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arXiv:quant-ph/9812045 · Physical Review A

Coherence properties of the stochastic oscillator

Martti Havukainen, Stig Stenholm

Published Jul 1, 1999• 3 citations
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Abstract

An oscillator with stochastic frequency is discussed as a model for evaluating the quantum coherence properties of a physical system. It is found that the choice of jump statistics has to be considered with care if unphysical consequences are to be avoided. We investigate one such model, evaluate the damping it causes, the decoherence rate and the correlations it results in, and the properties of the state for asymptotically long times. Also the choice of initial state and its effect on the time evolution of the correlations are discussed.

DOI
10.1103/physreva.60.621

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An oscillator with stochastic frequency is discussed as a model for evaluating the quantum coherence properties of a physical system. It is found that the choice of jump statistics has to be considered with care if unphysical consequences are to be avoided. We investigate one such model, evaluate the damping it causes, the decoherence rate and the correlations it results in, and the properties of the state for asymptotically long times. Also the choice of initial state and its effect on the time evolution of the correlations are discussed.
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The phase damping induced in the cyclotron mode of a trapped electron by the coupling to the axial fluctuations is studied analytically. This system, described as a nonlinear oscillator stochastically driven in frequency, allows testing the generality of some elements present in the phenomenology of decoherence. In our approach, the reduced density matrix is obtained by performing a statistical average from the propagator for each noise realization. For short times, the decay of the coherences presents a nonexponential form, rooted in the non-Gaussian character of the stochastic driving. For large times, the decay becomes purely exponential, the rate showing a complex dependence on the difference between the Fock indices. As the populations do not change, the asymptotic state corresponds to a nonthermalized statistical mixture.
We address the dynamics of nonclassicality for a quantum system interacting with a noisy fluctuating environment described by a classical stochastic field. As a paradigmatic example, we consider a harmonic oscillator initially prepared in a maximally nonclassical state, e.g., a Fock number state or a Schr\"odinger-cat-like state, and then coupled to either a resonant or a nonresonant external field. Stochastic modeling allows us to describe the decoherence dynamics without resorting to approximated quantum master equations and to introduce non-Markovian effects in a controlled way. A detailed comparison among different nonclassicality criteria and a thorough analysis of the decoherence time reveal a rich phenomenology whose main features may be summarized as follows: (i) Classical memory effects increase the survival time of quantum coherence and (ii) a detuning between the natural frequency of the system and the central frequency of the classical field induces revivals of quantum coherence.
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We introduce the quantitative measures characterizing the rates of decoherence and thermalization of quantum systems. We study the time evolution of these measures in the case of a quantum harmonic oscillator whose relaxation is described in the framework of the standard master equation, for various initial states (coherent, `cat', squeezed and number). We establish the conditions under which the true decoherence measure can be approximated by the linear entropy 1-Tr 2 . We show that at low temperatures and for highly excited initial states the decoherence process consists of three distinct stages with quite different time scales. In particular, the `cat' states preserve 50% of the initial coherence for a long time interval which increases logarithmically with the increase of the initial energy.
The article examines the problem of physical interpretation of quantum decoherence, which is traditionally described within phenomenological models of interaction between a quantum system and its environment. It is shown that standard approaches effectively reproduce the dynamics of phase damping but do not provide an explanation of the physical origin of coherence loss. A temporal interpretation of decoherence is proposed, in which phase damping is considered as a consequence of disruption of time synchronization in an inhomogeneous temporal environment. The paper demonstrates that the proposed approach is formally consistent with the density matrix formalism and Lindblad-type equations, reproducing standard exponential decoherence laws. It is substantiated that stochastic and non-Markovian effects can be interpreted as manifestations of temporal correlations and slow temporal drift. The practical value of the research lies in the possibility of applying the proposed interpretation to analyze noise, memory effects, and phase dynamics stability in complex information and quantum systems.