Macroscopic limit of a solvable dynamical model

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 weak-coupling, macroscopic (N\ensuremath{\rightarrow}\ensuremath{\infty}) limit, the system remains solvable and leads to interesting connections with the Jaynes-Cummings model, which 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 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 N\ensuremath{\rightarrow}\ensuremath{\infty} 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 increasingly loses coherence as the temperature of the detector increases. The problem of ``imperfect measurements'' is also briefly discussed.