Overdamped and amplifying meters in the quantum theory of measurement

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Abstract
We show that a quantum observable can be measured by coupling it to a meter which in turn interacts with a reservoir. The complete Hamiltonian is chosen so as to allow for an explicit exact solution of the quantum dynamics. In the continuum limit for the bath the solution displays irreversible behavior, two varieties of which, overdamping and amplification, turn out to be of special relevance. We establish the limit in which a suitable pointer variable (i) behaves effectively classically and (ii) acquires, through the measurement, a probability density of its eigenvalues with well-defined peaks each of which corresponds to one discrete eigenvalue of the measured observable. Under a slightly more restrictive condition the reduced density matrix of the object diagonalizes, during the measurement, in the eigenbasis of the measured observable.