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arXiv:1911.09027 · Physical Review A

Bogoliubov excitations in the quasiperiodic kicked rotor: Stability of a kicked condensate and the quasi–insulator-to-metal transition

Benoît Vermersch, Dominique Delande, Jean Claude Garreau

Published May 18, 2020• 8 citations
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Abstract

We study the dynamics of a Bose-Einstein condensate in the quasiperiodic kicked rotor described by a Gross-Pitaevskii equation confined in a toroidal trap. As the interactions are increased, Bogoliubov excitations appear and deplete the condensate; we characterize this instability by considering the population of the first Bogoliubov mode, and we show that it does not prevent, for small enough interaction strengths, the observation of the quasi--insulator-to-metal transition that replaces the Anderson transition of the noninteracting case. However, the predicted subdiffusion in momentum space is not observed in the stable region within experimentally accessible times. For higher interaction strengths, the condensate may be strongly depleted before this dynamical regimes set in.

DOI
10.1103/physreva.101.053625

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We study the dynamics of a Bose-Einstein condensate in the quasiperiodic kicked rotor described by a Gross-Pitaevskii equation confined in a toroidal trap. As the interactions are increased, Bogoliubov excitations appear and deplete the condensate; we characterize this instability by considering the population of the first Bogoliubov mode, and we show that it does not prevent, for small enough interaction strengths, the observation of the quasi--insulator-to-metal transition that replaces the Anderson transition of the noninteracting case. However, the predicted subdiffusion in momentum space is not observed in the stable region within experimentally accessible times. For higher interaction strengths, the condensate may be strongly depleted before this dynamical regimes set in.
We study the dynamics of Bogoliubov excitations of a Bose-Einstein condensate in the quasiperiodic kicked rotor. In the weakly interacting regime, the condensate is stable and both the condensate and the excitations undergo a phase transition from a quasilocalized to a diffusive regime. The corresponding critical exponents are identical for the condensate and the excitations, and compare very well with the value ν≈1.6ν\approx1.6ν≈1.6 for non-interacting particles.
A periodically kicked ring of a Bose-Einstein condensate is considered as a nonlinear generalization of the quantum kicked rotor, where the nonlinearity stems from the mean-field interactions between the condensed atoms. For weak interactions, periodic motion (antiresonance) becomes quasiperiodic (quantum beating) but remains stable. There exists a critical strength of interactions beyond which quasiperiodic motion becomes chaotic, resulting in an instability of the condensate manifested by exponential growth in the number of noncondensed atoms. In the stable regime, the system remains predominantly in the two lowest energy states and may be mapped onto a spin model, from which we obtain an analytic expression for the beat frequency and discuss the route to instability. We numerically explore a parameter regime for the occurrence of instability and reveal the characteristic density profile for both condensed and noncondensed atoms. The Arnold diffusion to higher energy levels is found to be responsible for the transition to instability. Similar behavior is observed for dynamically localized states (essentially quasiperiodic motions), where stability remains for weak interactions but is destroyed by strong interactions.
A periodically kicked ring of a Bose-Einstein condensate is considered as a nonlinear generalization of the quantum kicked rotor. For weak interactions between atoms, periodic motion (antiresonance) becomes quasiperiodic (quantum beating) but remains stable. There exists a critical strength of interactions beyond which quasiperiodic motion becomes chaotic, resulting in an instability of the condensate manifested by exponential growth in the number of noncondensed atoms. Similar behavior is observed for dynamically localized states (essentially quasiperiodic motions), where stability remains for weak interactions but is destroyed by strong interactions.
We study the dynamics of the many-body atomic kicked rotor with interactions at the mean-field level, governed by the Gross-Pitaevskii equation. We show that dynamical localization is destroyed by the interaction and replaced by a subdiffusive behavior. In contrast to results previously obtained from a simplified version of the Gross-Pitaevskii equation, the subdiffusive exponent does not appear to be universal. By studying the phase of the mean-field wave function, we propose an alternative approximation that describes correctly the dynamics at experimentally relevant times close to the start of subdiffusion, while preserving the reduced computational cost of the former approximation.
Bose-Einstein condensates subject to short pulses (``kicks'') from standing waves of light represent a nonlinear analog of the well-known chaos paradigm, the quantum kicked rotor. Previous studies of the onset of dynamical instability (i.e., exponential proliferation of noncondensate particles) suggested that the transition to instability might be associated with a transition to chaos. Here we conclude instead that instability is due to resonant driving of Bogoliubov modes. We investigate the Bogoliubov spectrum for both the quantum kicked rotor (QKR) and a variant, the double kicked rotor (QKR-2). We present an analytical model, valid in the limit of weak impulses which correctly gives the scaling properties of the resonances and yields good agreement with mean-field numerics.
We investigate the superfluid properties of a Bose–Einstein condensate (BEC) trapped in a one-dimensional periodic potential. We study, both analytically (in the tight binding limit) and numerically, the Bloch chemical potential, the Bloch energy and the Bogoliubov dispersion relation, and we introduce two different, density dependent, effective masses and group velocities. The Bogoliubov spectrum predicts the existence of sound waves, and the arising of energetic and dynamical instabilities at critical values of the BEC quasi-momentum which dramatically affect its coherence properties. We investigate the dependence of the dipole and Bloch oscillation frequencies in terms of an effective mass averaged over the density of the condensate. We illustrate our results with several animations obtained solving numerically the time-dependent Gross–Pitaevskii equation.