Macroscopic limit of a solvable dynamical model

TL;DR

This paper analyzes a solvable model of an ultrarelativistic particle interacting with a large array of two-level systems, revealing how macroscopic limits relate to quantum measurement and decoherence, with connections to the Jaynes-Cummings model.

Contribution

It introduces a solvable macroscopic limit of a dynamical model involving particle-molecule interactions, linking quantum measurement theory with the Jaynes-Cummings model.

Findings

Visibility of interference patterns decreases with temperature.

Thermal initial states cause more decoherence than ground states.

Superselection rules emerge in the macroscopic limit.

Abstract

The interaction between an ultrarelativistic particle and a linear array made up of $N$ two-level systems (^^ ^^ AgBr" molecules) is studied by making use of a modified version of the Coleman-Hepp Hamiltonian. Energy-exchange processes between the particle and the molecules are properly taken into account, and the evolution of the total system is calculated exactly both when the array is initially in the ground state and in a thermal state. In the macroscopic limit ($N \rightarrow \infty$), the system remains solvable and leads to interesting connections with the Jaynes-Cummings model, that describes the interaction of a particle with a maser. The visibility of the interference pattern produced by the two branch waves of the particle is computed, and the conditions under which the spin array in the $N \rightarrow \infty$ limit behaves as a ^^ ^^ detector" are investigated. The behavior of the visibility yields good insights into the issue of quantum measurements: It is found that, in the thermodynamical limit, a superselection-rule space appears in the description of the (macroscopic) apparatus. In general, an initial thermal state of the ^^ ^^ detector" provokes a more substantial loss of quantum coherence than an initial ground state. It is argued that a system decoheres more as the temperature of the detector increases. The problem of ^^ ^^ imperfect measurements" is also shortly discussed.