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Modern Physics Letters B

SOLVING MASTER EQUATION FOR CALDEIRA–LEGGETT MODEL BY VIRTUE OF THE THERMO ENTANGLED STATE REPRESENTATION AND THE IWOP TECHNIQUE

Qian-Fan Chen, QIAN YE, Zhihua Li, Hong-Yi Fan

Published May 8, 2012• 5 citations
Publisher

Abstract

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.

DOI
10.1142/s0217984912500856

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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.
Open quantum system, namely system-reservoir model, is described by a master equation of density operator. For example, the Caldeira-Leggett eqaution describes dissipative phenomenon of solid physics. Although some efforts have been made to derive the exact expression of this master equation, so far as we know, it has not been reported in the literature. The purpose of this paper is to provide a new approach to solving the Caldeira-Leggett equation, via this approach the explicit integral-form expression of ρ(t) can be obtained. The main point of this approach is to convert equation of density operator into an equation of density state vector, and then project density state vector into thermo entangled state representation and convert it into wave function by using the technique of integration within an ordered product of operators. Thus the master equation for Caldeira-Leggett model is converted into an differential equation of wave function. Wave function is also a function. The wave function can be obtained via the approach to solving the differential equation in mathematics. It can be converted into a density state vector and density operator. Using the technique of integration within an ordered product of operators again, the integra-form solution of the Caldeira-Leggett equation is obtained.
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.
We solve various master equations to obtain density operators' infinite operator-sum representation via a new approach, i.e., by virtue of the thermo-entangled state representation that has a fictitious mode as a counterpart mode of the system mode. The corresponding Kraus operators from the point of view of quantum channel are derived, whose normalization conditions are proved. Miscellaneous characters possessed by different quantum channels, such as decoherence, phase diffusion, damping, and amplification, can be shown explicitly in the entangled state representation of the density operators. Squeezing transformation is applied to the density operator for decoherence to generate a master equation for describing the phase sensitive process. Partial trace method for deriving new density operators is also introduced. Throughout our discussion, the technique of integration within an ordered product (IWOP) of operators is fully used.
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.
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.