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arXiv:2312.08159 · Physical Review A

Statistical and dynamical aspects of quantum chaos in a kicked Bose-Hubbard dimer

Chenguang Liang, Yu Zhang, Shu Chen

Published Mar 22, 2024• 7 citations
arXivHTMLPDF

Abstract

Systems of interacting bosons in double-well potentials, modeled by two-site Bose-Hubbard models, are of significant theoretical and experimental interest and attracted intensive studies in contexts ranging from many-body physics and quantum dynamics to the onset of quantum chaos. In this work we systematically study a kicked two-site Bose-Hubbard model (Bose-Hubbard dimer) with the on-site potential difference being periodically modulated. Our model can be equivalently represented as a kicked Lipkin-Meshkov-Glick model and thus displays different dynamical behaviors from the kicked-top model. By analyzing spectral statistics of Floquet operator, we unveil that the system undergoes a transition from regularity to chaos with increasing the interaction strength. Then based on semiclassical approximation and the analysis of R\'enyi entropy of coherent states in the basis of Floquet operator eigenstates, we reveal the local chaotic features of our model, which indicate the existence of integrable islands even in the deep chaotic regime. The semiclassical analysis also suggests that the system in chaotic regime may display different dynamical behavior depending on the choice of initial states. Finally, we demonstrate that dynamical signatures of chaos can be manifested by studying dynamical evolution of local operators and out-of-time-order correlation function as well as the entanglement entropy. Our numerical results exhibit the richness of dynamics of the kicked Bose-Hubbard dimer in both regular and chaotic regimes as the initial states are chosen as coherent spin states located in different locations of phase space.

DOI
10.1103/physreva.109.033316

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Systems of interacting bosons in double-well potentials, modeled by two-site Bose-Hubbard models, are of significant theoretical and experimental interest and attracted intensive studies in contexts ranging from many-body physics and quantum dynamics to the onset of quantum chaos. In this work we systematically study a kicked two-site Bose-Hubbard model (Bose-Hubbard dimer) with the on-site potential difference being periodically modulated. Our model can be equivalently represented as a kicked Lipkin-Meshkov-Glick model and thus displays different dynamical behaviors from the kicked-top model. By analyzing spectral statistics of Floquet operator, we unveil that the system undergoes a transition from regularity to chaos with increasing the interaction strength. Then based on semiclassical approximation and the analysis of R'enyi entropy of coherent states in the basis of Floquet operator eigenstates, we reveal the local chaotic features of our model, which indicate the existence of integrable islands even in the deep chaotic regime. The semiclassical analysis also suggests that the system in chaotic regime may display different dynamical behavior depending on the choice of initial states. Finally, we demonstrate that dynamical signatures of chaos can be manifested by studying dynamical evolution of local operators and out-of-time-order correlation function as well as the entanglement entropy. Our numerical results exhibit the richness of dynamics of the kicked Bose-Hubbard dimer in both regular and chaotic regimes as the initial states are chosen as coherent spin states located in different locations of phase space.
We study the quantum dynamics of the kicked Dicke model (KDM) in terms of the Floquet operator, and we analyze the connection between chaos and thermalization in this context. The Hamiltonian map is constructed by suitably taking the classical limit of the Heisenberg equation of motion to study the corresponding phase-space dynamics, which shows a crossover from regular to chaotic motion by tuning the kicking strength. The fixed-point analysis and calculation of the Lyapunov exponent (LE) provide us with a complete picture of the onset of chaos in phase-space dynamics. We carry out a spectral analysis of the Floquet operator, which includes a calculation of the quasienergy spacing distribution and structural entropy to show the correspondence to the random matrix theory in the chaotic regime. Finally, we analyze the thermodynamics and statistical properties of the bosonic sector as well as the spin sector, and we discuss how such a periodically kicked system relaxes to a thermalized state in accordance with the laws of statistical mechanics.
We study the quantum signatures of chaos by using the concept of bipartite fluctuations in the kicked two-site Bose-Hubbard model, which can be mapped to the well-studied kicked top model. We find that the evolution of bipartite fluctuations is quasiperiodic when the initial state centered in the regular regions, while it becomes irregular as the initial state centered in the chaotic regions. As a result, the quantum signatures of chaos can be identified in both the dynamical bipartite fluctuations and the long time averaged bipartite fluctuations. We further demonstrate that the regular versus irregular behaviors of bipartite fluctuations are induced by the different localization properties of the evolved state in regular and chaotic regimes. Finally, we discuss how to identify the signature of global chaos via bipartite fluctuations; and point out the relationship between bipartite fluctuations and the classical Lyapunov exponent.
In the large-NNN, classical limit, the Bose-Hubbard dimer undergoes a transition to chaos when its tunneling rate is modulated in time. We use exact and approximate numerical simulations to determine the features of the dynamically evolving state that are correlated with the presence of chaos in the classical limit. We propose the statistical distance between initially similar number distributions as a reliable measure to distinguish regular from chaotic behavior in the quantum dynamics. Besides being experimentally accessible, number distributions can be efficiently reconstructed numerically from binned phase-space trajectories in a truncated Wigner approximation. Although the evolving Wigner function becomes very irregular in the chaotic regions, the truncated Wigner method is nevertheless able to capture accurately the beyond-mean-field dynamics.
Quantum chaos in isolated quantum systems is intimately linked to thermalization and the rapid relaxation of observables. Although the spectral properties of the chaotic phase in the tilted Bose-Hubbard model have been well characterized, the corresponding dynamical signatures across the transition to regularity remain less explored . In this work, we investigate this transition by analyzing the time evolution of the survival probability, the single-site entanglement entropy, and the half-chain imbalance. Our results reveal a clear hierarchy in the sensitivity of these observables: the relaxation value of the entanglement entropy varies smoothly as a function of the Hamiltonian parameters across the chaos-regular transition, while the imbalance exhibits a more pronounced distinction. Most notably, the survival probability emerges as the most robust indicator of the transition between chaos and regularity. When appropriately scaled, all three observables converge onto a common behavior as a function of the Hamiltonian parameters for different numbers of sites and bosons,enabling a universal characterization of the transition between chaotic and regular dynamics.
We study the coherence dynamics of a kicked two-mode Bose-Hubbard model starting with an arbitrary coherent spin preparation. For preparations in the chaotic regions of phase space we find a generic behavior with Floquet participation numbers that scale as the entire N-particle Hilbert space, leading to a rapid loss of single-particle coherence. However, the chaotic behavior is not uniform throughout the chaotic sea and unique statistics is found for preparations at the vicinity of hyperbolic points that are embedded in it. This is contrasted with the low log(N) participation that is responsible for the revivals in the vicinity of isolated hyperbolic instabilities.
The quantum kicked top (QKT) is one of the most widely studied models in quantum chaos, providing a minimal yet powerful framework for exploring the relationship between classical nonlinear dynamics and quantum behavior. Unlike many chaotic systems with infinite-dimensional Hilbert spaces, the QKT possesses a finite-dimensional Hilbert space, making it analytically and numerically controllable while still showing a rich dynamical phenomena. In this chapter, we present a comprehensive introduction to the QKT as a paradigmatic model of quantum chaos. Starting from the classical kicked top, we derive the discrete nonlinear map governing the dynamics on the unit sphere and analyze its phase space structure through fixed points, stability analysis, bifurcations and Lyapunov exponents. We then discuss the role of symmetries, including rotational and time-reversal symmetry, and how their breaking modifies the dynamics. The quantum description is developed using Floquet theory, where the periodically driven spin system is represented by a unitary Floquet operator acting on a (2j+1)(2j+1)(2j+1)-dimensional Hilbert space. Within this framework, signatures of quantum chaos such as spectral statistics, entanglement generation and recurrences are discussed. The model also admits an interpretation as a system of interacting qubits, enabling explicit few-qubit realizations and direct connections with quantum information measures through reduced density matrices and entanglement entropy. By linking classical phase space structures with quantum dynamical indicators, the QKT provides a clear setting to investigate the emergence of chaotic behavior in the semiclassical limit. The chapter, therefore, highlights the quantum kicked top as a bridge between nonlinear classical dynamics, quantum chaos and modern quantum information science.