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Physical Review A

Classical motion of meter variables in the quantum theory of measurement

Fritz Haake, Marek Żukowski

Published Apr 1, 1993• 30 citations
Publisher

Abstract

We describe the measurement process for a model system whose Schr\"odinger equation yields to explicit solution. Any device for an effectively irreversible measurement must not just be a ``meter'' with a single degree of freedom; it must also contain a ``heat bath'' with many degrees of freedom. Such a bath may cause an effectively instantaneous decay of quantum coherences between macroscopically distinct meter readings, i.e., impart practically classical behavior to the meter. In turn, such a meter can be read without thereby suffering back reactions. We establish the limits in which a meter reading is uniquely related, with reasonable certainty, to a definite eigenvalue of the measured observable.

DOI
10.1103/physreva.47.2506

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We describe the measurement process for a model system whose Schr"odinger equation yields to explicit solution. Any device for an effectively irreversible measurement must not just be a meter'' with a single degree of freedom; it must also contain a heat bath'' with many degrees of freedom. Such a bath may cause an effectively instantaneous decay of quantum coherences between macroscopically distinct meter readings, i.e., impart practically classical behavior to the meter. In turn, such a meter can be read without thereby suffering back reactions. We establish the limits in which a meter reading is uniquely related, with reasonable certainty, to a definite eigenvalue of the measured observable.
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.
A model of quantum measurement is proposed, which aims to describe statistical mechanical aspects of this phenomenon, starting from a purely Hamiltonian formulation. The macroscopic measurement apparatus is modeled as an ideal Bose gas, the order parameter of which, that is, the amplitude of the condensate, is the pointer variable. It is shown that properties of irreversibility and ergodicity breaking, which are inherent in the model apparatus, ensure the appearance of definite results of the measurement, and provide a dynamical realization of wave-function reduction or collapse. The measurement process takes place in two steps: First, the reduction of the state of the tested system occurs over a time of order $\ensuremath{\Elzxh}{/(TN}^{1/4}),$ where T is the temperature of the apparatus, and N is the number of its degrees of freedom. This decoherence process is governed by the apparatus-system interaction. During the second step classical correlations are established between the apparatus and the tested system over the much longer time scale of equilibration of the apparatus. The influence of the parameters of the model on nonideality of the measurement is discussed. Schr\"odinger kittens, EPR setups, and information transfer are analyzed.
We report and give an alternative derivation of some results on a model for a quantum measurement studied in [1]. The measured microscopic system is coupled to the position of a macroscopic pointer, which itself interacts with its environment via its momentum. The entanglement between the system and the pointer produced by their mutual interaction is simultaneous with the decoherence of distinct pointer readings resulting from leakage of information to the environment. After a discussion on the various time scales in the model we calculate the matrix elements of the system-pointer density operator between eigenstates of the measured observable with distinct eigenvalues. In general, the decay with time of these coherences is neither exponential nor gaussian. We determine the decoherence (decay) time in terms of the strength of the system-pointer and pointer-environment couplings. This decoherence time does not depend upon the details of the pointer-bath coupling as soon as it is smaller than the bath correlation time (non-Markov regime). In contrast, in the Markov regime it depends strongly on whether this coupling is Ohmic or super-Ohmic.
We study a class of quantum measurement models. A microscopic object is entangled with a macroscopic pointer such that each eigenvalue of the measured object observable is tied up with a specific pointer deflection. Different pointer positions mutually decohere under the influence of a bath. Object-pointer entanglement and decoherence of distinct pointer readouts proceed simultaneously. Mixtures of macroscopically distinct object-pointer states may then arise without intervening macroscopic superpositions. Initially, object and apparatus are statistically independent while the latter has pointer and bath correlated according to a metastable local thermal equilibrium. We obtain explicit results for the object-pointer dynamics with temporal coherence decay in general neither exponential nor Gaussian. The decoherence time does not depend on details of the pointer-bath coupling if it is smaller than the bath correlation time, whereas in the opposite Markov regime the decay depends strongly on whether that coupling is Ohmic or super-Ohmic.
It is not possible to obtain information about the observable properties of a quantum system without a physical interaction between the system and an external meter. This physical interaction is described by a unitary transformation of the joint quantum state of the system and the meter, which means that the information transfer from the system to the meter depends on the initial quantum coherence of the meter. In the present paper, we analyze the measurement interaction in terms of the changes of the meter state caused by the interaction with the system. The sensitivity of the meter can then be defined by evaluating the distinguishability of meter states for different values of the target observable. It is shown that the sensitivity of the meter requires quantum coherences in the generator observable that determines the magnitude of the backaction of the meter on the system. The tradeoff between measurement resolution and backaction is decided by the relation between sensitivity and quantum coherent uncertainty in the external meter. No matter how macroscopic the device used as a meter is, it must necessarily be quantum coherent in the degrees of freedom that interact with the system being measured.