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Physical Review Letters

Low Energy Excitations of a Bose-Einstein Condensate: A Time-Dependent Variational Analysis

Víctor M. Pérez‐García, Humberto Michinel, J. I. Cirac, Maciej Lewenstein, P. Zoller

Published Dec 30, 1996• 422 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.77.5320

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We study the relation between the eigenfrequencies of the Bogoliubov excitations of Bose-Einstein condensates and the eigenvalues of the Jacobian stability matrix in a variational approach that maps the Gross-Pitaevskii equation to a system of equations of motion for the variational parameters. We do this for Bose-Einstein condensates with attractive contact interaction in an external trap and for a simple model of a self-trapped Bose-Einstein condensate with attractive 1/r1/r1/r interaction. The stationary solutions of the Gross-Pitaevskii equation and Bogoliubov excitations are calculated using a finite-difference scheme. The Bogoliubov spectra of the ground and excited state of the self-trapped monopolar condensate exhibit a Rydberg-like structure, which can be explained by means of a quantum-defect theory. On the variational side, we treat the problem using an ansatz of time-dependent coupled Gaussian functions combined with spherical harmonics. We first apply this ansatz to a condensate in an external trap without long-range interaction and calculate the excitation spectrum with the help of the time-dependent variational principle. Comparing with the full-numerical results, we find good agreement for the eigenfrequencies of the lowest excitation modes with arbitrary angular momenta. The variational method is then applied to calculate the excitations of the self-trapped monopolar condensates and the eigenfrequencies of the excitation modes are compared.
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The times of flight of an atomic Bose-Einstein condensate are theoretically investigated in the experimentally unexplored regime corresponding to detection close to the trap of the condensate. In this regime, there is no consensus on how to calculate the distribution of times of arrival onto the detector. For non-interacting particles, distinct theoretical predictions have been made in the past. This work analyses how these predictions are modified for an interacting Bose-Einstein condensate. For this purpose, a time-dependent Gross-Pitaevskii equation is solved analytically and numerically.