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arXiv:quant-ph/0106084 · Physica Scripta

The Pegg-Barnett Formalism and Covariant Phase Observables

PEKKA J. LAHTI, Juha-Pekka Pellonpää

Published Jan 1, 2002• 17 citations
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Abstract

We compare the Pegg-Barnett (PB) formalism with the covariant phase observable approach to the problem of quantum phase and show that PB-formalism gives essentially the same results as the canonical (covariant) phase observable. We also show that PB-formalism can be extended to cover all covariant phase observables including the covariant phase observable arising from the angle margin of the Husimi Q -function.

DOI
10.1238/physica.regular.066a00066

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We compare the Pegg-Barnett (PB) formalism with the covariant phase observable approach to the problem of quantum phase and show that PB-formalism gives essentially the same results as the canonical (covariant) phase observable. We also show that PB-formalism can be extended to cover all covariant phase observables including the covariant phase observable arising from the angle margin of the Husimi Q -function.
Abstract Covariant phase observables constitute a simple solution of the quantum phase problem of a single‐mode optical field. They share three important properties: their range of values is the phase interval [0,2π), they are covariant under shifts generated by the number operator (which is necessary for coherent state phase measurements), and they have the uniform phase distribution in number states. Moreover, some phase observables have been measured (e.g. the phase observable associated to the Q‐function). The canonical phase observable has some additional properties which distinguish it from other covariant phase observables: it generates number shifts, it is uniquely associated to the polar decomposition of the lowering operator (Dirac's idea [1]), it has a projection valued covariant dilation (Newton's extension [2]), and it has a projection valued discretization (the Pegg‐Barnett formalism [3]). The single‐mode covariant phase theory can easily be extended to the two‐mode theory of covariant phase difference observables. Finally, most of all phase theories have connections to the covariant phase theory.
The problem of defining a Hermitian quantum phase operator is nearly as old as quantum mechanics itself. Throughout the years, a number of solutions have been proposed, ranging from abstract operator formalisms to phase-space methods. In this work, we make an explicit connection between two of the most prominent approaches by proving that the probability distribution of phase in the Paul formalism follows exactly from the Pegg-Barnett formalism by combining the latter with the quantum-limited amplifier channel. Our findings suggest that the Paul framework may be viewed as a semiclassical limit of the Pegg-Barnett approach.
For a long time, about 60 years, the phase as an observable of the quantized light fields has r emained a fundamental problem with considerable difficulties on quantum mechanics and also q uantum optics.Recently, however, the hermitian phase operator and its eigenstates in a finite H ilbert space are found by Pegg and Barnett.In this paper, we provide a historical review and t he background of the problem on the phase operator, explore the Pegg-Barnett formalism, and s how the recent developements of this fields and its application to various light fields.Further problems on this formalism and other approaches are also discussed.
Covariant phase difference observables are determined in two different ways, by a direct computation and by a group theoretical method. A characterization of phase difference observables which can be expressed as the difference of two phase observables is given. The classical limits of such phase difference observables are determined and the Pegg-Barnett phase difference distribution is obtained from the phase difference representation. The relation of Ban’s theory to the covariant phase theories is exhibited.