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Acta Polytechnica

Measurement of a quantum particle position at two distant locations: a model

J. Dittrich

Published Aug 31, 2022• 0 citations
Publisher

Abstract

A simplified one-dimensional model of measurement of the position of a quantum particle by two distant detectors is considered. Detectors are modelled by quantum particles bounded in potential wells with just two bound states, prepared in the excited states. Their de-excitation due to the short range interaction with the measured particle is the signal for the detection. In the approximations of short time or small coupling between the particle and the measuring apparatuses, the simultaneous detection of the particle by both detectors is suppressed. The results extend to other models with two-level detectors.

DOI
10.14311/ap.2022.62.0445

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A simplified one-dimensional model of measurement of the position of a quantum particle by two distant detectors is considered. Detectors are modelled by quantum particles bounded in potential wells with just two bound states, prepared in the excited states. Their de-excitation due to the short range interaction with the measured particle is the signal for the detection. In the approximations of short time or small coupling between the particle and the measuring apparatuses, the simultaneous detection of the particle by both detectors is suppressed. The results extend to other models with two-level detectors.
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.
This is the second paper on a new formalism for relativistic quantum measurements. Here, we construct a fully relativistic model for detectors that takes into account the detector's state of motion, intrinsics dynamics, initial states and couplings to the measured field. The dual classical/quantum description of the detector is implemented by using a master-equation type of approximation for the coarse-grained pointer variables. Then we identify the probabilities that correspond to ideal measurements, i.e., measurements that are largely insensitive to modeling details of the apparatus. The Unruh-Dewitt and Glauber detectors are recovered at the appropriate limits. We employ our results to models of particle detection, photodetection and relativistic spin measurements, and we derive an ideal distribution for relativistic time-of-arrival probabilities.
A model for a one-dimensional quantum particle interacting with a point detector is developed and analyzed. The analysis leads to any simple, intuitively pleasing expression for the final density operator of the particle in terms of the freely propagating particle density operator, a projection operator representing the detector volume, and a probability density for the time at which the detector fires. This probability density is found explicitly to be the presence-time density given by ${\ensuremath{\mid}{\ensuremath{\psi}}_{f}\phantom{\rule{0.1em}{0ex}}(0,t)\ensuremath{\mid}}^{2}∕\ensuremath{\int}{\ensuremath{\mid}{\ensuremath{\psi}}_{f}\phantom{\rule{0.1em}{0ex}}(0,t)\ensuremath{\mid}}^{2}dt$ where ${\ensuremath{\psi}}_{f}\phantom{\rule{0.1em}{0ex}}(0,t)$ is the free particle wave function evaluated at the detector position (x=0)(x=0)(x=0). Moreover, it is shown that the resultant density operator transformation is a nonlinear transformation, not describable as a positive operator valued measure.
A typical quantum measurement process involves two steps. Step one is to separate the initial wavefunction into its spatial components which are then guided towards detectors at different locations. The second step involves the direct detection of the particle by a detector. Such a measurement process presupposes the existence of an interaction which can produce step one. The authors give a general mathematical formulation of the first step and establish the existence of a unitary evolution group which produces the separation. The results obtained support the view that quantum measurements are reducible to local position measurements.
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.
The von Neumann interaction between a particle and an apparatus has been considered in the measurement of the position of a particle when the interaction lasts for a finite amount of time. When the measurement has finite duration, both the motion of the pointer and the particle influences the result of the measurement. Provided that the particle is in an eigenstate of its position at the start of the measurement, the pointer will indicate the arithmetic average between the initial and final position of the particle. Furthermore, the probability that the pointer will indicate a given average value is equal to the transition probability for the undisturbed particle to experience the change in position. If the initial state of the pointer is a narrow wave packet, then for any initial state of the particle, the measurement yields, approximately, the undisturbed probability distribution for the position of the particle at the end of the measurement.