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Physica Scripta

On the occurrence of decoherence in nonrelativistic quantum mechanics

M. Dugić

Published Jan 1, 1996• 34 citations
Publisher

Abstract

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.

DOI
10.1088/0031-8949/53/1/001

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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Oct 15, 1982•1,410 citations
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.
Within the decoherence theory we investigate the physical background of the condition of the separability (diagonalizability in noncorrelated basis) of the interaction Hamiltonian of the composite system, "system plus environment". It proves that the condition of the separability may serve as a criterion for defining "system", but so that "system" cannot be defined unless it is simultaneously defined with its "environment". When extended to a set of the mutually interacting composite systems, this result implies that the separability conditions of the local interactions are mutually tied. The task of defining "system" (and "environment") via investigating the separability of the Hamiltonian is a sort of the inverse task of the decoherence theory. A simple example of doing the task is given.
We develop a general algebraic method for investigating the diagonalizability of a composite-system observable in the, so-called, noncorrelated basis of the Hilbert state space of a composite ("two-body") quantum system. The method consists in rewriting a composite-system obsrvable in linear form, for which one directly and uniquely obtains an answer concerning the diagonalizability. The method has been applied to the analysis of the concept of separability of the interaction part of the Hamiltonian of the composite system, "quantum system plus environment (apparatus)". The application leads to a new (more precise) operational definition of separability, which represents a necessary condition concerning the occurrence of decoherence in the, so-called, orthodox approach to the theory of decoherence.
We illustrate through numerical results a number of features of environment-induced decoherence under a broad class of apparatus-environment interactions in quantum measurements wherein the reduced system-apparatus density matrix evolves towards the nearest separable state and, in addition, there occurs a mixing in relevant groups of apparatus microstates (see below). The resulting final state is unique and correctly embodies the measurement statistics even in the absence of environment-induced superselection because of energy differences between these groups of states. The partial transpose remains non-positive throughout the process.
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.
We examine two exactly solvable models of decoherence---a central spin-system, (i) with and (ii) without a self-Hamiltonian, interacting with a collection of environment spins. In the absence of a self-Hamiltonian we show that in this model (introduced some time ago to illustrate environment-induced superselection) generic assumptions about the coupling strengths can lead to a universal (Gaussian) suppression of coherence between pointer states. On the other hand, we show that when the dynamics of the central spin is dominant a different regime emerges, which is characterized by a non-Gaussian decay and a dramatically different set of pointer states. We explore the regimes of validity of the Gaussian decay and discuss its relation to the spectral features of the environment and to the Loschmidt echo (or fidelity).
We show how the correlations of a quantum system with other quantum systems may cause one of its observables to behave in a classical manner. In particular, "reduction of the wave packet," postulated by von Neumann to explain definiteness of an outcome of an individual observation, can be explained when a realistic model of an apparatus is adopted. Instead of an isolated quantum apparatus with a number of states equal to the number of possible distinct outcomes of the measurement, discussed by von Neumann, we consider an apparatus interacting with other physical systems, described here summarily as "environment." The interaction of the quantum apparatus with the environment results in correlations. Correlations impose effective superselection rules which prevent apparatus from appearing in a superposition of states corresponding to different eigenvalues of the privileged pointer observable. It is the propagation of the correlations with the pointer basis states which is ultimately responsible for the choice of the pointer observable. It can be thought of as a process of amplification in which the state of many distinct physical systems becomes correlated with the pointer basis state. Whether these environment systems are regarded as a part of the apparatus setup, or as a part of its environment is irrelevant. What is crucial is the redundancy of the record concerning the pointer observable which is imprinted into the correlations. Eigenspaces of the pointer observable provide a natural basis for the pointer of the quantum apparatus and determine the to-be-measured observable of the quantum system. Decay of those elements of the apparatus-system density matrix, which are off-diagonal in the pointer observable, is caused by the natural evolution of the combined system-apparatus-environment object. For a hypothetical finite environment with NNN distinct eigenvalues of the apparatus-environment interaction Hamiltonian, off-diagonal terms will decay to become of the order of ${N}^{\ensuremath{-}\frac{1}{2}}$, and will recur only on a Poincar\'e time scale. No recurrences will be observed in realistic circumstances. As the correlations spread through the environment on a time scale typically much shorter than the recurrence time scale calculated for the environment already correlated with the pointer observable, the measurement becomes effectively irreversible. Relevance of this model of the measurement process for the understanding of the second law of thermodynamics and its relation to Bohr's "irreversible act of amplification" is briefly discussed. The emergence of the pointer observable can be interpreted as a clue about the resolution of the measurement problem in case of no environment. It points towards the possibility that properties of quantum systems have no absolute meaning. Rather, they must be always characterized with respect to other physical systems.