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Progress of Theoretical Physics

One-Particle Detectors with SU(1, 1) Coherent States in Quantum Measurement

K. Kakazu, A. Ogawa

Published Oct 1, 1991• 1 citation
Publisher

Abstract

The generalized coherent states based on SU(1, 1) are applied to quantum-measuring processes, in which each detector consists of a single particle. Differing from the Glauber coherent states, possible interactions are quite restricted. New definitions of the classical limit and classical operators are presented, where any polynomial of the position and momentum operators becomes classical. Although quantum interference itself becomes very large as the time after the interaction is increased, the statistical operator approaches its mixed-state part in a relative sense.

DOI
10.1143/ptp/86.4.943

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The generalized coherent states based on SU(1, 1) are applied to quantum-measuring processes, in which each detector consists of a single particle. Differing from the Glauber coherent states, possible interactions are quite restricted. New definitions of the classical limit and classical operators are presented, where any polynomial of the position and momentum operators becomes classical. Although quantum interference itself becomes very large as the time after the interaction is increased, the statistical operator approaches its mixed-state part in a relative sense.
The generalized coherent states based on SU(1, 1) are applied to quantum-measuring processes, in which each detector consists of a single particle. Differing from the Glauber coherent states, possible interactions are quite restricted. New definitions of the classical limit and classical operators are presented, where any polynomial of the position and momentum operators becomes classical. Although quantum interference itself becomes very large as the time after the interaction is increased, the statistical operator approaches its mixed-state part in a relative sense.
The measuring process of an object system and a detector are studied in the von Neumann model and one of its variants, in which the detector consists of a single particle. If the states of the particle after the interaction with the object system are described partly by coherent states, then the particle can play the role of a detector under certain conditions. The mass (or free Hamiltonian) of the detector does not affect the measuring process at all, while the mass of the object system plays an important role. In the von Neumann model, if the mass of the object system is quite large, the position measurement can be considered, and under certain conditions the statistical operator \ensuremath{\rho} tends to just its mixed-state part, i.e., quantum interference disappears. However, when the time after the interaction becomes very large, interference emerges again. On the other hand, if the mass of the object system is small, we can even consider the momentum measurement and show that \ensuremath{\rho} approaches its mixed-state part when the time becomes very large, i.e., \ensuremath{\rho} is equivalent to its mixed-state part. The momentum can also be measured almost completely in the variant model; the operator \ensuremath{\rho} becomes equivalent to its mixed-state part. The different quantum states distinguish the detectors from the object systems.
We present and discuss some models for the quantum measuring process. Our models indicate that the coherent state approach to classical limits in quantum theories becomes a powerful method for a unified description of various measuring processes. Each detector in the models has a parameter N , the number of its dynamical variables; the whole process of an interaction between an object system and the detector is calculable for any finite number N . It is shown that the detector states after the interaction are characterized by a macroscopic parameter, which is closely connected with the “reliability” of the detector. The reliability plays an essential role in eliminating the quantum interference between two states corresponding to two different positions of the pointer. Finally, the quantum mechanics of the detector reduces to classical mechanics on suitable “phase space” as N becomes large, as is required.
Quantum measurements of physical quantities are often described as ideal measurements. However, only a few measurements fulfil the conditions of ideal measurements. The aim of the present work is to describe real position measurements with detectors that are able to detect single particles. For this purpose, a detector model is developed that can describe the time dependence of the interaction between a non-relativistic particle and a detector. The example of a position measurement shows that this interaction can be described with the methods of quantum mechanics. At the beginning of a position measurement, the detector behaves as a target consisting of a large number of quantum mechanical systems. In the first reaction, the incident particle interacts with a single atom, electron or nucleus, but not with the whole detector. This reaction and all following reactions are quantum mechanical processes. At the end of the measurement, the detector can be considered as a classical apparatus. A detector is neither a quantum mechanical system nor a classical apparatus. The detector model explains why one obtains a well-defined result for each individual position measurement. It further explains that, in general, it is impossible to predict the outcome of an individual measurement.
The measurement process is considered through the interaction of the particles. The statistical operator of a single particle describes the classical behaviour while the interference term of the particles describes the quantum behaviour. A free single particle is treated classical. An apparatus consisting of two particles is treated quantum. The free particle is involved in the quantum superposition through the interaction with the apparatus. Collapse of the superposition state of the apparatus leads to the projection of the particle onto a definite state.