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arXiv:cond-mat/0109205 · Physical Review Letters

Experimental Observation of the Bogoliubov Transformation for a Bose-Einstein Condensed Gas

J. M. M. J. Vogels, K. Xu, Chandra Raman, J. R. Abo-Shaeer, Wolfgang Ketterle

Published Jan 28, 2002• 111 citations
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Abstract

Phonons with wave vector q/(planck constant) were optically imprinted into a Bose-Einstein condensate. Their momentum distribution was analyzed using Bragg spectroscopy with a high momentum transfer. The wave function of the phonons was shown to be a superposition of +q and -q free particle momentum states, in agreement with the Bogoliubov quasiparticle picture.

DOI
10.1103/physrevlett.88.060402

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Phonons with wave vector q/(planck constant) were optically imprinted into a Bose-Einstein condensate. Their momentum distribution was analyzed using Bragg spectroscopy with a high momentum transfer. The wave function of the phonons was shown to be a superposition of +q and -q free particle momentum states, in agreement with the Bogoliubov quasiparticle picture.
Phonon like excitations can be imprinted into a trapped Bose-Einstein condensate of cold atoms using light scattering. If the condensate is suddenly let to freely expand, the initial phonons lose their collective character by transferring their energy and momentum to the motion of individual atoms. The basic mechanisms of this evaporation process are investigated by using the Gross-Pitaevskii theory and dynamically rescaled Bogoliubov equations. Different regimes of evaporation are shown to occur depending on the phonon wavelength. Distinctive signatures of the evaporated phonons are visible in the density distribution of the expanded gas, thus providing a new type of spectroscopy of Bogoliubov excitations.
The behaviour of the momentum transferred to a trapped Bose-Einstein condensate by a two-photon Bragg pulse reflects the structure of the underlying Bogoliubov spectrum. In elongated condensates, axial phonons with different numbers of radial nodes give rise to a multibranch spectrum which can be resolved in Bragg spectroscopy, as shown by Steinhauer et al (2003 Phys. Rev. Lett. 90 060404). Here we present a detailed theoretical analysis of this process. We calculate the momentum transferred by numerically solving the time-dependent Gross-Pitaevskii (GP) equation. In the case of a cylindrical condensate, we compare the results with those obtained by linearizing the GP equation and using a quasiparticle projection method. This analysis shows how the axial-phonon branches affect the momentum transfer, in agreement with our previous interpretation of the observed data. We also discuss the applicability of this type of spectroscopy to typical available condensates, as well as the role of nonlinear effects.
We measure the response of an elongated Bose-Einstein condensate to a two-photon Bragg pulse. If the duration of the pulse is long, the total momentum transferred to the condensate exhibits a nontrivial behavior which reflects the structure of the underlying Bogoliubov spectrum. It is thus possible to perform a spectroscopic analysis in which axial phonons with a different number of radial nodes are resolved. The local density approximation is shown to fail in this regime, while the observed data agree well with the results of simulations based on the numerical solution of the Gross-Pitaevskii equation.
We propose an experiment, based on two consecutive Bragg pulses, to measure the momentum distribution of quasiparticle excitations in a trapped Bose gas at low temperature. With the first pulse one generates a bunch of excitations carrying momentum q, whose Doppler line is measured by the second pulse. We show that this experiment can provide direct access to the amplitudes u(q) and v(q) characterizing the Bogoliubov transformations from particles to quasiparticles. We simulate the behavior of the nonuniform gas by numerically solving the time dependent Gross-Pitaevskii equation.