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 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 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.
We consider the decoherence of phase-space histories in quantum Brownian motion models, consisting of a particle moving under a potential V(x) in contact with a heat bath of temperature T and dissipation constant $\ensuremath{\gamma}$ in the Markovian regime. The phase-space histories are described by quasiprojectors consisting of Gaussian density matrices smeared over large phase-space cells and are characterized by the size [$\ensuremath{\Gamma}$] of the phase-space cell together with the size [M] of the margin (the region at the boundary of $\ensuremath{\Gamma}$ in which the Weyl symbol of the projector goes from 1 to 0). We generalize earlier results of Hagedorn to show that an initial Gaussian density matrix remains approximately Gaussian under nonunitary time evolution deriving a bound giving the validity of this approximation. Following the earlier work of Omnes [J. Stat. Phys. 51, 351 (1989)] we use this result to compute the time evolution of the phase-space projectors under the master equation and show that histories of phase-space samplings approximately decohere, and that the probabilities for these histories are peaked about classical dissipative evolution, but with an element of unpredictability which is reflected in the increase of the size of the margin.
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.
Environment-induced destruction of quantum coherence is investigated in a simple model where the system is a harmonic oscillator, the environment is a collection of harmonic oscillators, and the interaction between them is linear in the position coordinate of the system. We study decoherence for initial states consisting of coherent superpositions of two Gaussian wave packets in either position or momentum. A new measure of the effectiveness of decoherence appropriate to the model and choice of initial conditions is proposed. By studying the dependence of the decoherence rate on the location of the initial peaks of the Wigner function, we clarify the sense in which position is a preferred observable even though position eigenstates are not the pointer states of this model. We analyze decoherence in the low-temperature regime and show that the usual "high-temperature" approximation is remarkably accurate in its domain of applicability. We also examine the relationship between the decoherence process and the frequency distribution of the environment oscillators (in particular, we focus attention on a specific "supra-Ohmic" environment). Implications of our results for the quantum to classical transition in various contexts are briefly explored.
Within the histories formalism the decoherence functional is a formal tool to investigate the emergence of classicality in isolated quantum systems, yet an explicit evaluation of it from first principles has not been reported. We provide such an evaluation for up to five-time histories based on exact numerical diagonalization of the Schrödinger equation. We find a robust emergence of decoherence for slow and coarse observables of a generic random matrix model and extract a finite-size scaling law by varying the Hilbert space dimension over 4 orders of magnitude. Specifically, we conjecture and observe an exponential suppression of coherent effects as a function of the particle number of the system. This suggests a solution to the preferred basis problem of the many-worlds interpretation (or the set selection problem of the histories formalism) within a minimal theoretical framework without relying on environmentally induced decoherence, quantum Darwinism, Markov approximations, low-entropy initial states, or ensemble averages. Published by the American Physical Society 2024
The act of measurement on a quantum state is supposed to "dephase" (dephasing refers to the phenomenon that the states lose phase coherence; then the phases get randomized in interaction with a bath of other oscillators, which is referred to as "decoherence"), then "decohere" and "collapse" (or more precisely "register" and "reduce") the state into one of several eigenstates of the operator corresponding to the observable being measured. This measurement process is sometimes described as outside standard quantum-mechanical evolution and not calculable from Schrödinger's equation. Progress has, however, been made in studying this problem with two main calculation tools-one uses a time-independent Hamiltonian, while a rather more general approach proving that decoherence occurs under some generic conditions. The two general approaches to the study of wave-function collapse are as follows. The first approach, called the "consistent" or "decoherent"' histories approach, studies microscopic histories that diverge probabilistically and explains collapse as an event in our particular history. The other, referred to as the "environmental decoherence" approach studies the effect of the environment upon the quantum system, to explain wave-function decoherence which is produced by irreversible effects of various sorts. However, as we know, wave-function collapse is not related to thermal connection with the environment, rather, it is inherent to how measurements are performed by macroscopic apparata. In the "environmental decoherence" approach, one studies decoherence using a Markovian-approximated Master equation to study the time-evolution of the reduced density matrix (post dephasing) and obtains the long-time dependence of the off-diagonal elements of this matrix. The calculation in this paper studies the evolution of a quantum system starting with "dephasing" followed by the effects of the environment with some differences from prior analyses. We start from the Schrödinger equation for the state of the system, with a time-dependent Hamiltonian that reflects the actual microscopic interactions that are occurring. Then we systematically solve (exactly) for the time-evolved state, without invoking a Markovian approximation when writing out the effective time-evolution equation, i.e., keeping the evolution unitary until the end. This approach is useful, and it shows that the system wave-function will explicitly "un-collapse" if the measurement apparatus is sufficiently small. However, in the limit of a macroscopic system, this "dephasing" quickly leads to "decoherence"-collapse is a temporary state that will simply take extremely long (of the order of multiple universe lifetimes) to reverse. This has been attempted previously and our calculation is particularly simple and calculable. We make some connections to the work by Linden et al. while doing so. The calculation in this paper has interesting implications for the interpretation of the Wigner's friend experiment, as well as the Mott experiment, which is explored in "Connection to some general theoretical results" and "Recurrence times" (especially the enumerated points in "Recurrence times"). The upshot is that as long as Wigner's friend is macroscopically large (or uses a macroscopically large measuring instrument), no one needs to worry that Wigner would see something different from his friend. Indeed, Wigner's friend does not even need to be conscious during the measurement that she conducts. It also allows one to reasonably interpret some of the more recent thought experiments proposed. In particular, as a result of the mathematical analysis, the short-time behavior of a collapsing system, at least the one considered in this paper, is not exponential. Instead, it is the usual Fermi-golden rule result. The long-term behavior is, of course, still exponential. This is a second novel feature of the paper-we connect the short-term Fermi-golden rule (quadratic-in-time behavior) transition probability to the exponential long-time behavior of a collapsing wave-function in one continuous mathematical formulation.