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arXiv:quant-ph/9811010

Exactly soluble models of decoherence

Joachim Kupsch

Published Nov 4, 1998• 1 citation
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Abstract

Superselection rules induced by the interaction with the environment are a basis to understand the emergence of classical observables within quantum theory. The aim of this article is to investigate the decoherence effects, which lead to superselection sectors, with the help of exactly soluble Hamiltonian models. Starting from the examples of Araki and of Zurek more general models with scattering are presented for which the projection operators onto the induced superselection sectors do no longer commute with the Hamiltonian. The example of an environment given by a free quantum field indicates that infrared divergence plays an essential role for the emergence of induced superselection sectors. For all models the induced superselection sectors are uniquely determined by the Hamiltonian, whereas the time scale of the decoherence depends crucially on the initial state of the total system.

DOI
10.48550/arxiv.quant-ph/9811010

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Superselection rules induced by the interaction with the environment are a basis to understand the emergence of classical observables within quantum theory. The aim of this article is to investigate the decoherence effects, which lead to superselection sectors, with the help of exactly soluble Hamiltonian models. Starting from the examples of Araki and of Zurek more general models with scattering are presented for which the projection operators onto the induced superselection sectors do no longer commute with the Hamiltonian. The example of an environment given by a free quantum field indicates that infrared divergence plays an essential role for the emergence of induced superselection sectors. For all models the induced superselection sectors are uniquely determined by the Hamiltonian, whereas the time scale of the decoherence depends crucially on the initial state of the total system.
Superselection rules induced by the interaction with a mass zero Boson field are investigated for a class of exactly soluble Hamiltonian models. The calculations apply as well to discrete as to continuous superselection rules. The initial state (reference state) of the Boson field is either a normal state or a KMS state. The superselection sectors emerge if and only if the Boson field is infrared divergent, i. e. the bare photon number diverges and the ground state of the Boson field disappears in the continuum. The time scale of the decoherence depends on the strength of the infrared contributions of the interaction and on properties of the initial state of the Boson system. These results are first derived for a Hamiltonian with conservation laws. But in the most general case the Hamiltonian includes an additional scattering potential, and the only conserved quantity is the energy of the total system. The superselection sectors remain stable against the perturbation by the scattering processes.
We investigate the conditions which can lead to the occurrence of "decoherence" (i.e., of "environment-induced superselection"), in a quantum system interacting with another, infinite quantum system (an apparatus, environment, etc.), in the context of the nonrelativistic quantum mechanics. We are particularly concerned with: a kind of interaction, its relative strength, initial state of the apparatus (environment), and compatibility of the self-Hamiltonian and the interaction Hamiltonian of the composite system, "system plus apparatus", including time dependent interactions. As necessary conditions concerning occurrence of decoherence appear the two requirements: (i) That the interaction Hamiltonian is of the separable kind , and (ii) That the complete Hamiltonian, Ĥ u satisfies commutator relation [Ĥ u ( t ), Ĥ u ( t ')] = 0 ( t and t ' arbitrary instants). Since non-separable interaction cannot produce the same, we conclude that occurrence of decoherence is neither a rule (as it is sometimes assumed in the context of the "environment theory"), nor an exceptional consequence of the dynamics of a system interacting with its environment (as it is sometimes claimed in the criticism of the environment theory). Exhaustiveness and generality of our analysis extend applicability of related results (which are independent on a particular model of (infinite) environment), at least as a qualitative test and/or quantitative estimation of the results of the calculations obtained by the use of the different methods (semiclassical methods, path-integration technique, etc.) in quantum many-particle physics.
According to Zurek, decoherence is a process resulting from the interaction between a quantum system and its environment; this process singles out a preferred set of states, usually called pointer basis, that determines which observables will receive definite values. This means that decoherence leads to a sort of selection which precludes all except a small subset of the states in the Hilbert space of the system from behaving in a classical manner: environment-induced-superselection (einselection) is a consequence of the process of decoherence. The aim of this paper is to present a new approach to decoherence, different from the mainstream approach of Zurek and his collaborators. We will argue that this approach offers conceptual advantages over the traditional one when problems of foundations are considered; in particular, from the new perspective, decoherence in closed quantum systems becomes possible and the preferred basis acquires a well founded definition.
The study of environmentally induced superselection and of the process of decoherence was originally motivated by the search for the emergence of classical behavior out of the quantum substrate, in the macroscopic limit [W. H. Zurek, Phys. Rev. D 24, 1516 (1981); 26, 1862 (1982)]. This limit, and other simplifying assumptions, have allowed the derivation of several simple results characterizing the onset of environmentally induced superselection; but these results are increasingly often regarded as a complete phenomenological characterization of decoherence in any regime. This is not necessarily the case: the examples presented in this paper counteract this impression by violating several of the simple general rules. This is relevant because decoherence is now beginning to be tested experimentally [C. Monroe et al., Science 272, 1131 (1996); M. Brune et al., Phys. Rev. Lett. 77, 4887 (1996)], and one may anticipate that, in at least some of the proposed applications (e.g., quantum computers), only the basic principle of ``monitoring by the environment'' will survive. The phenomenology of decoherence may turn out to be significantly different.