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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Investigating decoherence in a simple system

Andreas Albrecht

Published Dec 15, 1992• 82 citations
Publisher

Abstract

I present the results of some simple calculations designed to study the loss of quantum coherence. The relevant physical issues are briefly reviewed, and then a very simple "toy" model is analyzed. Exact solutions are found using numerical techniques. The type of decoherence exhibited by the model can be changed by varying a coupling strength. I study the system from two points of view. One, the Schmidt paths approach, is closely related to the conventional approach of studying decoherence by checking the form of the density matrix. The consistent histories approach is also used, and the relationship between the two approaches is explored.

DOI
10.1103/physrevd.46.5504

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I present the results of some simple calculations designed to study the loss of quantum coherence. The relevant physical issues are briefly reviewed, and then a very simple "toy" model is analyzed. Exact solutions are found using numerical techniques. The type of decoherence exhibited by the model can be changed by varying a coupling strength. I study the system from two points of view. One, the Schmidt paths approach, is closely related to the conventional approach of studying decoherence by checking the form of the density matrix. The consistent histories approach is also used, and the relationship between the two approaches is explored.
The results of some simple calculations designed to study quantum decoherence are presented. The physics of quantum decoherence are briefly reviewed, and a very simple 'toy' model is analyzed. Exact solutions are found using numerical techniques. The type of incoherence exhibited by the model can be changed by varying a coupling strength. The author explains why the conventional approach to studying decoherence by checking the diagonality of the density matrix is not always adequate. Two other approaches, the decoherence functional and the Schmidt paths approach, are applied to the toy model and contrasted to each other. Possible problems with each are discussed.
I study the quantum mechanics of a spin interacting with an "apparatus." Although the evolution of the whole system is unitary, the spin evolution is not. The system is chosen so that the spin exhibits loss of quantum coherence, or "wave function collapse," of the sort usually associated with a quantum measurement. The system is analyzed from the point of view of the spin density matrix (or "Schmidt paths"), and also using the consistent histories approach. These two points of view are contrasted with each other. Connections between the results and the form of the Hamiltonian are discussed in detail.
I briefly review the "decohering histories" or "consistent histories" formulation of quantum theory, due to Griffiths, Omnès, and Gell-Mann and Hartle (and the subject of my graduate work with George Sudarshan). I also sift through the many meanings that have been attached to decohering histories, with an emphasis on the most basic one: Decoherence of appropriate histories is needed to establish that quantum mechanics has the correct classical limit. Then I will describe efforts to find physical mechanisms that do this. Since most work has focused on density matrix versions of decoherence, I'll consider the relation between the two formulations, which historically has not been straightforward. Finally, I'll suggest a line of research that would use recent results by Sudarshan to illuminate this aspect of the classical limit of quantum theory.
We introduce and review the consistent (decoherent) histories approach to quantum mechanics due to Griffiths, to Gell-Mann and Hartle, and to Omnes. We will not attempt an in depth look at this approach as it would be impossible to treat it in such a short review. The emphasis is on understanding the broader meaning of the consistency and decoherence. The consistent history approach to quantum mechanics provides a precise conceptual framework for describing how a closed quantum system develops in time. The approach also provides a framework from which we may observe the emergence of an approximately classical domain for macroscopic systems, together with the conventional Copenhagen quantum mechanics for microscopic subsystems. We will study the formalism of decoherence and look at several approaches to decoherence. In examining a system of quantum Brownian motion and using the methods of Feynman and Vernon, we derive an equation of motion for the density matrix, the master equation of the system. This leads us to a brief overview of decoherence. PACS Nos.: 03.65.Ta, 03.65.–w, 03.65.Yz, 04.60.–m, 05.30.–d