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arXiv:quant-ph/0401174 · Chaos An Interdisciplinary Journal of Nonlinear Science

The semiclassical regime of the chaotic quantum-classical transition

Benjamin D. Greenbaum, Salman Habib, Kosuke Shizume, Bala Sundaram

Published Jul 20, 2005• 9 citations
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Abstract

An analysis of the semiclassical regime of the quantum-classical transition is given for open, bounded, one-dimensional chaotic dynamical systems. Environmental fluctuations-characteristic of all realistic dynamical systems-suppress the development of a fine structure in classical phase space and damp nonlocal contributions to the semiclassical Wigner function, which would otherwise invalidate the approximation. This dual regularization of the singular nature of the semiclassical limit is demonstrated by a numerical investigation of the chaotic Duffing oscillator.

DOI
10.1063/1.1979227

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An analysis of the semiclassical regime of the quantum-classical transition is given for open, bounded, one-dimensional chaotic dynamical systems. Environmental fluctuations-characteristic of all realistic dynamical systems-suppress the development of a fine structure in classical phase space and damp nonlocal contributions to the semiclassical Wigner function, which would otherwise invalidate the approximation. This dual regularization of the singular nature of the semiclassical limit is demonstrated by a numerical investigation of the chaotic Duffing oscillator.
We elucidate the basic physical mechanisms responsible for the quantum-classical transition in one-dimensional, bounded chaotic systems subject to unconditioned environmental interactions. We show that such a transition occurs due to the dual role of noise in regularizing the semiclassical Wigner function and averaging over fine structures in classical phase space. The results are interpreted in the context of applying recent advances in the theory of measurement and open systems to the semiclassical quantum regime. We use these methods to show how a local semiclassical picture is stabilized and can then be approximated by a classical distribution at later times. The general results are demonstrated explicitly via high-resolution numerical simulations of the quantum master equation for a chaotic Duffing oscillator.
We study the quantum dissipative Duffing oscillator across a range of system sizes and environmental couplings under varying semiclassical approximations. Using spatial complexity metrics based on Kullback-Leibler distances between phase-space attractors and temporal complexity metrics based on the Lyapunov exponent, we isolate the effect of the environment on quantum-classical differences. Moreover, we quantify the system sizes where quantum dynamics cannot be simulated using semiclassical or noise-added classical approximations. We find that a remarkable parametrically invariant meta-attractor emerges at a specific length scale and noise-added classical models deviate strongly from quantum dynamics below this scale. We also generalize the previous surprising result that classically regular orbits can have the greatest quantum-classical differences in the semiclassical regime and show that the dynamical growth of quantum-classical differences is not determined by the degree of classical chaos.
Except for the universe, all quantum systems are open, and according to quantum state diffusion theory, many systems localize to wavepackets in the neighbourhood of phase space points. This is due to decoherence from the interaction with the environment, and makes the quasiclassical limit of such systems both more realistic and simpler in many respects than the more familiar quasiclassical limit for closed systems. A linearized version of this theory leads to the correct classical dynamics in the macroscopic limit, even for nonlinear and chaotic systems. We apply the theory to the forced, damped Duffing oscillator, comparing the numerical results of the full and linearized equations, and argue that this can be used to make explicit calculations in the decoherent histories formalism of quantum mechanics.
The relationship between classically chaotic dynamics and the entanglement properties of the corresponding quantum system is examined in the semiclassical limit. Numerical results are computed for a modified kicked top, keeping the classical dynamics constant while investigating the entanglement for several versions of the corresponding quantum system characterized by a different value of the effective Planck constant ℏ eff. Our findings indicate that as ℏ eff → 0, the apparent signatures of classical chaos in the entanglement properties, such as characteristic oscillations in the time-dependence of the linear entropy, can also be obtained in the regular regime. These results suggest that entanglement is not a universal marker of chaotic dynamics of the corresponding classical system.
Except for the universe, all quantum systems are open, and according to quantum state diffusion theory, many systems localize to wave packets in the neighborhood of phase space points. This is due to decoherence from the interaction with the environment, and makes the quasiclassical limit of such systems both more realistic and simpler in many respects than the more familiar quasiclassical limit for closed systems. A linearized version of this theory leads to the correct classical dynamics in the macroscopic limit, even for nonlinear and chaotic systems. We apply the theory to the forced, damped Duffing oscillator, comparing the numerical results of the full and linearized equations, and argue that this can be used to make explicit calculations in the decoherent histories formalism of quantum mechanics. QMW Theory Preprint QMW-PH-95-33 1 1
We discuss some problems of dissipative chaos for open quantum systems in the framework of semiclassical and quantum distributions.For this goal, we propose a driven nonlinear oscillator with time-dependent coefficients, i.e. with time-dependent Kerr-nonlinearity and time-modulated driving field.This model showing both regular and chaotic dynamics in the classical limit is realized in several experimental schemes.Quantum dissipative chaos is analyzed on the base of numerical method of quantum trajectories.Three quantities are studied: the Wigner function of oscillatory mode from the point of view of quantum-assemble theory and both semiclassical Poincaré section and quantum Poincaré section calculated on a single quantum trajectory.The comparatively analysis of these distributions for various operational chaotic regimes of the models is performed, as well as scaling invariance in dissipative chaos and quantum interference effects assisted by chaos are discussed.