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arXiv:quant-ph/0309214 · Physical Review A

Chaos and quantum-classical correspondence via phase-space distribution functions

Jiangbin Gong, Paul W. Brumer

Published Dec 4, 2003• 20 citations
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Abstract

Quantum-classical correspondence in conservative chaotic Hamiltonian systems is examined using a uniform structure measure for quantal and classical phase-space distribution functions. The similarities and differences between quantum and classical time-evolving distribution functions are exposed by both analytical and numerical means. The quantum-classical correspondence of low-order statistical moments is also studied. The results shed considerable light on quantum-classical correspondence.

DOI
10.1103/physreva.68.062103

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Quantum-classical correspondence in conservative chaotic Hamiltonian systems is examined using a uniform structure measure for quantal and classical phase-space distribution functions. The similarities and differences between quantum and classical time-evolving distribution functions are exposed by both analytical and numerical means. The quantum-classical correspondence of low-order statistical moments is also studied. The results shed considerable light on quantum-classical correspondence.
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Quantum-classical correspondence is affirmed via performing Wigner function and a classical-quantum chaotic system containing random variables. The classical-quantum system is transformed into a Kolmogorov model for force and energy analysis. Combining different forces, the system is divided into two categories: conservative and non-conservative, revealing the mechanical characteristic of the classical-quantum system. The Casimir power, an analysis tool, is employed to find the key factors governing the orbital trajectory and the energy cycle of the system. Detailed analyses using the Casimir power and an energy transformation uncover the causes of the different dynamic behaviors, especially chaos. For the corresponding classical Hamiltonian system when Planck’s constant ħ → 0, the supremum bound of the system is derived analytically. Difference between the classical-quantum system and the classical Hamiltonian system is displayed through trajectories and energies. Quantum-classical correspondences are further demonstrated by comparing phase portrait, kinetic, potential and Casimir energies of the two systems.
Phase-space representations of the dynamics of the quantal and classical cat map are used to explore quantum-classical correspondence in a KKK system: As $\ensuremath{\Elzxh}\ensuremath{\rightarrow}0$, the classical chaotic behavior is shown to emerge smoothly and exactly. The quantum dynamics near the classical limit displays both exponential separation of adjacent distributions and long time relaxation, two characteristic features of classical chaotic motion.
We discuss the connection between the out-of-time-ordered correlator and the number of harmonics of the phase-space Wigner distribution function.In particular, we show that both quantities grow exponentially for chaotic dynamics, with a rate determined by the largest Lyapunov exponent of the underlying classical dynamics, and algebraically-linearly or quadratically-for integrable dynamics.It is then possible to use such quantities to detect in the time domain the integrability-to-chaos crossover in many-body quantum systems.
A direct classical analog of the quantum dynamics of intrinsic decoherence in Hamiltonian systems, characterized by the time dependence of the linear entropy of the reduced density operator, is introduced. The similarities and differences between the classical and quantum decoherence dynamics of an initial quantum state are exposed using both analytical and computational results. In particular, the classicality of early-time intrinsic decoherence dynamics is explored analytically using a second-order perturbative treatment, and an interesting connection between decoherence rates and the stability nature of classical trajectories is revealed in a simple approximate classical theory of intrinsic decoherence dynamics. The results offer deeper insights into decoherence, dynamics of quantum entanglement, and quantum chaos.
The quantum dynamics of a classically chaotic model are studied in the approach to the macroscopic limit. The quantum predictions are compared and contrasted with the classical predictions of both Newtonian and Liouville mechanics. The time-domain scaling of the optimal quantum-classical correspondence is analyzed in detail in the case of both classical theories. In both cases the correspondence for observable quantities is shown to break down on a time-scale that increases very slowly (logarithmically) with increasing system size. In the case of quantum-Liouville correspondence such a short time-scale does not imply a breakdown of correspondence since the largest quantum-Liouville differences reached on this time-scale decrease rapidly (as an inverse power) with increasing system size. Therefore the statistical properties of chaotic dynamics are, as expected, well described by quantum theory in the macroscopic limit. In contrast, the largest quantum-Newtonian differences reached on the log time-scale actually increase in proportion to the system size. Since the invariance properties of the Hamiltonian impose functional constraints on the time-varying chaotic coordinates, it is possible to show, moreover, that if the quantum predictions are believed to describe the coordinates of individual chaotic systems, then they also predict macroscopic violations of any kinematic or dynamic constants of the motion. These results for chaotic systems indicate that a valid description of the time-varying properties of individual macroscopic bodies is not available within the standard interpretive framework of quantum theory.