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Quantum Chaos in Open Systems: A Quantum State Diffusion Analysis

Todd A. Brun, Ian Colin Percival, Rüdiger Schack

Published Sep 26, 1995• 54 citations
OpenAlex

Abstract

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

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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
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.
In this work the decoherence formalism of quantum mechanics is explored and applied to a number of interesting problems in quantum physics. The boundary between quantum and classical physics is examined, and demonstration made that quantum histories corresponding to classical equations of motion become more probable for a broad class of models, including linear and nonlinear models of Brownian motion. The link between noise, dissipation, and decoherence is studied. This work is then applied to systems which classically exhibit dissipative chaotic dynamics. A theory is explicated for treating these systems, and the ideas are applied to a particular model of the forced, damped Duffing oscillator, which is chaotic for certain parameter values. Differences between classical and quantum chaos are examined, particularly differences arising in the structure of fractal strange attractors, and the conceptual difficulties in framing standard notions of chaos in a quantum system. A brief discussion of previous work on quantum chaos is included, and the differences between Hamiltonian and dissipative chaos pointed out; a somewhat different interpretation of quantum chaos from the standard one is suggested. A class of histories for quantum systems, in phase space rather than configuration space, is studied. Different ways of representing projections in phase space are discussed, and expressions for the probability of phase space histories are derived; conditions for such histories to decohere are also estimated in the semiclassical limit.
In a wide-open quantum system, the effect of the Hamiltonian is negligible by comparison with the effect of the environment. For open systems, this is the opposite limit to closed or isolated systems. The quantum state diffusion model provides equations for the localization or reduction of quantum states of wide-open systems. The ensemble localization of an operator is defined, and it is proved that the rate of selflocalization of a selfadjoint environment operator towards one of its eigenstates is no less than 2. A bound is also obtained for the rate of selflocalization of some non-selfadjoint operators, which localize to minimum indeterminacy wave packets. The theory is presented for quasiclassical systems. For a sufficient number of independent environment operators, the states localize asymptotically to wave packets with Heisenberg indeterminacy products close to the minimum, which look to classical eyes like phase space points. To zeroth order in h(cross), the time-dependent WKB theory of quantum state diffusion due to a single operator shows localization or reduction within and between fixed classical sheets or Lagrangian manifolds. To first order, the sheets themselves diffuse. The rate of localization in an ensemble is determined by commutation terms with either sign and by correlation terms which always increase the localization. For the quasiclassical case the latter dominate, and this leads to a purely classical theory of localization, with a picture based on the diffusion of phase space densities. This means that state diffusion dynamics, like Hamiltonian dynamics, has a purely classical form, in which Planck's constant plays no role.
Quantum chaos---the study of quantized nonintegrable Hamiltonian systems---is an extremely well-developed and sophisticated field. By contrast, very little work has been done in looking at quantum versions of systems which classically exhibit {\\it dissipative} chaos. Using the decoherence formalism of Gell-Mann and Hartle, I find a quantum mechanical analog of one such system, the forced damped Duffing oscillator. I demonstrate the classical limit of the system, and discuss its decoherent histories. I show that using decoherent histories, one can define not only the quantum map of an entire density operator, but can find an analog to the Poincar\\'e map of the individual trajectory. Finally, I argue the usefulness of this model as an example of quantum dissipative chaos, as well as of a practical application of the decoherence formalism to an interesting problem.
The quantum-classical correspondence for dynamics of the nonlinear classically chaotic systems is analysed. The problem of quantum chaos consists of two parts: the quasiclassical quantisation of the chaotic systems and attempts to understand the classical chaos in terms of quantum mechanics. The first question has been partially solved by the Gutzwiller semiclassical trace formula for the eigenvalues of chaotic systems, while the classical chaos may be derived from quantum equations only introducing the decoherence process due to interaction with system's environment or intermediate frequent measurement. We may conclude that continuously observable quasiclassical system evolves essentially classically-like.
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.