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arXiv:1804.10755 · Physical Review A

Parametric resonance of a Bose-Einstein condensate in a ring trap with periodically driven interactions

Chen-Xi Zhu, Wei Yi, Guang‐Can Guo, Zheng-Wei Zhou

Published Feb 13, 2019• 13 citations
arXivHTMLPDF

Abstract

We study the instability of a ring Bose-Einstein condensate under a periodic modulation of interatomic interactions. The condensate exhibits temporal and spatial patterns induced by the parametric resonance, which can be characterized by Bogoliubov quasiparticle excitations in the Floquet basis. As the ring geometry significantly limits the number of excitable Bogoliubov modes, we are able to capture the nonlinear dynamics of the condensate using a three-mode model. We further demonstrate the robustness of the temporal and spatial patterns against disorder, which is attributed to the mode-locking mechanism under the ring geometry. Our results can be observed in cold atomic systems and are also relevant to physical systems described by the nonlinear Schr\"odinger equation.

DOI
10.1103/physreva.99.023619

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We study the instability of a ring Bose-Einstein condensate under a periodic modulation of interatomic interactions. The condensate exhibits temporal and spatial patterns induced by the parametric resonance, which can be characterized by Bogoliubov quasiparticle excitations in the Floquet basis. As the ring geometry significantly limits the number of excitable Bogoliubov modes, we are able to capture the nonlinear dynamics of the condensate using a three-mode model. We further demonstrate the robustness of the temporal and spatial patterns against disorder, which is attributed to the mode-locking mechanism under the ring geometry. Our results can be observed in cold atomic systems and are also relevant to physical systems described by the nonlinear Schr"odinger equation.
The study of collective excitations is a crucial tool for understanding many-body quantum systems. For instance, collective excitations play a central role in the exploration of superfluidity and other quantum macroscopic phenomena in Bose and Fermi systems. In this work we present a variational and a numerical study of a parametrically driven Bose-Einstein condensate confined in a cylindrical harmonic trap in which the aspect ratio can be varied from a prolate (cigar-shaped) to an oblate (pancake-shaped) system. The excitation can be applied by periodically modulating the harmonic frequencies of the trap or alternatively the interatomic interaction strength at a frequency that matches that of the system's breathing mode. As a result, we observe the formation of dynamical density patterns that depend on the geometry of the trap: a fringe pattern in a prolate system and a ring pattern in an oblate one. By decomposing the total energy into its kinetic, potential, and interaction terms, we show that the onset of these patterns coincides with the redistribution of kinetic energy in the weakly trapped directions of the sample, indicating the three-dimensional nature of the studied phenomena. Finally, our analysis shows that the difference between the two excitation mechanisms lies in the system stability. Modulating the trap destabilizes the system quicker than modulating the interactions, leading to earlier formation of the patterns.
Abstract We investigate spatiotemporal periodic patterns in harmonically trapped Bose–Einstein condensates (BECs) driven by a periodic modulation of the interaction. Resonant with the breathing mode, we show the emergence of a square lattice pattern containing two orthonormal stripes. We study the time evolutions of the lattice patterns for different driving strengths and dissipations. We find that its spatial periodicity and temporal oscillating frequency match the Bogoliubov dispersion, which is the intrinsic property of the system and relevant to the parametric amplification of elementary excitations. In the circumstances of strong driving strength and low dissipation, we further observe the triad interaction and the resulting superlattice state, which are well explained by the nonlinear amplitude equation for superimposed stripes. These results shed light on unexplored nonlinear spatiotemporal dynamics of two-dimensional patterns in harmonically trapped BECs that can pave the way for engineering exotic patterns by state-of-the-art experiments.
We investigate the dynamical formation of nonlinear patterns in one-dimensional ring condensates under bichromatic periodic modulation of the interaction strength. The stability phase diagram of the condensate's homogeneous density state is analytically derived through a suitable biharmonic variant of the Mathieu equation and computing the associated Floquet spectrum. It reveals the complex interplay between the driving parameters, i.e., amplitude, frequencies, and the so-called frequencies' mixing angle, which dictate the instability onset and the selective enhancement of higher-order resonance tongues, thus offering precise control over the excited modes. These results are in agreement with time-dependent mean-field simulations evidencing the emergence of density wave modulations of specific momenta, while enabling a deeper understanding of the nonlinear stage of the relevant instability. Further insights on the ensuing unstable nonlinear dynamics are provided through a reduced {five-mode} model which captures the instability onset, the oscillatory behavior of the mode populations and the phase-space dynamics, in agreement with the mean-field predictions. Our study highlights the versatility of bichromatic driving to generate and control complex nonlinear patterns that are within reach in present day ultracold atom experiments.
Instabilities of vortex-ring-bright coherent structures in harmonically trapped two-component three-dimensional Bose-Einstein condensates are studied numerically within the coupled Gross-Pitaevskii equations and interpreted analytically. Interestingly, the filled vortex core with a sufficiently large amount of the bright component is observed to reduce the parametric interval of stability of the vortex ring. We have identified the mechanisms of several linear instabilities and one nonlinear parametric instability in this connection. Two of the linear instabilities are qualitatively different from ones reported earlier and are associated with azimuthal modes of m=0m=0m=0 and m=1m=1m=1, i.e., deviations of the vortex from the stationary ring shape. Our nonlinear parametric resonance instability occurs between the m=0m=0m=0 and m=2m=2m=2 modes and signals the exchange of energy between them.
Abstract We report parametric resonances (PRs) in a numerical investigation of a driven one-dimensional, interacting, and disordered Bose–Einstein condensate (BEC) confined in different traps. The BEC is excited by an oscillating Gaussian obstacle along a broad range of driving frequencies Ω. The PRs are detected via a quantity that is closely related to the time-average of the kinetic energy. The significant result of this work is that the trapping geometry plays a major role in defining the values of Ω at which PRs arise and controls their response to disorder. As such, it reveals the interplay of trapping geometry and disorder in these resonances. The dynamics of the modal coefficient C 0 ( t ) as well as that of the phase-mismatch δ ( t ) between the C 0 ( t ) and C 1 ( t ) are examined at and away from PR. At PR, | C 0 ( t )| is generally found to be lower in magnitude than away from it, demonstrating that the atoms leave the n = 0 ground state towards higher states. In the harmonic oscillator trap, the dynamic pattern of δ ( t ) is found to be quite robust against changes in the disorder strength contrary to the box potential. This is because in the box the ratio of the random-potential and kinetic energies is higher than in the harmonic trap signaling that the influence of disorder is weaker in the latter.
We investigate the dynamics of a Bose-Einstein condensate in a one-dimensional ring-shaped lattice with the Peierls phase and site-dependent modulations, where the condensate is confined in a single deep trap and the interparticle interaction strength is modulated by a time-periodic driving field. The system has a finite spectrum, which limits the excitation regimes, and the Peierls phase typically induces imbalanced complex hopping amplitudes in each direction, leading to nonzero net particle currents along the lattice chain, which can hold nearly persistent even when the driving field is turned off after half of the period. The configuration provides a specific way for the coherent control of particle currents in many-body quantum systems with the help of an external driving field and promotes the possible applications in future closed-loop atom circuits.