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arXiv:quant-ph/0501138 · Physical Review A

Self-induced decoherence approach: Strong limitations on its validity in a simple spin bath model and on its general physical relevance

Maximilian Schlosshauer

Published Jul 18, 2005• 28 citations
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Abstract

The ``self-induced decoherence'' (SID) approach suggests that (1) the expectation value of any observable becomes diagonal in the eigenstates of the total Hamiltonian for systems endowed with a continuous energy spectrum, and that (2) this process can be interpreted as decoherence. We evaluate the first claim in the context of a simple spin bath model. We find that even for large environments, corresponding to an approximately continuous energy spectrum, diagonalization of the expectation value of random observables does in general not occur. We explain this result and conjecture that SID is likely to fail also in other systems composed of discrete subsystems. Regarding the second claim, we emphasize that SID does not describe a physically meaningful decoherence process for individual measurements, but only involves destructive interference that occurs collectively within an ensemble of presupposed ``values'' of measurements. This leads us to question the relevance of SID for treating observed decoherence effects.

DOI
10.1103/physreva.72.012109

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The self-induced decoherence'' (SID) approach suggests that (1) the expectation value of any observable becomes diagonal in the eigenstates of the total Hamiltonian for systems endowed with a continuous energy spectrum, and that (2) this process can be interpreted as decoherence. We evaluate the first claim in the context of a simple spin bath model. We find that even for large environments, corresponding to an approximately continuous energy spectrum, diagonalization of the expectation value of random observables does in general not occur. We explain this result and conjecture that SID is likely to fail also in other systems composed of discrete subsystems. Regarding the second claim, we emphasize that SID does not describe a physically meaningful decoherence process for individual measurements, but only involves destructive interference that occurs collectively within an ensemble of presupposed values'' of measurements. This leads us to question the relevance of SID for treating observed decoherence effects.
Our account of the problem of the classical limit of quantum mechanics involves two elements. The first one is self-induced decoherence, conceived as a process that depends on the own dynamics of a closed quantum system governed by a Hamiltonian with continuous spectrum; the study of decoherence is addressed by means of a formalism used to give meaning to the van Hove states with diagonal singularities. The second element is macroscopicity: when the macroscopic limit is applied to the Wigner transformation of the diagonal state resulting from decoherence, the description of the quantum system becomes equivalent to the description of an ensemble of classical trajectories on phase space weighted by their corresponding probabilities.
In this paper we argue that the emergence of the classical world from the underlying quantum reality involves two elements: self-induced decoherence and macroscopicity. Self-induced decoherence does not require the openness of the system and its interaction with the environment: a single closed system can decohere when its Hamiltonian has continuous spectrum. We show that, if the system is macroscopic enough, after self-induced decoherence it can be described as an ensemble of classical distributions weighted by their corresponding probabilities. We also argue that classicality is an emergent property that arises when the behavior of the system is described from an observational perspective.
The fact that the Environment Induced Decoherence approach offers no general criterion to decide where to place the "cut" between system and environment has been considered as a serious conceptual problem of the proposal. In this letter we argue that this is actually a pseudo-problem, which is dissolved by the fact that decoherence is a phenomenon relative to the relevant observables selected by the measuring arrangement. We also show that, when the spin-bath model is studied from this perspective, certain unexpected results are obtained, as that of a system decohering in interaction with a very small environment.
According to a usual reading, decoherence is a process resulting from the interaction between a small system and its large environment where information and energy are dissipated. The particular models treated in the literature on the subject reinforce this idea since, in general, the behavior of a particle immersed in a large "bath" composed by many particles is studied. The aim of this letter is to warn against this usual simplified reading. By means of the analysis of a well-known model, we will show that decoherence may occur in a system interacting with an environment consisting of only one particle.
A general method for obtaining the decoherence time in self-induced decoherence is presented. In particular, it is shown that such a time can be computed from the poles of the resolvent or of the initial conditions in the complex extension of the Hamiltonian's spectrum. Several decoherence times are estimated: ${10}^{\ensuremath{-}13}--{10}^{\ensuremath{-}15}\phantom{\rule{0.3em}{0ex}}\mathrm{s}$ for microscopic systems, and ${10}^{\ensuremath{-}37}--{10}^{\ensuremath{-}39}\phantom{\rule{0.3em}{0ex}}\mathrm{s}$ for macroscopic bodies. For the particular case of a thermal bath, the order of magnitude of our results agrees with that obtained by the einselection (environment-induced decoherence) approach.