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CRC Press eBooks

Continuous Quantum Nondemolition Measurements

Michael B. Mensky

Published Oct 19, 2017• 0 citations
Publisher

Abstract

This chapter discusses the type of measurement in the framework of the path-integral method. It begins from the simplest case of measurement of the linear momentum of a free particle, which allows one to see all the typical features characteristic of quantum nondemolition (QND) measurements. The idea of QND measurement is to measure a variable such that the unavoidable disturbance of the complementary one does not disturb the evolution of the chosen variable. The weight factor describes monitoring of the momentum, which for a free particle turns out to be a QND variable. This can be seen from the formula for the probability distribution of the measurement results. Usual relation between linear momentum and velocity is violated in the modified dynamics. From the physical point of view this is a consequence of measurement, of the influence of a measuring device.

DOI
10.1201/9781315140292-6

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This chapter discusses the type of measurement in the framework of the path-integral method. It begins from the simplest case of measurement of the linear momentum of a free particle, which allows one to see all the typical features characteristic of quantum nondemolition (QND) measurements. The idea of QND measurement is to measure a variable such that the unavoidable disturbance of the complementary one does not disturb the evolution of the chosen variable. The weight factor describes monitoring of the momentum, which for a free particle turns out to be a QND variable. This can be seen from the formula for the probability distribution of the measurement results. Usual relation between linear momentum and velocity is violated in the modified dynamics. From the physical point of view this is a consequence of measurement, of the influence of a measuring device.
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In this work a family of quantum nondemolition variables for the case of a particle caught in a Paul trap is obtained. Afterwards, in the context of the so called restricted path integral formalism, a continuous measuring process for this family of parameters is considered, and then the corresponding propagators are calculated. In other words, the time evolution of a particle in a Paul trap, when the corresponding quantum nondemolition parameter is being continuously monitored, is deduced. The probabilities associated with the possible measurement outputs are also obtained, and in this way new theoretical results emerge, which could allow us to confront the predictions of this restricted path integral formalism with the readouts of some future experiments. 1
The problem of chaotic behavior in quantum mechanics is investigated against the background of the theory of quantum-nondemolition (QND) measurements. The analysis is based on two relevant features: The outcomes of a sequence of QND measurements are unambiguously predictable, and these measurements actually can be performed on one single system without perturbing its time evolution. Consequently, QND measurements represent an appropriate framework to analyze the conditions for the occurrence of ``deterministic randomness'' in quantum systems. The general arguments are illustrated by a discussion of a quantum system with a time evolution that possesses nonvanishing algorithmic complexity.