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arXiv:1707.04055 · Physical Review X

Parametric Excitation of a Bose-Einstein Condensate: From Faraday Waves to Granulation

J. H. V. Nguyen, M. C. Tsatsos, Di Luo, Axel U. J. Lode, Gustavo Deczka Telles, Vanderlei Salvador Bagnato, Randall G. Hulet

Published Mar 25, 2019• 102 citations
arXivHTMLPDF

Abstract

We explore, both experimentally and theoretically, the response of an elongated Bose-Einstein condensate to modulated interactions. We identify two distinct regimes differing in modulation frequency and modulation strength. Longitudinal surface waves are generated either resonantly or parametrically for modulation frequencies near the radial trap frequency or twice the trap frequency, respectively. The dispersion of these waves, the latter being a Faraday wave, is well-reproduced by a mean-field theory that accounts for the 3D nature of the elongated condensate. In contrast, in the regime of lower modulation frequencies we find that no clear resonances occur, but with increased modulation strength, the condensate forms an irregular granulated distribution that is outside the scope of a mean-field approach. We find that the granulated condensate is characterized by large quantum fluctuations and correlations, which are well-described with single-shot simulations obtained from wavefunctions computed by a beyond mean-field theory at zero temperature, the multiconfigurational time-dependent Hartree for bosons method.

DOI
10.1103/physrevx.9.011052

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We explore, both experimentally and theoretically, the response of an elongated Bose-Einstein condensate to modulated interactions. We identify two distinct regimes differing in modulation frequency and modulation strength. Longitudinal surface waves are generated either resonantly or parametrically for modulation frequencies near the radial trap frequency or twice the trap frequency, respectively. The dispersion of these waves, the latter being a Faraday wave, is well-reproduced by a mean-field theory that accounts for the 3D nature of the elongated condensate. In contrast, in the regime of lower modulation frequencies we find that no clear resonances occur, but with increased modulation strength, the condensate forms an irregular granulated distribution that is outside the scope of a mean-field approach. We find that the granulated condensate is characterized by large quantum fluctuations and correlations, which are well-described with single-shot simulations obtained from wavefunctions computed by a beyond mean-field theory at zero temperature, the multiconfigurational time-dependent Hartree for bosons method.
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.
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.
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 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.
Granulation of quantum matter -- the formation of persistent small-scale patterns -- is realized in the images of quasi-one-dimensional Bose-Einstein condensates perturbed by a periodically modulated interaction. Our present analysis of a mean-field approximation suggests that granulation is caused by the gradual transformation of phase undulations into density undulations. This is achieved by a suitably large modulation frequency, while for low enough frequencies the system exhibits a quasi-adiabatic regime. We show that the persistence of granulation is a result of the irregular evolution of the phase of the wavefunction representing an irreversible process. Our model predictions agree with numerical solutions of the Schrödinger equation and experimental observations. The numerical computations reveal the emergent many-body correlations behind these phenomena via the multi-configurational time-dependent Hartree theory for bosons (MCTDHB).