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arXiv:0707.2912 · Journal of Physics A Mathematical and Theoretical

Frictional quantum decoherence

Bruno Bellomo, Stephen Mark Barnett, John Jeffers

Published Jul 19, 2007• 7 citations
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Abstract

The dynamics associated with a measurement-based master equation for quantum Brownian motion are investigated. A scheme for obtaining time evolution from general initial conditions is derived. This is applied to analyze dissipation and decoherence in the evolution of both a Gaussian and a Schrödinger cat initial state. Dependence on the diffusive terms present in the master equation is discussed with reference to both the coordinate and momentum representations.

DOI
10.1088/1751-8113/40/31/019

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The dynamics associated with a measurement-based master equation for quantum Brownian motion are investigated. A scheme for obtaining time evolution from general initial conditions is derived. This is applied to analyze dissipation and decoherence in the evolution of both a Gaussian and a Schrödinger cat initial state. Dependence on the diffusive terms present in the master equation is discussed with reference to both the coordinate and momentum representations.
We analyze the evolution of a quantum Brownian particle starting from an initial state that contains correlations between this system and its environment. Using a path-integral approach, we obtain a master equation for the reduced density matrix of the system finding relatively simple expressions for its time-dependent coefficients. We examine the evolution of delocalized initial states (Schr\"odinger cats) investigating the effectiveness of the decoherence process. Analytic results are obtained for an Ohmic environment (Drude's model) at zero temperature.
In the present work, I identify the pointer states of quantum Brownian motion as rotated Gaussian states in phase space. By means of a particular piecewise deterministic unraveling, the pointer state unraveling, I determine the trajectories of the position and momentum expectation of the pointer states. They turn out to be described by a diffusion process in phase space and, in the semiclassical limit, they turn into the Langevin equation of classical Brownian motion. For collisional decoherence, the non-dissipative limit of the quantum linear Boltzmann equation, which describes the dynamics of a marker particle in an ideal gaseous environment, I examine the width of the pointer states and thereby extend previous results to more than one dimension. Crucially, it follows that pointer states do not exist for all parameter values in that case. By expanding the Lindblad operators of the quantum linear Boltzmann equation, I derive a new Lindblad master equation, dissipative collisional decoherence, which shows a decoherence behavior similar to that of collisional decoherence and also incorporates frictional as well as diffusion effects. Moreover this equation exhibits soliton-like solutions as candidates for the pointer states.
Decoherence and dissipation in quantum systems has been studied extensively in the context of Quantum Brownian Motion. Effective decoherence in coarse grained quantum systems has been a central issue in recent efforts by Zurek and by Hartle and Gell-Mann to address the Quantum Measurement Problem. Although these models can yield very general classical phenomenology, they are incapable of reproducing relevant characteristics expected of a local environment on a quantum system, such as the characteristic dependence of decoherence on environment spatial correlations. I discuss the characteristics of Quantum Brownian Motion in a local environment by examining aspects of first principle calculations and by the construction of phenomenological models. Effective quantum Langevin equations and master equations are presented in a variety of representations. Comparisons are made with standard results such as the Caldeira-Leggett master equation.
Much of the discussion of decoherence has been in terms of a particle moving in one dimension that is placed in an initial superposition state (a Schrödinger "cat" state) corresponding to two widely separated wave packets. Decoherence refers to the destruction of the interference term in the quantum probability function. Here, we stress that a quantitative measure of decoherence depends not only on the specific system being studied but also on whether one is considering coordinate, momentum or phase space. We show that this is best illustrated by considering Wigner phase space where the measure is again different. Analytic results for the time development of the Wigner distribution function for a two-Gaussian Schrodinger "cat" state have been obtained in the high-temperature limit (where decoherence can occur even for negligible dissipation) which facilitates a simple demonstration of our remarks.