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arXiv:2103.00748 · Physical review. E

Nonlinear dynamics and quantum chaos of a family of kicked p -spin models

Manuel H. Muñoz-Arias, Pablo M. Poggi, Ivan Deutsch

Published May 17, 2021• 27 citations
arXivHTMLPDF

Abstract

We introduce kicked p-spin models describing a family of transverse Ising-like models for an ensemble of spin-1/2 particles with all-to-all p-body interaction terms occurring periodically in time as delta-kicks. This is the natural generalization of the well-studied quantum kicked top (p=2) [Haake, Kuś, and Scharf, Z. Phys. B 65, 381 (1987)10.1007/BF01303727]. We fully characterize the classical nonlinear dynamics of these models, including the transition to global Hamiltonian chaos. The classical analysis allows us to build a classification for this family of models, distinguishing between p=2 and p>2, and between models with odd and even p's. Quantum chaos in these models is characterized in both kinematic and dynamic signatures. For the latter, we show numerically that the growth rate of the out-of-time-order correlator is dictated by the classical Lyapunov exponent. Finally, we argue that the classification of these models constructed in the classical system applies to the quantum system as well.

DOI
10.1103/physreve.103.052212

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We introduce kicked p-spin models describing a family of transverse Ising-like models for an ensemble of spin-1/2 particles with all-to-all p-body interaction terms occurring periodically in time as delta-kicks. This is the natural generalization of the well-studied quantum kicked top (p=2) [Haake, Kuś, and Scharf, Z. Phys. B 65, 381 (1987)10.1007/BF01303727]. We fully characterize the classical nonlinear dynamics of these models, including the transition to global Hamiltonian chaos. The classical analysis allows us to build a classification for this family of models, distinguishing between p=2 and p>2, and between models with odd and even p's. Quantum chaos in these models is characterized in both kinematic and dynamic signatures. For the latter, we show numerically that the growth rate of the out-of-time-order correlator is dictated by the classical Lyapunov exponent. Finally, we argue that the classification of these models constructed in the classical system applies to the quantum system as well.
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.
While classical chaos has been successfully characterized with consistent theories and intuitive techniques, such as with the use of Lyapunov exponents, quantum chaos is still poorly understood, as well as its relation with multi-partite entanglement and information scrambling. We consider a benchmark system, the kicked top model, which displays chaotic behaviour in the classical version, and proceed to characterize the quantum case with a thorough diagnosis of the growth of chaos and entanglement in time. As a novel tool for the characterization of quantum chaos, we introduce for this scope the quasi-probability distribution behind the out-of-time-ordered correlator (OTOC). We calculate the cumulative nonclassicality of this distribution, which has already been shown to outperform the simple use of OTOC as a probe to distinguish between integrable and nonintegrable Hamiltonians. To provide a thorough comparative analysis, we contrast the behavior of the nonclassicality with entanglement measures, such as the tripartite mutual information of the Hamiltonian as well as the entanglement entropy. We find that systems whose initial states would lie in the "sea of chaos" in the classical kicked-top model, exhibit, as they evolve in time, characteristics associated with chaotic behavior and entanglement production in closed quantum systems. We corroborate this indication by capturing it with this novel OTOC-based measure.
We explore the dynamics of entanglement in classically chaotic systems by considering a multiqubit system that behaves collectively as a spin system obeying the dynamics of the quantum kicked top. In the classical limit, the kicked top exhibits both regular and chaotic dynamics depending on the strength of the chaoticity parameter kappa in the Hamiltonian. We show that the entanglement of the multiqubit system, considered for both the bipartite and the pairwise entanglement, yields a signature of quantum chaos. Whereas bipartite entanglement is enhanced in the chaotic region, pairwise entanglement is suppressed. Furthermore, we define a time-averaged entangling power and show that this entangling power changes markedly as kappa moves the system from being predominantly regular to being predominantly chaotic, thus sharply identifying the edge of chaos. When this entangling power is averaged over all states, it yields a signature of global chaos. The qualitative behavior of this global entangling power is similar to that of the classical Lyapunov exponent.
The problem ``what is the quantum signature of a classically chaotic system'' is studied for the periodically kicked top. We find that the quantum variances initially grow exponentially if the corresponding classical description is chaotic. The rate of growth is connected to the corresponding classical Jacobi matrix and, thereby, to the local, classical, transient expansion rate. These connections were recently established in an analysis of the kicked pendulum for the correspondence between quantum Husimi-O'Connell-Wigner distributions and classical Gaussian ensembles. Here, we present closely related results for the kicked top by using generalized coherent states. An explanation is given for why this quantum signature of classical chaos was missed in earlier studies of the kicked top.
The double kicked top (DKT) is studied as an extension of the standard quantum kicked top (QKT) model. The model allows us to study the transition from time-reversal symmetric to broken time-reversal symmetric dynamics. A transformation in the kick strength parameter space (k,k^{'})→(k_{r},k_{θ}) reveals interesting features. The transformed kicked strength parameter k_{r} drives a higher growth of chaos and is equivalent to the standard QKT, whereas k_{θ} leads to a weaker growth. Fixed points and their stability are analyzed both analytically and computationally by using the largest Lyapunov exponent and the Kolmogorov-Sinai entropy. Exact solutions are obtained for 2- to 4-qubit version of the DKT, including eigenvalues, eigenvectors, and entanglement dynamics. Criteria for periodicity of the entanglement dynamics is obtained. Quantum correlations are investigated in both deep quantum and semiclassical regime. Signatures of phase-space structure are numerically shown in the long-time averages of the quantum correlations. The homoclinic point and its associated state are also investigated within the semiclassical regime. Our model can be realised experimentally as an extension of the standard QKT.
We study the dynamics of a quantum spin ensemble controlled by trains of ultrashort pulses. To model disturbances of the kicks, we consider that the spins are submitted to different kick trains which follow regular, random, stochastic, or chaotic dynamical processes. These are described by dynamical systems on a torus. We study the quantum decoherence and the population relaxation of the spin ensemble induced by these classical dynamical processes disturbing the kick trains. For chaotic kick trains we show that the decoherence and the relaxation processes exhibit a signature of chaos directly related to the Lyapunov exponents of the dynamical system. This signature is a horizon of coherence, i.e., a preliminary duration without decoherence followed by a rapid decoherence process.