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arXiv:gr-qc/9303022

Inconsistency between alternative approaches to Quantum Decoherence in special systems

Jason Twamley

Published Mar 19, 1993• 2 citations
arXivHTMLPDF

Abstract

We study the decoherence properties of a certain class of Markovian quantum open systems from both the Decohering Histories and Environment Induced Superselection paradigms. The class studied includes many familiar quantum optical cases. For this class, we show that there always exists a basis which leads to {\em exactly} consistent histories for any coarse graining {\em irrespective} of the initial conditions. The magnitude of the off--diagonal elements of the reduced density matrix $ρ$ in this basis however, depends on the initial conditions. Necessary requirements for classicality as advanced by the two paradigms are thus in direct conflict in these systems.

DOI
10.48550/arxiv.gr-qc/9303022

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arXiv:gr-qc/9210010•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 15, 1993•930 citations
We study the decoherence properties of a certain class of Markovian quantum open systems from both the Decohering Histories and Environment Induced Superselection paradigms. The class studied includes many familiar quantum optical cases. For this class, we show that there always exists a basis which leads to {\em exactly} consistent histories for any coarse graining {\em irrespective} of the initial conditions. The magnitude of the off--diagonal elements of the reduced density matrix ρρρ in this basis however, depends on the initial conditions. Necessary requirements for classicality as advanced by the two paradigms are thus in direct conflict in these systems.
We examine the decoherence properties of a quantum open system as modeled by a quantum optical system in the Markov regime. We look for decoherence in both the environment-induced superselection (EIS) and consistent histories frameworks. We propose a general measure of the coherence of the reduced density matrix and find that EIS decoherence occurs in a number of bases for this model. The degree of "diagonality" achieved increases with bath temperature. We evaluate the decoherence functional of consistent histories for coarse-grained phase-space two-time projected histories. Using the measures proposed by Dowker and Halliwell we find that the consistency of the histories improves with increasing bath temperature, time, and final grain size and decreases with initial grain size. The peaking increases with increasing grain size and decreases with increasing bath temperature. Adopting the above-proposed measure of "coherence" to the decoherence functional gives similar results. The results agree in general with expectations while the anomalous dependence of the consistency on the initial grain size is discussed.
We prove that for an open system, in the Markovian regime, it is always possible to construct an infinite number of nontrivial sets of histories that exactly satisfy the probability sum rules. In spite of being perfectly consistent, these sets manifest a very nonclassical behavior: they are quite unstable under the addition of an extra instant to the list of times defining the history. To eliminate this feature---the implications of which we discuss---and to achieve the stability that characterizes the quasiclassical domain, it is necessary to separate the instants which define the history by time intervals significantly larger than the typical decoherence time. In this case environment induced superselection is very effective. The quasiclassical domain is defined by predictably evolving perferred states, ``pointer projectors,'' which give rise to consistent preferred sets of histories.
We propose a new approach to coarse-grained description of quantum evolution that provides an explicit recipe to construct and evaluate multi-time decoherent histories in a controlled way, applicable to non-Markovian and integrable systems. Specifically, we study local interaction quench of a local degree of freedom (an open quantum system) within a noninteracting integrable environment. This setting allows us to identify the environmental degrees of freedom that irreversibly store records of the system's past. These modes emerge sequentially in time and define the projectors required for decoherent histories. We show numerically that the off-diagonal elements of the decoherence functional are exponentially suppressed relative to a significance threshold.
We study a formulation of quantum mechanics in which the central notion is that of a quantum-mechanical history---a sequence of events at a succession of times. The primary aim is to identify sets of ``decoherent'' (or ``consistent'') histories for the system. These are quantum-mechanical histories suffering negligible interference with each other, and, therefore, to which probabilities may be assigned. These histories may be found for a given system using the so-called decoherence functional. When the decoherence functional is exactly diagonal, probabilities may be assigned to the histories, and all probability sum rules are satisfied exactly. We propose a condition for approximate decoherence, and argue that it implies that most probability sum rules will be satisfied to approximately the same degree. We also derive an inequality bounding the size of the off-diagonal terms of the decoherence functional. We calculate the decoherence functional for some simple one-dimensional systems, with a variety of initial states. For these systems, we explore the extent to which decoherence is produced using two different types of coarse graining. The first type of coarse graining involves imprecise specification of the particle's position. The second involves coupling the particle to a thermal bath of harmonic oscillators and ignoring the details of the bath (the Caldeira-Leggett model). We argue that both types of coarse graining are necessary in general. We explicitly exhibit the degree of decoherence as a function of the temperature of the bath, and of the width to within which the particle's position is specified. We study the diagonal elements of the decoherence functional, representing the probabilities for the possible histories of the system. To the extent that the histories decohere, we show that the probability distributions are peaked about the classical histories of the system, with the distribution of their initial positions and momenta given by a smeared version of the Wigner function. We discuss this result in connection with earlier uses of the Wigner function in this context. We find that there is a certain amount of tension between the demands of decoherence and peaking about classical paths.
The origin of the phenomenological deterministic laws that approximately govern the quasiclassical domain of familiar experience is considered in the context of the quantum mechanics of closed systems such as the universe as a whole. A formulation of quantum mechanics is used that predicts probabilities for the individual members of a set of alternative coarse-grained histories that decohere, which means that there is negligible quantum interference between the individual histories in the set. We investigate the requirements for coarse grainings to yield decoherent sets of histories that are quasiclassical, i.e., such that the individual histories obey, with high probability, effective classical equations of motion interrupted continually by small fluctuations and occasionally by large ones. We discuss these requirements generally but study them specifically for coarse grainings of the type that follows a distinguished subset of a complete set of variables while ignoring the rest. More coarse graining is needed to achieve decoherence than would be suggested by naive arguments based on the uncertainty principle. Even coarser graining is required in the distinguished variables for them to have the necessary inertia to approach classical predictability in the presence of the noise consisting of the fluctuations that typical mechanisms of decoherence produce. We describe the derivation of phenomenological equations of motion explicitly for a particular class of models. Those models assume configuration space and a fundamental Lagrangian that is the difference between a kinetic energy quadratic in the velocities and a potential energy. The distinguished variables are taken to be a fixed subset of coordinates of configuration space. The initial density matrix of the closed system is assumed to factor into a product of a density matrix in the distinguished subset and another in the rest of the coordinates. With these restrictions, we improve the derivation from quantum mechanics of the phenomenological equations of motion governing a quasiclassical domain in the following respects: Probabilities of the correlations in time that define equations of motion are explicitly considered. Fully nonlinear cases are studied. Methods are exhibited for finding the form of the phenomenological equations of motion even when these are only distantly related to those of the fundamental action. The demonstration of the connection between quantum-mechanical causality and causality in classical phenomenological equations of motion is generalized. The connections among decoherence, noise, dissipation, and the amount of coarse graining necessary to achieve classical predictability are investigated quantitatively. Routes to removing the restrictions on the models in order to deal with more realistic coarse grainings are described.