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Communications in Theoretical Physics

Decoherence, Orbit, and Classical Linear Frequency Entropy: Quantum-Classical Correspondence

Shihui Zhang, Jie Quan-Lin, Wang Zhuo

Published Aug 1, 2009• 1 citation
Publisher

Abstract

The decoherence process is analyzed for an open quantum system that is classically chaotic, with a classical linear frequency entropy developed to measure the stability of classical motion. Investigation shows that the decoherence measured by the rate of quantum linear entropy production varies significantly with both the underlying classical orbits and the classical linear frequency entropy. Such correspondence is also supported by the further investigation on the Loschmidt Echo.

DOI
10.1088/0253-6102/52/2/06

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The decoherence process is analyzed for an open quantum system that is classically chaotic, with a classical linear frequency entropy developed to measure the stability of classical motion. Investigation shows that the decoherence measured by the rate of quantum linear entropy production varies significantly with both the underlying classical orbits and the classical linear frequency entropy. Such correspondence is also supported by the further investigation on the Loschmidt Echo.
We study the decoherence process for an open quantum system that is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators). We carefully analyze the time dependence of the rate of entropy production showing that it has 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) it is fixed by dynamical properties of the system (and is related to the Lyapunov exponent). The nature of the transition time between both regimes is investigated and the issue of quantum to classical correspondence is addressed. Finally, the impact of the interaction with the environment on coherent tunneling is analyzed.
We discuss some recent results concerning the decoherence in controlled quantum open systems within the mathematical setting corresponding to motion reversal experiments (the Loschmidt echo).We compare the case of randomly chosen sequence of unitary dynamical maps with the case of a constant dynamics corresponding to a classically chaotic evolution.The interplay between chaos and decoherence is illustrated by the new numerical results on the quantum Arnold cat map perturbed by a measurement process.Open problems related to the simple operational characterization of the decoherence strength are discussed.
Decoherence causes entropy increase that can be quantified using, e.g., the purity sigma=Trrho(2). When the Hamiltonian of a quantum system is perturbed, its sensitivity to such perturbation can be measured by the Loschmidt echo M(t). It is given by the squared overlap between the perturbed and unperturbed state. We describe the relation between the temporal behavior of sigma(t) and the average Mmacr;(t). In this way we show that the decay of the Loschmidt echo can be analyzed using tools developed in the study of decoherence. In particular, for systems with a classically chaotic Hamiltonian the decay of sigma and Mmacr; has a regime where it is dominated by the Lyapunov exponents.
This paper is devoted to analyzing the connections between entanglement dynamics and classical trajectories with a model of coupled rotor.In the numerical simulation of quantum evolution and the frequency spectrum analysis of classical orbit,we find that entanglement production depends on classical orbit and such phenomenon is closely related to the power spectra of classical trajectories.Classical power spectrum with a more complicated structure corresponds to faster entanglement.With the classical power spectrum,we introduce a new quantity,i.e.,the frequency entropy,to measure the stability of classical orbit.The results show that there is good correspondence between the classical frequency entropies and the maximum von Neumann entropies.