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Physical Review Letters

Solvable dynamical model for a quantum measurement process

Hiromichi Nakazato, Saverio Pascazio

Published Jan 4, 1993• 68 citations
Publisher

Abstract

A model Hamiltonian describing an energy-exhange process between an ultrarelativistic particle and a one-dimensional spin array is proposed and solved exactly. Interesting relations with the quantum measurement problem are discussed.
DOI
10.1103/physrevlett.70.1

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An exactly solvable model for a quantum measurement is discussed which is governed by hamiltonian quantum dynamics. The z‐component ŝz of a spin −12 is measured with an apparatus, which itself consists of magnet coupled to a bath. The initial state of the magnet is a metastable paramagnet, while the bath starts in a thermal, gibbsian state. Conditions are such that the act of measurement drives the magnet in the up or down ferromagnetic state according to the sign of sz of the tested spin. The quantum measurement goes in two steps. On a timescale 1/N the off‐diagonal elements of the spin’s density matrix vanish due to a unitary evolution of the tested spin and the N apparatus spins; on a larger but still short timescale this is made definite by the bath. Then the system is in a ‘classical’ state, having a diagonal density matrix. The registration of that state is a quantum process which can already be understood from classical statistical mechanics. The von Neumann collapse and the Born rule are derived rather than postulated.
An exactly solvable model for a quantum measurement is discussed, that integrates quantum measurements with classical measurements. The z-component of a spin-1/2 test spin is measured with an apparatus, that itself consists of magnet of N spin-1/2 particles, coupled to a bath. The initial state of the magnet is a metastable paramagnet, while the bath starts in a thermal, gibbsian state. Conditions are such that the act of measurement drives the magnet in the up or down ferromagnetic state, according to the sign of s_z of the test spin. The quantum measurement goes in two steps. On a timescale 1/\sqrt{N} the collapse takes place due to a unitary evolution of test spin and apparatus spins; on a larger but still short timescale this collapse is made definite by the bath. Then the system is in a `classical' state, having a diagonal density matrix. The registration of that state is basically a classical process, that can already be understood from classical statistical mechanics.
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.
The measurement process of observables in a quantum system comes out to be an unsovable problem which started in the early times of the development of the theory. In the present note we consider the measured system part of an open system interacting with the measuring device and show under which ideal conditions the measure process may ideally work. Our procedure leads to the conclusion that there is no hope that any experimental procedure will be able to lead to a clean solution of the process, except maybe in very specific cases. The reasons for this situation are deeply rooted in the fundamental properties of quantum theory.