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FARADAY WAVES IN ONE-DIMENSIONAL BOSE-EINSTEIN CONDENSATES

Alexandru I. Nicolin, Mihaela Carina Raportaru

Published Jan 1, 2011• 2 citations
OpenAlex

Abstract

Faraday waves in Bose-Einstein condensates are analysed in a one-dimensional setting by variational means. A model that describes the bulk part of a condensate through a Gaussian envelope and incorporates the Faraday wave as a surface wave grafted on the envelope is used to derive a set of equations for Faraday waves in one-dimensional Bose-Einstein condensates.

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Faraday waves in Bose-Einstein condensates are analysed in a one-dimensional setting by variational means. A model that describes the bulk part of a condensate through a Gaussian envelope and incorporates the Faraday wave as a surface wave grafted on the envelope is used to derive a set of equations for Faraday waves in one-dimensional Bose-Einstein condensates.
In this study we introduce a variational model for the dynamics of Faraday waves in longitudinally-inhomogeneous Bose-Einstein condensates subject to anisotropic transverse confinement and periodic modulations of the scattering length. We derive a set of variational equations under the assumption that the period of the Faraday wave is smaller than the longitudinal width of the condensate and determine analytically the dispersion relation for large condensates through a Mathieu-type ana- lysis of the variational equations. Finally, we show the emergence of Faraday waves in a realistic anisotropic 87 Rb Bose-Einstein condensate and discuss possible extensions of the variational treatment to binary condensates.
We numerically studied Faraday waves in Bose-Einstein Condensates(BECs) trapped by anisotropic harmonic potentials using the three-dimensional Gross-Pitaevskii equation.The response of BECs was studied by modulating the interaction parameter twice the trap frequency in the tight confinement direction.In the case of elongated BECs, breathing modes were excited in the tight confinement direction, followed by Faraday waves in the weak confinement direction.First, we discuss the appearance of breathing modes and/or Faraday waves which depended strongly on the modulation frequencies.Second, we investigate the dependence of dynamics on the anisotropy of the potentials.Faraday waves appeared only for the elongated BECs.
We show by extensive numerical simulations and analytical variational calculations that elongated binary nonmiscible Bose-Einstein condensates subject to periodic modulations of the radial confinement exhibit a Faraday instability similar to that seen in one-component condensates. Considering the hyperfine states of 87{}^{87}87Rb condensates, we show that there are two experimentally relevant stationary-state configurations: one in which the components form a dark-bright symbiotic pair (the ground state of the system) and one in which the components are segregated (first excited state). For each of these two configurations, we show numerically that far from resonances the Faraday waves excited in the two components are of similar periods, emerge simultaneously, and do not impact the dynamics of the bulk of the condensate. We derive analytically the period of the Faraday waves using a variational treatment of the coupled Gross-Pitaevskii equations combined with a Mathieu-type analysis for the selection mechanism of the excited waves. Finally, we show that for a modulation frequency close to twice that of the radial trapping, the emergent surface waves fade out in favor of a forceful collective mode that turns the two condensate components miscible.
Faraday and resonant density waves emerge in Bose-Einstein condensates as a result of harmonic driving of the system. They represent nonlinear excitations and are generated due to the interaction-induced coupling of collective oscillation modes and the existence of parametric resonances. Using a mean-field variational and a full numerical approach, we studied density waves in dipolar condensates at zero temperature, where breaking of the symmetry due to anisotropy of the dipole-dipole interaction (DDI) plays an important role. We derived variational equations of motion for the dynamics of a driven dipolar system and identify the most unstable modes that correspond to the Faraday and resonant waves. Based on this, we derived the analytical expressions for spatial periods of both types of density waves as functions of the contact and the DDI strength. We compared the obtained variational results with the results of extensive numerical simulations that solve the dipolar Gross-Pitaevskii equation in 3D, and found a very good agreement.
Motivated by recent experiments on Faraday waves in Bose-Einstein condensates we investigate both analytically and numerically the dynamics of cigar-shaped Bose-condensed gases subject to periodic modulation of the strength of the transverse confinement. We offer a fully analytical explanation of the observed parametric resonance, based on a Mathieu-type analysis of the non-polynomial Schr\"odinger equation. The theoretical prediction for the pattern periodicity versus the driving frequency is directly compared to the experimental data, yielding good qualitative and quantitative agreement between the two. These results are corroborated by direct numerical simulations of both the one-dimensional non-polynomial Schr\"odinger equation and of the fully three-dimensional Gross-Pitaevskii equation.
We investigate analytically the dynamics of a trapped, quasi-one-dimensional Bose-Einstein condensate subject to resonant and nonresonant periodic modulation of the transverse confinement. The dynamics of the condensate is described variationally through a set of coupled ordinary differential equations, and the period of the excited waves is determined analytically using a Mathieu-type analysis. For a modulation frequency equal to that of the radial confinement we show that the predicted period of the resonant wave is in agreement with the existing experimental results. Finally, we present a detailed comparison between the resonant waves and the Faraday waves that emerge outside of resonance.