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arXiv:cond-mat/0204517 · Physical Review Letters

Faraday Patterns in Bose-Einstein Condensates

Kęstutis Staliūnas, Stefano Longhi, Germán J. de Valcárcel

Published Nov 4, 2002• 215 citations
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Abstract

Temporal periodic modulation of the interatomic s-wave scattering length in Bose-Einstein condensates is shown to excite subharmonic patterns in the atom density through a parametric resonance. The dominant wavelength of the spatial structures is primarily selected by the excitation frequency but also affected by the depth of the spatial modulation via a nonlinear resonance. These phenomena represent analogues of the Faraday patterns excited in vertically vibrated liquids.

DOI
10.1103/physrevlett.89.210406

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Temporal periodic modulation of the interatomic s-wave scattering length in Bose-Einstein condensates is shown to excite subharmonic patterns in the atom density through a parametric resonance. The dominant wavelength of the spatial structures is primarily selected by the excitation frequency but also affected by the depth of the spatial modulation via a nonlinear resonance. These phenomena represent analogues of the Faraday patterns excited in vertically vibrated liquids.
A mechanism leading to spontaneous breaking of the space symmetry, and to the appearance of spatial patterns in the most recently created state of matter, the Bose-Einstein condensate (BEC), is proposed. The pattern formation are due to a periodic modulation in time of the scattering length of the condensed atoms. Experimentally this can be achieved via the Feshbach resonance, or, for low dimensional (1D or 2D) condensates, also by modulation of the trapping potential along a spatial coordinate normal to the "working" space. The parametric excitation of patterns in condensates by considering the mean-field Gross-Pitaevskii equation, and its dissipative generalization are investigated.
We investigate parametric excitation and pattern formation in a harmonically trapped two-component Bose-Einstein condensate. We assume the condensate to be in the miscible phase, but near the miscible-immiscible phase transition, where total-density and spin-density excitations are decoupled. By periodically modulating the atomic scattering lengths, Faraday patterns can be generated in both density and spin channels. In an elongated condensate, the pattern in the spin channel corresponds to a one-dimensional standing wave with the two components exhibiting an out-of-phase density oscillation, where the modulation frequency and the oscillation period are related to the velocity of the spin sound. After the spin pattern is fully developed, the system quickly enters a nonlinear destabilization regime. For a pancake-shaped condensate, a two-dimensional Faraday pattern is generated with an interesting lll-fold rotational symmetry. The number of nodes along the radial and angular directions increases with larger modulation frequencies. We also compare the growth rates of spin Faraday patterns generated with different modulation protocols, which are accessible to current experiments.
Abstract We consider Bose–Einstein condensates with two- and three-body interactions periodically varying in time. Two models of time-dependent three-body interactions, with quadratic and quartic dependence on the two-body atomic scattering length a s , are studied. It is shown that parametric instabilities in the condensate lead to the generation of Faraday waves (FWs), with wavelengths depending on the background scattering length, as well as on the frequency and amplitude of the modulations of a s . From an experimental perspective, this opens a new possibility to tune the period of Faraday patterns by varying not only the frequency of modulations and background scattering length, but also the amplitude of the modulations. The latter effect can be used to estimate the parameters of three-body interactions from the FW experimental results. Theoretical predictions are confirmed by numerical simulations of the corresponding extended Gross–Pitaevskii equation.
Excitation of Faraday patterns in the weak confinement space of low-dimensional Bose-Einstein condensates (BEC) by a periodic modulation of the trap frequency, ${\ensuremath{\Omega}}_{\text{tight}}$, in the tight confinement space is shown. For slow modulation the low-dimensional dynamics of the BEC in the weak confinement space is described by a Gross-Pitaevskii equation with time modulated nonlinearity coefficient. For increasing modulation frequencies a noticeable reduction of the pattern formation threshold is observed near $2{\ensuremath{\Omega}}_{\text{tight}}$, which is related to the parametric excitation of the internal breathing mode in the tight confinement space. These predictions could be relevant for the experimental excitation of Faraday patterns in BEC.