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Journal of the Physical Society of Japan

Time Evolution of a Quantum System in Contact with a Nearly Gaussian-Markoffian Noise Bath

Yoshitaka Tanimura, Ryogo Kubo

Published Jan 15, 1989• 1,008 citations
Publisher

Abstract

A test system is assumed to interact with a heat bath consisting of harmonic oscillators or an equivalent bath with a proper frequency spectrum producing a Gaussian-Markoffian random perturbation. The effect of reaction of the test system to the bath is considered in the high temperature approximation. Elimination of the bath using the influence functional method of Feynman and Vernon yields a continuous fraction expression for the reduced density matrix of the test system. The result affords a basis to clarify the relationship between the stochastic and the dynamical approaches to treat the problem of partial destruction of quantum coherence of a system interacting with its environment.

DOI
10.1143/jpsj.58.101

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A test system is assumed to interact with a heat bath consisting of harmonic oscillators or an equivalent bath with a proper frequency spectrum producing a Gaussian-Markoffian random perturbation. The effect of reaction of the test system to the bath is considered in the high temperature approximation. Elimination of the bath using the influence functional method of Feynman and Vernon yields a continuous fraction expression for the reduced density matrix of the test system. The result affords a basis to clarify the relationship between the stochastic and the dynamical approaches to treat the problem of partial destruction of quantum coherence of a system interacting with its environment.
A general formulation is given for the second order optical process of a three-level system in contact with a heat bath consisting of harmonic oscillators with a proper frequency spectrum producing a Markoffian random perturbation. Elimination of the bath using the influence functional method of Feynman and Vernon yields a continuous fraction expression and the result affords a basis to clarify the relation between the stochastic and dynamical approaches to treat the partial destruction of the quantum coherence of the optical process. As an example, a three-level system with intermediate-state interaction is treated in some detail.
Two-time correlation functions of physical operators of a system interacting with a heat bath producing a Gaussian-Markoffian noise are treated with the use of Feynman-Vernon's influence functional formalism. The Laplace transforms of correlation functions are shown to be expressed in terms of continued fractions of relevant hyperoperators. The result is valid to all orders of the system-bath interaction. In the limitation of motional narrowing, it reduces to those obtained in conventional methods.
In many physical situations the behavior of a quantum system is affected by interaction with a larger environment. We develop, using the method of an influence functional, how to deduce the density matrix of the quantum system incorporating the effect of environment. After introducing the characterization of the environment by spectral weight, we first devise schemes to approximate the spectral weight, and then a perturbation method in field theory models, in order to approximately describe the environment. All of these approximate models may be classified as extended Ohmic models of dissipation whose differences are in the high frequency part. The quantum system we deal with in the present work is a general class of harmonic oscillators with an arbitrary time-dependent frequency. The late time behavior of the system is well described by an approximation that employs a localized friction in the dissipative part of the correlation function appearing in the influence functional. The density matrix of the quantum system is then determined in terms of a single classical solution obtained with the time-dependent frequency. With this one can compute the entropy, the energy distribution function, and other physical quantities of the system in a closed form. A specific application is made to the case of a periodically varying frequency. This dynamical system has a remarkable property when the environmental interaction is switched off: The effect of the parametric resonance gives rise to an exponential growth of the populated number in higher excitation levels, or particle production in field theory models. The effect of the environment is investigated for this dynamical system and it is demonstrated that there exists a critical strength of the friction for the parametric effect. In the model of a periodically oscillating field coupled to a system quantum field, it is verified that the parametric effect occurs in medium, with a somewhat diminished rate, if the relaxation time scale of the system field towards thermalization given by the friction term is larger than the time scale of the coherent parametric amplification. The effect persists until the back reaction against the periodic oscillation stops particle production. The resulting energy distribution of produced particles described by a universal function deviates from the thermal one, having an average energy that exponentially increases with time. The dynamical system driven by the parametric oscillator thus maintains and does not lose its quantum nature even in thermal bath. In the present work analytic formulas of how physical quantities behave at late times both in the high and in the low temperature regions are given, along with the results of a numerical computation displaying time evolution.
We study the effects of dissipation and decoherence induced on a harmonic oscillator by the coupling to a chaotic system with two degrees of freedom. Using the Feynman-Vernon approach and treating the chaotic system semiclassically, we show that the effects of the low-dimensional chaotic environment are in many ways similar to those produced by thermal baths. The classical correlation and response functions play important roles in both classical and quantum formulations. Our results are qualitatively similar to the high-temperature regime of the Caldeira-Leggett model.
We consider a model of environment-induced dissipationless decoherence of a quantum system where the system is coupled to the bath degrees of freedom via the system Hamiltonian itself. We solve exactly for the reduced density operator of the system for an arbitrary spectral density of the thermal bath and also write down an exact master equation in the Lindblad form. We compare and contrast the above results with those obtained by considering the system frequencies to be randomly modulated as in stochastic models. We observe that a coupling to the bath as above necessarily induces a Kerr-like coherent contribution in the reduced dynamics of the system. This Kerr-like term is a reflection of the quantum nature of the bath and cannot be obtained from stochastic models. For the special case of a harmonic oscillator we consider the influence of decoherence and its relation with phase diffusion. Our numerical results exhibit oscillations in the evolution of system variables which overall is a signature of the quantum nature of the environment.
A microscopic approach treating the quantum dissipation process presented by Sun and Yu (Phys. Rev. A49 (1994) 592; A51 (1995) 1845) is invoked to construct the wavefunction of the composite system—the model for a harmonic oscillator interacting with a many-oscillator bath under the rotating wave approximation. It shows the back-action of the system on the bath. In particular, the dynamic evolution of the wavefunction for the composite system maintains a factorized form in its wavefunction. In the limited temperature, the reduced density matrix for the system is also calculated to clarify the influence of Brownian motion on the system.