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arXiv:1808.07741 · Physical review. E

Quantum signatures of chaos, thermalization, and tunneling in the exactly solvable few-body kicked top

Shruti Dogra, Vaibhav Madhok, Arul Lakshminarayan

Published Jun 19, 2019• 28 citations
arXivHTMLPDF

Abstract

Exactly solvable models that exhibit quantum signatures of classical chaos are both rare as well as important-more so in view of the fact that the mechanisms for ergodic behavior and thermalization in isolated quantum systems and its connections to nonintegrability are under active investigation. In this work, we study quantum systems of few qubits collectively modeled as a kicked top, a textbook example of quantum chaos. In particular, we show that the three- and four-qubit cases are exactly solvable and yet, interestingly, can display signatures of ergodicity and thermalization. Deriving analytical expressions for entanglement entropy and concurrence, we see agreement in certain parameter regimes between long-time average values and ensemble averages of random states with permutation symmetry. Comparing with results using the data of a recent transmons-based experiment realizing the three-qubit case, we find agreement for short times, including a peculiar steplike behavior in correlations of some states. In the case of four qubits we point to a precursor of dynamical tunneling between what in the classical limit would be two stable islands. Numerical results for larger number of qubits show the emergence of the classical limit including signatures of a bifurcation.

DOI
10.1103/physreve.99.062217

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Exactly solvable models that exhibit quantum signatures of classical chaos are both rare as well as important-more so in view of the fact that the mechanisms for ergodic behavior and thermalization in isolated quantum systems and its connections to nonintegrability are under active investigation. In this work, we study quantum systems of few qubits collectively modeled as a kicked top, a textbook example of quantum chaos. In particular, we show that the three- and four-qubit cases are exactly solvable and yet, interestingly, can display signatures of ergodicity and thermalization. Deriving analytical expressions for entanglement entropy and concurrence, we see agreement in certain parameter regimes between long-time average values and ensemble averages of random states with permutation symmetry. Comparing with results using the data of a recent transmons-based experiment realizing the three-qubit case, we find agreement for short times, including a peculiar steplike behavior in correlations of some states. In the case of four qubits we point to a precursor of dynamical tunneling between what in the classical limit would be two stable islands. Numerical results for larger number of qubits show the emergence of the classical limit including signatures of a bifurcation.
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.
Quantum correlations reflect the quantumness of a system and are useful resources for quantum information and computational processes. Measures of quantum correlations do not have a classical analog and yet are influenced by classical dynamics. In this work, by modeling the quantum kicked top as a multiqubit system, the effect of classical bifurcations on measures of quantum correlations such as the quantum discord, geometric discord, and Meyer and Wallach Q measure is studied. The quantum correlation measures change rapidly in the vicinity of a classical bifurcation point. If the classical system is largely chaotic, time averages of the correlation measures are in good agreement with the values obtained by considering the appropriate random matrix ensembles. The quantum correlations scale with the total spin of the system, representing its semiclassical limit. In the vicinity of trivial fixed points of the kicked top, the scaling function decays as a power law. In the chaotic limit, for large total spin, quantum correlations saturate to a constant, which we obtain analytically, based on random matrix theory, for the Q measure. We also suggest that it can have experimental consequences.
Quantum kicked top is a fundamental model for time-dependent, chaotic Hamiltonian system and has been realized in experiments as well. As the quantum kicked top can be represented as a system of qubits, it is also popular as a test bed for the study of measures of quantum correlations, such as entanglement, quantum discord, and other multipartite entanglement measures. Further, earlier studies on kicked top have led to a broad understanding of how these measures are affected by the classical dynamical features. In this work, relying on the invariance of quantum correlation measures under local unitary transformations, it is shown exactly these measures display periodic behavior either as a function of time or as a function of the chaos parameter in this system. As the kicked top has been experimentally realized using cold atoms as well as superconducting qubits, it is pointed out that these periodicities must be factored in while choosing experimental parameters so that repetitions can be avoided.
We identify signatures of chaos in the dynamics of discord in a multiqubit system collectively modelled as a quantum kicked top. The evolution of discord between any two qubits is quasiperiodic in regular regions, while in chaotic regions the quasiperiodicity is lost. As the initial wave function is varied from the regular regions to the chaotic sea, a contour plot of the time-averaged discord remarkably reproduces the structures of the classical stroboscopic map. We also find surprisingly opposite behavior of two-qubit discord versus entanglement of the two qubits as measured by the concurrence. Our results provide evidence of signatures of chaos in dynamically generated discord.
We analyze the interplay of chaos, entanglement, and decoherence in a system of qubits whose collective behavior is that of a quantum kicked top. The dynamical entanglement between a single qubit and the rest can be calculated from the mean of the collective spin operators. This allows the possibility of efficiently measuring entanglement dynamics in an experimental setting. We consider a deeply quantum regime and show that signatures of chaos are present in the dynamical entanglement for parameters accessible in an experiment that we propose using cold atoms. The evolution of the entanglement depends on the support of the initial state on regular versus chaotic Floquet eigenstates, whose phase-space distributions are concentrated on the corresponding regular or chaotic eigenstructures. We include the effect of decoherence via a realistic model and show that the signatures of chaos in the entanglement dynamics persist in the presence of decoherence. In addition, the classical chaos affects the decoherence rate itself.
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.