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arXiv:0705.4164 · Physical Review A

Formation of a condensed state with a macroscopic number of phonons in ultracold Bose gases

Yu. Kagan, L. A. Manakova

Published Aug 7, 2007• 13 citations
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Abstract

A mechanism for the formation of a different type of stationary state with macroscopical number of phonons in condensed atomic gases is proposed. This mechanism is based on generating longitudinal phonons as a result of parametric resonance caused by a permanent modulation of the transverse trap frequency in an elongated trap. The phonon-phonon interaction predetermines the self-consistent evolution which is completed with macroscopic population of one from all levels within the energy interval of parametric amplification. This level proves to be shifted to the edge of this interval. All other levels end the evolution with zero population.

DOI
10.1103/physreva.76.023601

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A mechanism for the formation of a different type of stationary state with macroscopical number of phonons in condensed atomic gases is proposed. This mechanism is based on generating longitudinal phonons as a result of parametric resonance caused by a permanent modulation of the transverse trap frequency in an elongated trap. The phonon-phonon interaction predetermines the self-consistent evolution which is completed with macroscopic population of one from all levels within the energy interval of parametric amplification. This level proves to be shifted to the edge of this interval. All other levels end the evolution with zero population.
We present a numerical study of the dynamical effects following a sudden change of the transverse trapping frequency in an elongated Bose-Einstein condensate, which induces periodic oscillations of the radial density. At early times, we observe an exponential growth of the number of resonant longitudinal phonons, in agreement with the predictions of the Bogoliubov--de Gennes treatment. We then observe an ordered sequence of phenomena induced by the nonlinearities of the system. The first is a loss of the nonseparability of the resonant phonon pairs. This is followed by the saturation of the exponential growth and a strong depletion of condensed atoms. Notably, these effects are well described by effective one-dimensional dynamics, and are hardly affected by the damping of the radial oscillations. Finally, the atomic spectrum becomes broad, featureless and almost incoherent, in agreement with experimental results. The link between this sequence of events and the preheating scenario in inflationary cosmology is striking, as is the similarity of techniques used to study them.
We analyze the periodic tunneling of a Bose-Einstein condensate in a double-well potential which has an oscillating energy barrier. We show that the dynamics of the Bose condensate critically depends on the frequency ωωω of the oscillating energy barrier. In the regime of periodic macroscopic quantum tunneling (PMQT) with frequency ωJω_JωJ​, the population imbalance of the condensate in the two wells can be enhanced under the condition of parametric resonance ω=2ωJω= 2 ω_Jω=2ωJ​. Instead, in the regime of macroscopic quantum self-trapping (MQST), we find that MQST can be reduced or suppressed under the condition of parametric resonance between the frequency ωωω of the energy barrier and the frequency ωSTω_{ST}ωST​ of oscillation through the barrier of the very small fraction of particles which remain untrapped during MQST.
We present a numerical study of the dynamical effects following a sudden change of the transverse trapping frequency in an elongated Bose-Einstein condensate, which induces periodic oscillations of the radial density. At early times, we observe an exponential growth of the number of resonant longitudinal phonons, in agreement with the predictions of the Bogoliubov-de Gennes treatment. We then observe an ordered sequence of phenomena induced by the nonlinearities of the Gross-Pitaevskii equation. The first is a loss of the nonseparability of the resonant phonon pairs. This is followed by the saturation of the exponential growth and a strong depletion of condensed atoms. Notably, these effects are well-described by effective 1D dynamics, and are hardly affected by the damping of the radial oscillations. Finally, the atomic spectrum becomes broad, featureless and almost incoherent, in agreement with experimental results obtained in [J.-C. Jaskula et al., Phys. Rev. Lett. 109, 220401 (2012)]. The link between this sequence of events and the preheating scenario in inflationary cosmology is striking.
Trapped atoms cooled down to temperatures below the Bose-Einstein condensation temperature are considered. Stationary solutions to the Gross-Pitaevskii equation (GPE) define the topological coherent modes, representing nonground-state Bose-Einstein condensates. These modes can be generated by means of alternating fields whose frequencies are in resonance with the transition frequencies between two collective energy levels corresponding to two different topological modes. The theory of resonant generation of these modes is generalized in several aspects: Multiple-mode formation is described; a shape-conservation criterion is derived, imposing restrictions on the admissible spatial dependence of resonant fields; evolution equations for the case of three coherent modes are investigated; the complete stability analysis is accomplished; the effects of harmonic generation and parametric conversion for the topological coherent modes are predicted. All considerations are realized both by employing approximate analytical methods as well as by numerically solving the GPE. Numerical solutions confirm all conclusions following from analytical methods.