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Physical Review Letters

Large Amplitude Oscillations of a Bose Condensate

Augusto Smerzi, Stefano Fantoni

Published May 12, 1997• 53 citations
Publisher

Abstract

We analyze the response of condensed trapped atoms to external driving magnetic fields. Solving the time-dependent Gross-Pitaewskii equation within a new, accurate algorithm, we found frequencies and relaxation times of collective oscillations in excellent agreement with the experimental results. Using simple scaling arguments we make quantitative predictions for future experiments.

DOI
10.1103/physrevlett.78.3589

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We analyze the response of condensed trapped atoms to external driving magnetic fields. Solving the time-dependent Gross-Pitaewskii equation within a new, accurate algorithm, we found frequencies and relaxation times of collective oscillations in excellent agreement with the experimental results. Using simple scaling arguments we make quantitative predictions for future experiments.
We study the frequency response of large-amplitude oscillations of a trapped Bose condensed gas. On the basis of the Thomas?Fermi approximation, we deduce the hydrodynamical equation including the excitation source of the velocity drive, and obtain the analytical expression for the frequency as a function of velocity fluctuation amplitude, trap geometry, and symmetry of modes. By solving the expansion equation after switching off the trap, we find a simple relationship between the velocity amplitude and the oscillation amplitude. The theoretical calculations on the response frequencies agree well with the existing experimental observations.
Trapped Bose-condensed atomic gases are remarkable because, in spite of the fact that these are very dilute systems, they exhibit robust coherent dynamic behaviour when perturbed. These quantum “wisps of matter” are a new phase of highly coherent matter. While binary collisions are very infrequent, the large coherent mean field associated with the Bose condensate ensures that interactions play a crucial role in determining the collective response of these trapped superfluid gases. In our discussion of the theory of collective oscillations of atomic condensates, the macroscopic Bose wavefunction Φ( r , t ) plays a central role. This wavefunction is the BEC order parameter. As discussed in Chapter 1, the initial attempts at defining this order parameter began with the pioneering work of London (1938a), were further developed by Bogoliubov (1947) and finally extended to deal with any Bose superfluid using the general quantum field theoretic formalism developed by Beliaev (1958a). Almost all this early theoretical work was limited to T = 0 where, in a dilute weakly interacting Bose gas, all the atoms are in the condensate. The first extension of these ideas to nonuniform Bose condensates was by Pitaevskii (1961) and, independently, by Gross (1961), which led to the now famous Gross–Pitaevskii (GP) equation of motion for Φ( r , t ). Before the discovery of BEC in trapped gases, the time-dependent GP equation was mainly used to study vortices in Bose superfluids, which involve a spatially nonuniform ground state. Apart from this application, the GP equation was largely unknown. The situation changed overnight in 1995 with the creation of trapped nonuniform Bose condensates in atomic gases.
The linearized time-dependent coupled Gross-Pitaevskii equations describing the long-wavelength excitations of Bose-Einstein condensed binary mixtures are solved in harmonic traps for the case where only the binary phase is present. The dispersion law contains two branches whose dependence on the quantum numbers of the modes is the same as for a one-component condensate, but with different prefactors, depending on the ratios of the three sss-wave scattering lengths of the two atomic species. In the general case where the binary phase in the trap coexists with one or both one-component phases, the mode spectrum depends on the geometry of the interphase boundaries due to the boundary conditions there. Measurements of the oscillation frequencies as in recent experiments with modulated traps would yield very detailed information on this system.
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 linearised time dependent coupled Gross-Pitaevskii equations describing the long wavelength excitations of Bose-condensed binary mixtures are solved, in the bulk and in harmonic traps for the case where only the binary phase is present. In the former case we obtain two zero-sound branches.In the latter case the dispersion law also contains two branches whose dependence on the quantum numbers of the modes is the same as for a one-component condensate,but with different prefactors, depending on the ratios of the three s-wave scattering lengths of the two atomic species. In the general case where the binary phase in the trap coexists with one or both one-component phases, the mode spectrum depends on the geometry of the interphase boundaries due to the boundary conditions there. Measurements of the oscillation frequencies as in recent experiments with modulated traps would yield very detailed information on this system.