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arXiv:1806.06184 · Physical review. E

Periodicity of quantum correlations in the quantum kicked top

Udaysinh T. Bhosale, M. S. Santhanam

Published Nov 28, 2018• 31 citations
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Abstract

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.

DOI
10.1103/physreve.98.052228

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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.
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.
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.
We present an analysis of the entangling quantum kicked top focusing on the few qubit case and the initial condition dependence of the time-averaged entanglement S_{Q} for spin-coherent states. We show a very strong connection between the classical phase space and the initial condition dependence of S_{Q} even for the extreme case of two spin-1/2 qubits. This correlation is not related directly to chaos in the classical dynamics. We introduce a measure of the behavior of a classical trajectory which correlates far better with the entanglement and show that the maps of classical and quantum initial-condition dependence are both organized around the symmetry points of the Hamiltonian. We also show clear (quasi-)periodicity in entanglement as a function of number of kicks and of kick strength.
Quantum chaotic kicked top model is implemented experimentally in a two-qubit system comprising of a pair of spin-1/2 nuclei using nuclear magnetic resonance techniques. The essential nonlinear interaction was realized using indirect spin-spin coupling, while the linear kicks were realized using radio-frequency pulses. After a variable number of kicks, quantum state tomography was employed to reconstruct the single-qubit reduced density matrices, using which we could extract von Neumann entropies and Husimi distributions. These measures enabled the study of correspondence with classical phase space as well as probing distinct features of quantum chaos, such as symmetries and temporal periodicity in the two-qubit kicked top.
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.