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Quantum Electronics and Laser Science Conference

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 Jun 25, 2003• 7 citations
Publisher

Abstract

Summary form only given. Phonons with wavevector q//spl planck/-were optically imprinted into a Bose-Einstein condensate. Their momentum distribution was analyze using Bragg spectroscopy with a high momentum transfer. The wavefunction of these elementary excitations was shown to be a superposition of +q and -q free particle momentum states, in agreement with the Bogoliubov quasiparticle picture.

DOI
10.1109/qels.2002.1031245

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Summary form only given. Phonons with wavevector q//spl planck/-were optically imprinted into a Bose-Einstein condensate. Their momentum distribution was analyze using Bragg spectroscopy with a high momentum transfer. The wavefunction of these elementary excitations 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 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.
We analyze the mean-field equilibrium states of a homogeneous Bose-Einstein condensate of boson particles in different internal states, possibly in the presence of an internal Josephson interaction, and we describe the quantum fluctuations by constructing an explicit form of the Bogoliubov transformation that characterizes the quasiparticle excitations of the system. The derivation of the Bogoliubov transformation is based on a simple geometrical picture which leads to a convenient parametrization of the transformation. Our calculations of physical properties, such as depletions and phonon velocities, as well as our discussion of stability, indicate the sense and the extent to which the internal Josephson interaction controls the many-body physics of multicomponent phase coherent matter.
Using two-photon Bragg spectroscopy, we study the energy of particlelike excitations in a strongly interacting homogeneous Bose-Einstein condensate, and observe dramatic deviations from Bogoliubov theory. In particular, at large scattering length a the shift of the excitation resonance from the free-particle energy changes sign from positive to negative. For an excitation with wave number q, this sign change occurs at a≈4/(πq), in agreement with the Feynman energy relation and the static structure factor expressed in terms of the two-body contact. For a≳3/q we also see a breakdown of this theory, and better agreement with calculations based on the Wilson operator product expansion. Neither theory explains our observations across all interaction regimes, inviting further theoretical efforts.