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Journal of Physics A Mathematical and General

Phase observables, phase operators and operator orderings

Juha-Pekka Pellonpää

Published Sep 14, 2001• 16 citations
Publisher

Abstract

We represent quantum phase observables as phase shift covariant normalized positive operator measures. The phase operators are the first moment operators of the phase observables. A phase operator determines the associated phase observable uniquely. We show that the Cahill–Glauber s -ordered phase operators are determined by phase shift covariant generalized operator measures, which are ordinary operator measures whenever Re s < 0 and phase observables when s ≤ −1. The Wigner–Weyl quantized phase operator is not determined by any phase observable. We investigate the classical limit of covariant (generalized) operator measures in coherent states.

DOI
10.1088/0305-4470/34/38/314

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We represent quantum phase observables as phase shift covariant normalized positive operator measures. The phase operators are the first moment operators of the phase observables. A phase operator determines the associated phase observable uniquely. We show that the Cahill–Glauber s -ordered phase operators are determined by phase shift covariant generalized operator measures, which are ordinary operator measures whenever Re s < 0 and phase observables when s ≤ −1. The Wigner–Weyl quantized phase operator is not determined by any phase observable. We investigate the classical limit of covariant (generalized) operator measures in coherent states.
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The questions studied in this thesis are centered around the moment operators of a quantum observable, the latter being represented by a normalized positive operator measure. The moment operators of an observable are physically relevant, in the sense that these operators give, as averages, the moments of the outcome statistics for the measurement of the observable. The main questions under consideration in this work arise from the fact that, unlike a projection valued observable of the von Neumann formulation, a general positive operator measure cannot be characterized by its first moment operator. The possibility of characterizing certain observables by also involving higher moment operators is investigated and utilized in three different cases: a characterization of projection valued measures among all the observables is given, a quantization scheme for unbounded classical variables using translation covariant phase space operator measures is presented, and, finally, a mathematically rigorous description is obtained for the measurements of rotated quadratures and phase space observables via the high amplitude limit in the balanced homodyne and eight-port homodyne detectors, respectively. In addition, the structure of the covariant phase space operator measures, which is essential for the above quantization, is analyzed in detail in the context of a (not necessarily unimodular) locally compact group as the phase space.
We consider questions related to a quantization scheme in which a classical variable f:Ω→R on a phase space Ω is associated with a (preferably unique) semispectral measure Ef, such that the moment operators of Ef are required to be of the form Γ(fk), with Γ a suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in the Hilbert space of the quantum system. In particular, we investigate the situation where the map Γ is implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ω is first taken to be an abstract measurable space, then a locally compact unimodular group, and finally R2, where we determine explicitly the relevant operators Γ(fk) for certain variables f, in the case where the quantization map Γ is implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.
Abstract Coherent states, in Wigner representation of quantum mechanics which is classical in structure with Wigner function playing the role of classical phase space probability distribution, and standard quantum mechanical expression for average values of corresponding Weyl quantum operators, general expression for matrix elements of the operator corresponding to phase of the oscillator by Weyl procedure, are obtained in the |n〉 basis. The general expression for matrix elements in the same basis of any other operator is also obtained. As the only mathematical technique necessary in the proposed procedure is a simple change of variables to polar coordinates in the corresponding integrals, this way to introduce Weyl phase operator, and some other operators in Weyl quantization, greatly simplifies necessary derivations and calculations.