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Physical Review A

Quantum-classical correspondence of chaotic dynamics in a continuously driven system

Ke Li, Yuxiang Liu, Sheng-Chang Li

Published Nov 1, 2024• 2 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.110.052402

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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.
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.
Identifying quantum chaos in Floquet systems with mixed classical phase spaces requires robust experimentally accessible signatures. We investigate the time-averaged quantum Fisher information (QFI) and the fidelity out-of-time-ordered correlator (FOTOC) of continuous periodically driven systems. We uncover two distinct power-law scalings of the time-averaged QFI with respect to system size: the standard quantum limit and the Heisenberg limit. These scalings correspond to initial states localized in the regular and chaotic regions of the classical phase space, respectively. Remarkably, we show that the time-averaged FOTOC can accurately identify the transition from mixed phase space to fully chaotic sea (at a critical driving strength Bx≈5.3) as verified by the maximum Lyapunov exponent. The results suggest that both time-averaged QFI and FOTOC can be used as excellent quantum signatures for probing mixed phase-space structures and our adopted protocol for measuring the FOTOC eliminates the need for time-reversal operations. This work establishes a practical framework for investigating quantum chaos in Floquet systems, bridging theoretical insights with experimental applications.
We study a periodically driven macrospin system with anisotropic long-range interactions and collective dissipation, described by a Lindblad master equation. In the thermodynamic limit (N→∞N\to\inftyN→∞), a mean-field treatment yields classical equations of motion, whose dynamics are characterized via the maximal Lyapunov exponent (MLE). Focusing on the thermodynamic limit, we map out chaotic, quasiperiodic, and periodic phases via bifurcation diagrams, MLEs, and Fourier spectra of evolved observables, identifying classic period-doubling bifurcations and fractal boundaries in the regions of attractors. Finite-size quantum simulations in the Dicke basis reveal that while both quantum and classical systems exhibit diverse dynamical phases, finite-size effects suppress some behaviors present in the thermodynamic limit. The sign of λmaxλ_{\mathrm{max}}λmax​ serves as a key indicator of convergence between quantum and classical dynamics, which agree over timescales up to the Lyapunov time. Analysis of the density matrix shows that convergence occurs only when its nonzero elements are sharply localized. However, the nonconvergence does not imply a fundamental difference between quantum and classical dynamics: in chaotic regimes, although the evolution orbits of quantum and classical systems show significant differences, quantum evolution becomes mixed and diffusively explores the Hilbert space, signaling quantum chaos, which can be confirmed by the delocalized nature of the density matrix.
We investigate both the classical and quantum dynamics of a spin-1 Bose–Einstein condensate (BEC) system subject to periodic driving of quadratic Zeeman splitting. Without driving, the system is integrable. Here, we found that increasing the driving strength will drive the system to chaos. Classically, we demonstrate chaotic dynamics by examining the Poincaré section and Lyapunov exponent. While in the quantum regime, the level statistic and Husimi distribution are studied, and they also clearly indicate the onset of quantum chaos. The dynamics of atom number and out-of-time-ordered correlator (OTOC) are also studied. In the integrable region, the variance of atom number and OTOC both remain very small during the time evolution; while in the chaos region, a quick increase is observed in the early time. This work presents a quantum many-body system showing chaotic behavior and can be readily realized in experiment.
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.
Nonlinear systems, including atom–field interaction, are investigated due to their fundamental applications in quantum mechanics and rapidly growing fields of quantum communication, especially secure communication with chaotic dynamics. In this paper, we study the chaotic dynamics of a system consisting of an atomic Bose–Einstein condensate interacting with a quantized radiation field in a high-quality cavity with a periodically modulated length. The frequency modulation is adapted by a periodic time-dependent atom–field coupling strength. We use a semiclassical approach to decouple the atoms and field variables and then numerically solve the corresponding nonlinear dynamical equations of the system. Generally, the dynamics of the system sensitively depends on its initial conditions, thereby long-term prediction is impossible. We show that the system demonstrates the emergence of classical dynamical chaos from quantum electrodynamics. The chaotic behavior of energy transfer in the system can be enhanced by increasing the depth of frequency modulation. The strange attractor clearly illustrates that the system presents extremely exotic dynamics over a wide range of parameters. This implies that the dynamical quantities oscillate irregularly, never exactly repeating but always remaining in a bounded region of the phase space.