Decoherent histories for a particle coupled to a von Neumann apparatus

TL;DR

This paper analyzes decoherent histories of a quantum particle coupled to a measurement apparatus using the Gell-Mann and Hartle formalism, deriving exact and approximate decoherence conditions and probabilities for a driven harmonic oscillator system.

Contribution

It provides an exact expression for the decoherence functional in a coupled particle-apparatus system and explores conditions for exact and approximate decoherence with quantitative bounds.

Findings

Exact decoherence occurs with initial position states of the particle or pointer.

Approximate decoherence is achieved with narrow initial states, allowing probability estimates.

Derived bounds for decoherence functional and probabilities in the narrow state regime.

Abstract

Using the Gell-Mann and Hartle formalism of generalized quantum mechanics of closed systems, we study coarse-grained decoherent histories. The system under consideration is one-dimensional and consists of a particle coupled to a von Neumann apparatus that measures its position. The particle moves in a quadratic potential; in particular we consider a driven harmonic oscillator. The real line is divided into intervals of the same length, and coarse-grained histories are defined by the arithmetic average of the initial and final position of the particle to be within one of these intervals. The position of the pointer correlates with this arithmetic average. In addition a constant term is added to this average, which is a result of the presence of a driving force on the oscillator. We investigate decoherence for such histories via the decoherence functional for the particle-apparatus. An exact expression for this functional has been derived. If the particle or the pointer of the apparatus is in an exact position state initially, then exact decoherence ensues, and the relative probability for such histories is obtained. If the initial state of the particle, which is centered at some arbitrary point in the $x$-axis, is narrow compared to the width of the intervals on the $x$-axis, then we obtain approximate decoherence. The same result follows if the initial state of the pointer is narrow. In these two situations, we evaluate expressions for the decoherence functional qualitatively and quantitatively. In the latter case we obtain the first two terms in an expansion in powers of the width of the initial state of the particle or the pointer respectively. Approximate relative probabilities can be assigned in this case, and we have obtained an expansion up to second order in powers of the width of the initial state for an expression for an upper bound of these probabilities.