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arXiv:cond-mat/0003402 · Physical Review Letters

Collective Oscillations of Two Colliding Bose-Einstein Condensates

P. Maddaloni, Michele Modugno, C. Fort, F. Minardi, M. Inguscio

Published Sep 18, 2000• 152 citations
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Abstract

Two ${}^{87}\mathrm{Rb}$ condensates ( $F\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$, ${m}_{f}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$, and ${m}_{f}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$) are produced in highly displaced harmonic traps and the collective dynamical behavior is investigated. The mutual interaction between the two condensates is evidenced in the center-of-mass oscillations as a frequency shift of $6.4(3)%$. Calculations based on a mean-field theory well describe the observed effects of periodical collisions both on the center-of-mass motion and on the shape oscillations.

DOI
10.1103/physrevlett.85.2413

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We have investigated the center-of-mass oscillations of a Rb87 Bose-Einstein condensate in an elongated magneto-static trap. We start from a trapped condensate and we transfer part of the atoms to another trapped level, by applying a radio-frequency pulse. The new condensate is produced far from its equilibrium position in the magnetic potential, and periodically collides with the parent condensate. We discuss how both the damping and the frequency shift of the oscillations are affected by the mutual interaction between the two condensates, in a wide range of trapping frequencies. The experimental data are compared with the prediction of a mean-field model.
We study the dynamics of two interacting Bose-Einstein condensates, by numerically solving two coupled Gross-Pitaevskii equations at zero temperature. We consider the case of a sudden transfer of atoms between two trapped states with different magnetic moments: the two condensates are initially created with the same density profile, but are trapped into different magnetic potentials, whose minima are vertically displaced by a distance much larger than the initial size of both condensates. Then the two condensates begin to perform collective oscillations, undergoing a complex evolution, characterized by collisions between the two condensates. We investigate the effects of their mutual interaction on the center-of-mass oscillations and on the time evolution of the aspect ratios. Our theoretical analysis provides a useful insight into the recent experimental observations by Maddaloni et al. [Phys. Rev. Lett. 85, 2413 (2000)].
We experimentally investigate the collective excitation of 87 Rb Bose–Einstein condensates confined in a cigar-shaped magnetic trap (QUIC trap). Using a method of magnetic perturbation, the center-of-mass oscillation of the condensate is excited, so that the radial trapping frequency of the QUIC trap can be precisely determined. A high-order excitation, characterized by a fast shape oscillation, also occurs simultaneously, with a noticeable damping in the oscillation amplitude compared with the oscillation of the center of mass. The measured oscillation frequencies, associated with these two low-energy excitation modes, agree well with theoretical predictions based on the Gross–Pitaevskii equation.
We calculate the effect of the interaction between an optically active material and a Bose-Einstein condensate on the collective oscillations of the condensate. We provide explicit expressions for the frequency shift of the center-of-mass oscillation in terms of the potential generated by the substrate and of the density profile of the gas. The form of the potential is discussed in detail and various regimes (van der Waals--London, Casimir-Polder, and thermal regimes) are identified as a function of the distance of atoms from the surface. Numerical results for the frequency shifts are given for the case of a sapphire dielectric substrate interacting with a harmonically trapped condensate of 87Rb^{87}\mathrm{R}\mathrm{b}87Rb atoms. We find that at distances of $4--8\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m}$, where thermal effects become visible, the relative frequency shifts produced by the substrate are of the order ${10}^{\ensuremath{-}4}$ and hence accessible experimentally. The effects of nonlinearities due to the finite amplitude of the oscillation are explicitly discussed. Predictions are also given for the radial breathing mode.
We theoretically study binary Bose-Einstein condensates trapped in a single-well harmonic potential to probe the dynamics of collective atomic motion. The idea is to choose tunable scattering lengths through Feshbach resonances such that the ground-state wave function for two types of the condensates are spatially immiscible where one of the condensates, located at the center of the potential trap, can be effectively treated as a potential barrier between bilateral condensates of the second type of atoms. In the case of small wave-function overlap between bilateral condensates, one can parametrize their spatial part of the wave functions in the two-mode approximation together with the time-dependent population imbalance zzz and the phase difference $\ensuremath{\phi}$ between two wave functions. The condensate in the middle can be approximated by a Gaussian wave function with the displacement of the condensate center $\ensuremath{\xi}$. As driven by the time-dependent displacement of the central condensate, we find the Josephson oscillations of the collective atomic motion between bilateral condensates as well as their anharmonic generalization of macroscopic self-trapping effects. In addition, with the increase in the wave-function overlap of bilateral condensates by properly choosing tunable atomic scattering lengths, the chaotic oscillations are found if the system departs from the state of a fixed point. The Melnikov approach with a homoclinic solution of the derived $z, \ensuremath{\phi}$, and $\ensuremath{\xi}$ equations can successfully justify the existence of chaos. All results are consistent with the numerical solutions of the full time-dependent Gross-Pitaevskii equations.
We investigate the collective excitations of a one-dimensional Bose-Einstein condensate with repulsive interaction between atoms in a quadratic plus quartic trap. Using variational approaches, the coupled equations of motions for the center-of-mass coordinate of the condensate and its width are derived. Then, two low-energy excitation modes are obtained analytically. The frequency shift induced by the anharmonic distortion, and the collapse and revival of the collective excitations originating from the nonlinear coupling between the two modes, are discussed.