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arXiv:quant-ph/9910004

Exact Solutions of the Caldeira-Leggett Master Equation: A Factorization Theorem For Decoherence

Shasanka Mohan Roy, Anu Venugopalan

Published Oct 1, 1999• 7 citations
arXivHTMLPDF

Abstract

Exact solutions of the Caldeira-Leggett Master equation for the reduced density matrix for a free particle and for a harmonic oscillator system coupled to a heat bath of oscillators are obtained for arbitrary initial conditions. The solutions prove that the Fourier transform of the density matrix at time t with respect to (x + x')/2, where x and x' are the initial and final coordinates, factorizes exactly into a part depending linearly on the initial density matrix and a part independent of it. The theorem yields the exact initial state dependence of the density operator at time t and its eventual diagonalization in the energy basis.

DOI
10.48550/arxiv.quant-ph/9910004

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Exact solutions of the Caldeira-Leggett Master equation for the reduced density matrix for a free particle and for a harmonic oscillator system coupled to a heat bath of oscillators are obtained for arbitrary initial conditions. The solutions prove that the Fourier transform of the density matrix at time t with respect to (x + x')/2, where x and x' are the initial and final coordinates, factorizes exactly into a part depending linearly on the initial density matrix and a part independent of it. The theorem yields the exact initial state dependence of the density operator at time t and its eventual diagonalization in the energy basis.
In the present work, the exact formal solution of the Caldeira–Leggett quantum master equation for the reduced density matrix of an oscillator is obtained by virtue of the thermo entangled state representation and the technique of integration within an ordered product (IWOP) of operators. Also, the solution for high temperature approximation is derived. When the initial state is a coherent state, it will evolve into a displaced-squeezed chaotic state (a mixed state) which shows that decoherence is involved in the Caldeira–Leggett equation due to the interactions between the system and environment.
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.
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.
We calculate the reduced density matrix for a system of coupled harmonic oscillators in a bosonic heat bath using the Born-Markov approximation and show that the expectation values of position and momentum evolve like classical quantities. We consider the cases where every oscillator is coupled to its own reservoir and where all oscillators are coupled to the same heat bath, pointing out the differences between these two models. In particular, for systems coupled to a common reservoir, we present an uncommon dissipation mechanism, which disappears, when the dynamics of the system are gouverned by the low frequency modes of the environment (Caldeira-Leggett limit). We also address the interesting phenomenon of reservoir-induced interactions.
A systematic procedure for deriving the master equation of a dissipative system is reported in the framework of stochastic description. For the Caldeira-Leggett model of the harmonic-oscillator bath, a detailed and elementary derivation of the bath-induced stochastic field is presented. The dynamics of the system is thereby fully described by a stochastic differential equation, and the desired master equation would be acquired with statistical averaging. It is shown that the existence of a closed-form master equation depends on the specificity of the system as well as the feature of the dissipation characterized by the spectral density function. For a dissipative harmonic oscillator it is observed that the correlation between the stochastic field due to the bath and the system can be decoupled, and the master equation naturally results. Such an equation possesses the Lindblad form in which time-dependent coefficients are determined by a set of integral equations. It is proved that the obtained master equation is equivalent to the well-known Hu-Paz-Zhang equation based on the path-integral technique. The procedure is also used to obtain the master equation of a dissipative harmonic oscillator in time-dependent fields.
Using the exact path integral solution for the damped harmonic oscillator it is shown that in general there does not exist an exact dissipative Liouville operator describing the dynamics of the oscillator for arbitrary initial bath preparations. Exact nonstationary Liouville operators can be found only for particular preparations. Three physically meaningful examples are examined. An exact master equation is derived for thermal initial conditions. Second, the Liouville operator governing the time evolution of equilibrium correlations is obtained. Third, factorizing initial conditions are studied. Additionally, one can show that there are approximate Liouville operators independent of the initial preparation describing the long-time dynamics under appropriate conditions. The general form of these approximate master equations is derived and the coefficients are determined for special cases of the bath spectral density including the Ohmic, Drude, and weak coupling cases. The connection with earlier work is discussed.