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arXiv:cond-mat/0010157 · Physical Review Letters

Experimental Observation of Beliaev Coupling in a Bose-Einstein Condensate

E. Hodby, Onofrio M. Maragò, Gerald Hechenblaikner, Christopher J. Foot

Published Mar 12, 2001• 79 citations
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Abstract

We report the first experimental observation of Beliaev coupling between collective excitations of a Bose-Einstein condensed gas. Beliaev coupling is not predicted by the Gross-Pitaevskii equation and so this experiment tests condensate theory beyond the mean field approximation. Measurements of the amplitude of a high frequency scissors mode show that the Beliaev process transfers energy to a lower-lying mode and then back and forth between these modes, unlike Landau processes which lead to a monotonic decrease in amplitude. To enhance the Beliaev process we adjusted the geometry of the magnetic trapping potential to give a frequency ratio of 2 to 1 between the two scissors modes.

DOI
10.1103/physrevlett.86.2196

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We report the first experimental observation of Beliaev coupling between collective excitations of a Bose-Einstein condensed gas. Beliaev coupling is not predicted by the Gross-Pitaevskii equation and so this experiment tests condensate theory beyond the mean field approximation. Measurements of the amplitude of a high frequency scissors mode show that the Beliaev process transfers energy to a lower-lying mode and then back and forth between these modes, unlike Landau processes which lead to a monotonic decrease in amplitude. To enhance the Beliaev process we adjusted the geometry of the magnetic trapping potential to give a frequency ratio of 2 to 1 between the two scissors modes.
Summary form only given. The microscopic theory of interacting Bose gases developed more than 50 years ago by Bogoliubov plays an essential role in the description of these systems. However, processes such as Beliaev damping are not predicted by the mean field approximation. Recently, the observation of Beliaev damping for a scissors mode of a BEC of rubidium atoms has been reported allowing the consideration of theory beyond the mean field. The Beliaev damping process is, in a sense, analogous to parametric down conversion of light, where one photon incident on a nonlinear crystal with second order susceptibility is split into two new photons of lower frequencies. Parametric down conversion has long been considered as an important source of squeezed states. In this paper we point out that similar processes can be observed in BEC.
Bose–Einstein condensation represents a quintessential manifestation of quantum mechanics at macroscopic scales, where a dilute gas of bosonic atoms undergoes a phase transition to form a coherent matter wave. This study presents a comprehensive theoretical investigation of collective excitations and critical phenomena in interacting Bose–Einstein condensates (BECs). Beginning with the Gross–Pitaevskii equation , we develop the Bogoliubov theory of excitations yielding the spectrum , where is the free-particle energy and characterizes contact interactions. The spectrum reveals phonon behavior at low momenta with sound velocity , transitioning to free-particle dispersion at momenta exceeding the inverse healing length . Collective modes in harmonically trapped condensates are analyzed in the Thomas–Fermi regime, yielding breathing mode frequency and quadrupole frequency . Damping mechanisms including Landau damping ( ) and Beliaev damping ( ) are characterized. The Lee–Huang–Yang correction extends the equation of state beyond mean field. Critical phenomena near are analyzed within the 3D XY universality class, with critical exponents , , and the interaction-induced shift with . All predictions show excellent agreement with experimental measurements on ultracold atomic gases.
The dynamics of a trapped Bose-condensed gas at finite temperatures is described by a generalized Gross-Pitaevskii equation for the condensate order parameter and a semiclassical kinetic equation for the thermal cloud, solved using N-body simulations. The two components are coupled by mean fields as well as collisional processes that transfer atoms between the two. We use this scheme to investigate scissors modes in anisotropic traps as a function of temperature. Frequency shifts and damping rates of the condensate mode are extracted, and are found to be in good agreement with recent experiments.
We study the energies and decay of elementary excitations in weakly interacting Bose-Einstein condensates within a finite-temperature gapless second-order theory. The energy shifts for the high-lying collective modes turn out to be systematically negative compared with the Hartree-Fock-Bogoliubov-Popov approximation and the decay of the low-lying modes is found to exhibit collapse and revival effects. In addition, perturbation theory is used to qualitatively explain the experimentally observed Beliaev decay process of the scissors mode.
We study nonlinear mixing effects among quadrupole modes and scissors modes in a harmonically trapped Bose-Einstein condensate. Using a perturbative technique in conjunction with a variational approach with a Gaussian trial wave function for the Gross-Pitaevskii equation, we find that mode mixing occurs selectively. Our perturbative approach is useful in gaining a qualitative understanding of the recent experiment [M. Yamazaki et al., J. Phys. Soc. Jpn. 84, 44001 (2015)], exhibiting a beating phenomenon of the scissors mode as well as a modulation phenomenon of the low-lying quadrupole mode by the high-lying quadrupole mode frequency. Within the second-order treatment of the nonlinear mode coupling terms, our approach predicts all the spectral peaks obtained by the numerical simulation of the Gross-Pitaevskii equation.
We report the observation of the scissors mode of a Bose-Einstein condensed gas of 87Rb atoms in a magnetic trap, which gives direct evidence of superfluidity in this system. The scissors mode of oscillation is excited by a sudden rotation of the anisotropic trapping potential. For a gas above T(c) (normal fluid) we detect the occurrence of oscillations at two frequencies, with the lower frequency corresponding to the rigid body value of the moment of inertia. Well below T(c) the condensate oscillates at a single frequency, without damping, as expected for a superfluid.