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IntechOpen eBooks

Introductory Chapter: Dynamical Symmetries and Quantum Chaos

Paul Bracken

Published Mar 11, 2020• 1 citation
Publisher
DOI
10.5772/intechopen.88551

Related papers

Dynamical symmetries in quantum theory

Evelyn Aronson, I.A. Malkin, Vladimir I. Man’ko

•Jul 1, 1974•2 citations

Quantum Kicked Top: A Paradigmatic Model

Avadhut V. Purohit, Udaysinh T. Bhosale

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 $(2j+1)$-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.

quant-pharXiv:2604.12345•Apr 15, 2026•0 citations

Influence of dynamical Pauli effect and dynamical symmetry breaking on quantum chaos

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In this paper, we study the influence of quantum effects on chaotic dynamics, especially the influence of the Pauli effect and dynamical symmetry breaking on chaotic motions. We apply the semiquantal theory to the Sp(6) fermion symmetry model in nuclear collective motion. We demonstrate that quantum chaos appears when dynamical symmetry is broken. We further show that the dynamical Pauli effect enhances quantum chaos.

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In this paper, we study the influence of quantum effects to chaotic dynamics, especially the influence of Pauli effect and dynamical symmetry breaking to chaotic motions. We apply the semiquantal theory to the Sp(6) fermion symmetry model in nuclear collective motion. We demonstrate that quantum chaos appears when dynamical symmetry is broken. We further show that dynamical Pauli effect enhances quantum chaos.

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Quantum mechanics and dynamical symmetries

Victor Guillemin, Shlomo Sternberg

Colloquium Publications - American Mathematical Society/Colloquium Publications•Sep 15, 2006•0 citations
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.