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arXiv:quant-ph/0301053 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Exact solution of the Hu-Paz-Zhang master equation

G. W. Ford, Robert F. O’Connell

Published Oct 24, 2001• 143 citations

Abstract

The Hu-Paz-Zhang equation is a master equation for an oscillator coupled to a linear passive bath. It is exact within the assumption that the oscillator and bath are initially uncoupled. Here an exact general solution is obtained in the form of an expression for the Wigner function at time t in terms of the initial Wigner function. The result is applied to the motion of a Gaussian wave packet and to that of a pair of such wave packets. A serious divergence arising from the assumption of an initially uncoupled state is found to be due to the zero-point oscillations of the bath and not removed in a cutoff model. As a consequence, worthwhile results for the equation can only be obtained in the high temperature limit, where zero-point oscillations are neglected. In that limit closed form expressions for wave packet spreading and attenuation of coherence are obtained. These results agree within a numerical factor with those appearing in the literature, which apply for the case of a particle at zero temperature that is suddenly coupled to a bath at high temperature. On the other hand very different results are obtained for the physically consistent case in which the initial particle temperature is arranged to coincide with that of the bath.

DOI: 10.1103/physrevd.64.105020

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The Hu-Paz-Zhang equation is a master equation for an oscillator coupled to a linear passive bath. It is exact within the assumption that the oscillator and bath are initially uncoupled. Here an exact general solution is obtained in the form of an expression for the Wigner function at time t in terms of the initial Wigner function. The result is applied to the motion of a Gaussian wave packet and to that of a pair of such wave packets. A serious divergence arising from the assumption of an initially uncoupled state is found to be due to the zero-point oscillations of the bath and not removed in a cutoff model. As a consequence, worthwhile results for the equation can only be obtained in the high temperature limit, where zero-point oscillations are neglected. In that limit closed form expressions for wave packet spreading and attenuation of coherence are obtained. These results agree within a numerical factor with those appearing in the literature, which apply for the case of a particle at zero temperature that is suddenly coupled to a bath at high temperature. On the other hand very different results are obtained for the physically consistent case in which the initial particle temperature is arranged to coincide with that of the bath.

Time evolution of a harmonic oscillator linearly coupled to a heat bath is compared for three classes of initial states for the bath modes-grand canonical ensemble, number states, and coherent states. It is shown that for a wide class of number states the behavior of the oscillator is similar to the case of the equilibrium bath. If the bath modes are initially in coherent states, then the variances of the oscillator coordinate and momentum, as well as its entanglement to the bath, asymptotically approach the same values as for the oscillator at zero temperature and the average coordinate and momentum show a Brownian-like behavior. We derive an exact master equation for the characteristic function of the oscillator valid for arbitrary factorized initial conditions. In the case of the equilibrium bath this equation reduces to an equation of the Hu-Paz-Zhang type, while for the coherent states bath it leads to an exact stochastic master equation with a multiplicative noise.

We derive the Hu-Paz-Zhang master equation for a Brownian particle linearly coupled to a bath of harmonic oscillators on the plane with spatial noncommutativity. The results obtained are exact to all orders in the noncommutative parameter. As a by-product we derive some miscellaneous results such as the equilibrium Wigner distribution for the reservoir of noncommutative oscillators, the weak coupling limit of the master equation and a set of sufficient conditions for strict purity decrease of the Brownian particle. Finally, we consider a high-temperature Ohmic model and obtain an estimate for the time scale of the transition from noncommutative to ordinary quantum mechanics. This scale is considerably smaller than the decoherence scale.

Hu, Paz, and Zhang [B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D 45, 2843 (1992)] have derived an exact master equation for quantum Brownian motion in a general environment via path integral techniques. Their master equation provides a very useful tool to study the decoherence of a quantum system due to the interaction with its environment. In this paper, we give an alternative and elementary derivation of the Hu-Paz-Zhang master equation, which involves tracing the evolution equation for the Wigner function. We also discuss the master equation in some special cases. \textcopyright{} 1996 The American Physical Society.

We investigate the exact master equation of Hu, Paz, and Zhang for a quantum harmonic oscillator at zero temperature with a Lorentz-Drude type Ohmic spectral density. This master equation plays an important role in the study of quantum Brownian motion and in various applications. In this paper, we give an analytical evaluation of the coefficients of this non-Markovian master equation without Lindblad form, which allows us to investigate consistencies of the solutions, the positivity of the stationary density operator, and the boundaries of the model's parameters.

Abstract After replacing the reduced density matrix of a quantum harmonic oscillator linearly coupled to a thermal bath by its ‘characteristic function’ (a non-negative definite function defined on the Heisenberg-Weyl group), the master equation of the reduced density matrix can be transformed into a first-order linear partial differential equation of the characteristic function. We derive stationary solutions for the Caldeira–Leggett (CL) master equation as well as for the late-time Hu–Paz–Zhang master equation, and examine the positivity problem for these solutions. In order to illustrate to what extent classical properties of quantum states of the damping oscillator emerge dynamically due to coupling to the environment, we discuss the problem of how and under what conditions a steady state predicted by the CL master equation could be expressed as a statistical mixture of Gaussian packets having a uniform position variance. Similar studies are carried out for the solutions of a Lindblad-form generalization of the CL master equation, where the positivity problem is certainly absent. Our results show that in the low-temperature regime the steady states predicted by the Lindblad-type master equation may depend on the damping strength of the bath and there are cases which may not show Gaussian-packet decomposition.