Parameter scaling in a novel measure of quantum-classical difference for decohering chaotic systems (original) (raw)
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Parameter Scaling In the Decoherent Quantum-Classical Transition for Chaotic Systems
Physical review letters, 2003
The quantum to classical transition has been shown to depend on a number of parameters. Key among these are a scale length for the action,h, a measure of the coupling between a system and its environment, D, and, for chaotic systems, the classical Lyapunov exponent, λ. We propose computing a measure, reflecting the proximity of quantum and classical evolutions, as a multivariate function of (h, λ, D) and searching for transformations that collapse this hyper-surface into a function of a composite parameter ζ =h α λ β D γ . We report results for the quantum Cat Map, showing extremely accurate scaling behavior over a wide range of parameters and suggest that, in general, the technique may be effective in constructing universality classes in this transition.
Quantum-classical comparison in chaotic systems
Physical Review E, 1996
We introduce a characteristic time of a classical chaotic dynamics, represented by the coherence time of the local maximum expansion direction. For a quantum system whose classical limit follows the above chaotic dynamics, the ratio between this time and the decorrelation time ͑of the order of the reciprocal of the maximum Liapunov exponent͒ rules the ratio between nonclassical ͑Moyal͒ and classical ͑Liouville͒ terms in the evolution of the density matrix. We show that such a ratio does not provide a complete criterion for quantumclassical correspondence.
Exponential Divergence and Long Time Relaxation in Chaotic Quantum Dynamics
Physical Review Letters, 1996
Phase space representations of the dynamics of the quantal and classical cat map are used to explore quantum--classical correspondence in a K-system: as hbarto0\hbar \to 0hbarto0, 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.
Semiclassics of the Chaotic Quantum-Classical Transition
Physical Review E, 2007
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 novel 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.
Decoherence and the rate of entropy production in chaotic quantum systems
Physical Review Letters, 2000
We show that for an open quantum system which is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators) the rate of entropy production has, as a function of time, two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system-environment coupling strength). For longer times (but before equilibration) there is a regime where the entropy production rate is fixed by the Lyapunov exponent. The nature of the transition time between both regimes is investigated.
Nonmonotonicity in the Quantum-Classical Transition: Chaos Induced by Quantum Effects
Physical Review Letters, 2008
The transition from classical to quantum behavior for chaotic systems is understood to be accompanied by the suppression of chaotic effects as the relative size ofh is increased. We show evidence to the contrary in the behavior of the quantum trajectory dynamics of a dissipative quantum chaotic system, the double-well Duffing oscillator. The classical limit in the case considered has regular behavior, but as the effectiveh is increased we see chaotic behavior. This chaos then disappears deeper into the quantum regime, which means that the quantum-classical transition in this case is non-monotonic inh.
The semiclassical regime of the chaotic quantum-classical transition
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 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.
Quantum chaos in open systems: a quantum state diffusion analysis
Journal of Physics A: …, 1996
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.
1993
Using the decoherence formalism of Gell-Mann and Hartle, a quantum system is found which is the equivalent of the classical chaotic Duffing oscillator. The similarities and the differences from the classical oscillator are examined; in particular, a new concept of quantum maps is introduced, and alterations in the classical strange attractor due to the presence of scale- dependent quantum effects are studied. Classical quantities such as the Lyapunov exponents and fractal dimension are examined, and quantum analogs are suggested. These results are generalized into a framework for quantum dissipative chaos, and there is a brief discussion of other work in this area.