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arXiv:2012.12429 · Physical Review A

Multiparticle entanglement dynamics of quantum chaos in a Bose-Einstein condensate

Sheng-Chang Li, Luca Pezzè, Augusto Smerzi

Published May 17, 2021• 15 citations
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Abstract

We study the particle-entanglement dynamics witnessed by the quantum Fisher information (QFI) of a trapped Bose-Einstein condensate governed by the kicked rotor Hamiltonian. The dynamics is investigated with a beyond-mean-field approach. We link the time scales of the validity of this approximation in both classical regular and chaotic regions, with the maximum Lyapunov exponents of the classical system. This establishes an effective connection between the classical chaos and the QFI. We finally study the critical point of a quantum phase transition using the beyond-mean-field approximation by considering a two-mode bosonic Josephson junction with attractive interparticle interaction.

DOI
10.1103/physreva.103.052417

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We study the particle-entanglement dynamics witnessed by the quantum Fisher information (QFI) of a trapped Bose-Einstein condensate governed by the kicked rotor Hamiltonian. The dynamics is investigated with a beyond-mean-field approach. We link the time scales of the validity of this approximation in both classical regular and chaotic regions, with the maximum Lyapunov exponents of the classical system. This establishes an effective connection between the classical chaos and the QFI. We finally study the critical point of a quantum phase transition using the beyond-mean-field approximation by considering a two-mode bosonic Josephson junction with attractive interparticle interaction.
We investigate both classical and quantum chaotic dynamics of a two-mode Bose-Einstein condensate driven by a continuously periodic field. We calculate the quantities, including quantum Fisher information (QFI), fidelity, and linear entropy, to characterize the quantum chaos and compare these quantities with the maximum Lyapunov exponent (MLE) of the corresponding classical system. The behaviors of the quantum quantities and the classical quantity show a strong correspondence regardless of the classical regular and chaotic regions in the parameter space. A quantitative link between the MLE and the short-time dynamics of the QFI is established. Particularly, we find that our continuously driven model is significantly superior to other models in accelerating entanglement generation through chaos, which is evidenced by the shorter time required for the QFI to reach the Heisenberg limit. We briefly discuss the underlying mechanism of this finding and its application in saving time costs and improving measurement accuracy as well.
Quantum Fisher information,derived from the classical Fisher information, is closely related to the quantum entanglement in quantum information. The entanglement and the quantum information which are both associated with the classical phase space are investigated in a two-component Bose-Einstein condensate impacted by the impulses. The results reveal that the states regardless of disorder of the phase space after the first impulse are entangled. However, the quantum information is very sensitive to the state centred in the classical phase space, concretely, the value of the quantum information centred in the chaotic region is greater than in the regular region. By employing the good quantum-classical correspondence, we conclude that the quantum information can serve as a signature of the quantum chaos.
The mean-field limit of a bosonic quantum many-body system is described by (mostly) nonlinear equations of motion which may exhibit chaos very much in the spirit of classical particle chaos, i.e., by an exponential separation of trajectories in Hilbert space with a rate given by a positive Lyapunov exponent $\ensuremath{\lambda}$. The question now is whether $\ensuremath{\lambda}$ imprints itself onto measurable observables of the underlying quantum many-body system even at finite particle numbers. Using a Bose-Einstein condensate expanding in a shallow potential landscape as a paradigmatic example for a bosonic quantum many-body system, we show that the system loses its coherence at an exponentially fast rate. Furthermore, we show that the rate is given by the Lyapunov exponent associated with the chaotic mean-field dynamics. Finally, we demonstrate that this chaos-induced loss of coherence imprints itself onto the visibility of interference fringes in the total density after time of flight, thus, opening the possibility to measure $\ensuremath{\lambda}$ and with it the interplay between chaos and nonequilibrium quantum matter in a real experiment.
It is widely recognized that entanglement generation and dynamical chaos are intimately related in semiclassical models via the process of decoherence. In this paper, we propose a unifying framework which directly connects the bipartite and multipartite entanglement growth to the quantifiers of classical and quantum chaos. In the semiclassical regime, the dynamics of the von Neumann entanglement entropy, the spin squeezing, the quantum Fisher information, and the out-of-time-order square commutator are governed by the divergence of nearby phase-space trajectories via the local Lyapunov spectrum, as suggested by previous conjectures in the literature. General analytical predictions are confirmed by detailed numerical calculations for two paradigmatic models, relevant in atomic and optical experiments, which exhibit a regular-to-chaotic transition: the quantum kicked top and the Dicke model.
We investigate the coherence dynamics of a two-mode Bose-Einstein condensate coupled with the environment in the mean-field approximation. We give an analytical result of the time-average coherence in the absence of system-environment coupling, and find that the time-average coherence attains its maximum value at the critical point which corresponds to the boundary between the self-trapping regime and the Josephson oscillation regime. The effect of noise on dynamical coherence is also considered by analyzing the couplings of the condensate to the environment. With the first kind of coupling, the coherence finally stabilizes at a fixed value. Meanwhile, we show that the presence of the noise can even enhance the coherence for some particular interaction. With the second kind of coupling as collisional dephasing, the noise leads a sudden transition of the coherence which then subjects to an exponential decay.
We study the dynamics of a three-mode bosonic system with mode-changing interactions. For large mode occupations the short-time dynamics is well described by classical mean-field equations allowing us to study chaotic dynamics in the classical system and its signatures in the corresponding quantum dynamics. By introducing a symmetry-breaking term we tune the classical dynamics from integrable to strongly chaotic, which we demonstrate by calculating Poincar\'e sections and Lyapunov exponents. The corresponding quantum system features level statistics that change from Poissonian in the integrable to Wigner-Dyson in the chaotic case. We investigate the behavior of out-of-time-ordered correlators (OTOCs), specifically the squared commutator, for initial states located in regular and chaotic regions of the classical mixed phase space and find marked differences between the two cases. The short-time behavior is well captured by semi-classical truncated Wigner simulations directly relating these features to properties of the underlying classical mean-field dynamics. We discuss a possible experimental realization of this model system in a Bose-Einstein condensate of rubidium atoms, which allows reversing the sign of the Hamiltonian required for measuring OTOCs.