunctions have a vanishing Laplacian. The shift in the frequency of the lowest quadrupole mode for an axially symmetric trap is large enough that it should be measurable in future experiments.
We present exact results in the Thomas-Fermi regime for the statics and dynamics of a harmonically trapped Bose-Einstein condensate that has dipole-dipole interactions in addition to the usual s-wave contact interactions. Remarkably, despite the nonlocal and anisotropic nature of the dipolar interactions, the density profile in a general time-dependent harmonic trap is an inverted parabola. The evolution of the condensate radii is governed by local, ordinary differential equations, and as an example we calculate the monopole and quadrupole shape oscillation frequencies.
The monopole and quadrupole oscillations of a Bose-Einstein condensate (BEC) gas trapped in a spherically symmetric potential are calculated using the time-dependent Hartree approximation which enables us to use an orthonormal single-particle basis for the description of both the condensate and noncondensate components. It is found that the monopole mode has a much smaller damping rate than the quadrupole mode. The obtained temperature dependence of the frequency shifts and the damping rates of the collective oscillations are found to be consistent with experimental data obtained for BEC gases in anisotropic trapping potentials. A comparison between the time-dependent Hartree approximation and the time-dependent Hartree-Fock approximation is also made and it is found that there is a rather small difference.
Collective modes of the ground and excited states of Bose?Einstein condensates are calculated by diagonalization of the Bogoliubov?de Gennes matrix in cylindrical coordinates. For an isotropic magnetic trap, the first breathing mode frequencies of the ground and dark soliton excited states are found to be in excellent agreement with a direct solution of the Gross?Pitaevskii equation with a time-dependent driving potential. In addition, the matrix diagonalization yields densities and frequencies for second breathing modes and higher order multipole modes. For an axially symmetric anisotropic trap, a considerable number of collective modes of the ground and dark soliton excited states are calculated. Prolate traps are found to have many more low-frequency multipolar modes than found in oblate traps for both ground and dark soliton excited states.