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Physical Review E

Resonant wave formation in Bose-Einstein condensates

Alexandru I. Nicolin

Published Nov 4, 2011• 53 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.84.056202

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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.
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.
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.
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 study the dynamics of Faraday waves for Bose-Einstein condensates(BECs) trapped by anisotropic potentials using the three-dimensional Gross-Pitaevskii equation. In previous studies, Faraday waves were excited by periodic modulation of the interaction or potential; in contrast, this study systematically addresses the excitations of the two methods. When the interaction is modulated with a modulation frequency resonant with Faraday waves, the breathing mode along the tight confinement direction is excited, and the Faraday waves appear in the direction of weak confinement. A modulation frequency that is not resonant with Faraday waves does not excite Faraday waves. Thus, the dynamics depend on modulation frequencies. The behavior of the total energy and its decomposition characterize the dynamics. The excitation of Faraday waves depends on the anisotropy of the potentials as well; Faraday waves are excited only for elongated BECs. We compare the differences of the dynamics in modulation methods. There are no qualitative differences between the modulation of the interaction and potential. When the interaction and potential are simultaneously modulated, Faraday waves are excited but they do not necessarily work additively. To understand this phenomenon as a dynamical system, we choose a few dynamical variables and follow their trajectory in a phase space. The trajectory characteristics of Faraday waves and the breathing mode show that the methods of modulation are not very relevant; determining the target mode to excite is important.
We study the emergence of Faraday waves in cigar-shaped collisionally inhomogeneous Bose-Einstein condensates subject to periodic modulation of the radial confinement. Considering a Gaussian-shaped radially inhomogeneous scattering length, we show through extensive numerical simulations and detailed variational treatment that the spatial period of the emerging Faraday waves increases as the inhomogeneity of the scattering length gets weaker, and that it saturates once the width of the radial inhomogeneity reaches the radial width of the condensate. In the regime of strongly inhomogeneous scattering lengths, the radial profile of the condensate is akin to that of a hollow cylinder, while in the weakly inhomogeneous case the condensate is cigar shaped and has a Thomas-Fermi radial density profile. Finally, we show that when the frequency of the modulation is close to the radial frequency of the trap, the condensate exhibits resonant waves which are accompanied by a clear excitation of collective modes, while for frequencies close to twice that of the radial frequency of the trap, the observed Faraday waves set in forcefully and quickly destabilize condensates with weakly inhomogeneous two-body interactions.
Faraday waves in a cigar-shaped Bose-Einstein condensate are created. It is shown that periodically modulating the transverse confinement, and thus the nonlinear interactions in the BEC, excites small amplitude longitudinal oscillations through a parametric resonance. It is also demonstrated that even without the presence of a continuous drive, an initial transverse breathing mode excitation of the condensate leads to spontaneous pattern formation in the longitudinal direction. Finally, the effects of strongly driving the transverse breathing mode with large amplitude are investigated. In this case, impact-oscillator behavior and intriguing nonlinear dynamics, including the gradual emergence of multiple longitudinal modes, are observed.