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arXiv:quant-ph/0301055 · Physical Review A

Quantum measurement and decoherence

G. W. Ford, John T. Lewis, Robert F. O’Connell

Published Aug 8, 2001• 76 citations
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Abstract

Distribution functions defined in accordance with the quantum theory of measurement are combined with results obtained from the quantum Langevin equation to discuss decoherence in quantum Brownian motion. Closed form expressions for wave packet spreading and the attenuation of coherence of a pair of wave packets are obtained. The results are exact within the context of linear passive dissipation. It is shown that, contrary to widely accepted current belief, decoherence can occur at high temperature in the absence of dissipation. Expressions for the decoherence time with and without dissipation are obtained that differ from those appearing in earlier discussions.

DOI
10.1103/physreva.64.032101

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Distribution functions defined in accordance with the quantum theory of measurement are combined with results obtained from the quantum Langevin equation to discuss decoherence in quantum Brownian motion. Closed form expressions for wave packet spreading and the attenuation of coherence of a pair of wave packets are obtained. The results are exact within the context of linear passive dissipation. It is shown that, contrary to widely accepted current belief, decoherence can occur at high temperature in the absence of dissipation. Expressions for the decoherence time with and without dissipation are obtained that differ from those appearing in earlier discussions.
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.
Ford, Lewis, and O'Connell [Phys. Rev. A 64, 032101 (2001)] have recently discussed a thought experiment in which a Brownian particle is subjected to a double-slit measurement . Analyzing the decay of the emerging interference pattern, they derive a decoherence rate that is much faster than previous results and even persists in the limit of vanishing dissipation. This result is based on the definition of a certain attenuation factor, which they analyze for short times. In this note, we point out that this attenuation factor captures the physics of decoherence only for times larger than a certain time tmix{t}_{\mathrm{mix}}tmix​, which is the time it takes until the two emerging wave packets begin to overlap. Therefore, the strategy of Ford et al. of extracting the decoherence time from the regime t<tmixt<{t}_{\mathrm{mix}}t<tmix​ is in our opinion not meaningful. If one analyzes the attenuation factor for t>tmixt>{t}_{\mathrm{mix}}t>tmix​, one recovers familiar behavior for the decoherence time; in particular, no decoherence is seen in the absence of dissipation. We confirm the latter conclusion by calculating the off-diagonal elements of the reduced density matrix.
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.
The prototypical Schr\\"{o}dinger cat state, i.e., an initial state corresponding to two widely separated Gaussian wave packets, is considered. The decoherence time is calculated solely within the framework of elementary quantum mechanics and equilibrium statistical mechanics. This is at variance with common lore that irreversible coupling to a dissipative environment is the mechanism of decoherence. Here, we show that, on the contrary, decoherence can in fact occur at high temperature even for vanishingly small dissipation.
We first review the usefulness of the Wigner distribution functions (WDFs), associated with Lindblad and pre-master equations, for analysing a host of problems in quantum optics where dissipation plays a major role, an arena where weak coupling and long-time approximations are valid. However, we also show their limitations for the discussion of decoherence, which is generally a short-time phenomenon with decay rates typically much smaller than typical dissipative decay rates. We discuss two approaches to the problem both of which use a quantum Langevin equation as a starting-point: (a) use of a reduced WDF but in the context of an exact master equation (b) use of a WDF for the complete system corresponding to entanglement at all times.