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Physical Review A

Antinormal ordering of Susskind-Glogower quantum phase operators

Constantin Brif, Y. Ben-Aryeh

Published Sep 1, 1994• 21 citations
Publisher

Abstract

A quantum phase description is developed by the use of the antinormal ordering of Susskind-Glogower phase operators. The theory is studied in the infinite-dimensional Hilbert space and it is closely related to the Pegg-Barnett quantum phase description in the finite-dimensional Hilbert space with a proper limiting procedure. Various applications of the antinormal ordering technique are presented.

DOI
10.1103/physreva.50.2727

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A quantum phase description is developed by the use of the antinormal ordering of Susskind-Glogower phase operators. The theory is studied in the infinite-dimensional Hilbert space and it is closely related to the Pegg-Barnett quantum phase description in the finite-dimensional Hilbert space with a proper limiting procedure. Various applications of the antinormal ordering technique are presented.
Abstract In quantum mechanics the position and momentum operators are related to each other via the Fourier transform. In the same way, here we show that the so-called Pegg–Barnett phase operator can be obtained by the application of the discrete Fourier transform to the number operators defined in a finite-dimensional Hilbert space. Furthermore, we show that the structure of the London–Susskind–Glogower phase operator, whose natural logarithm gives rise to the Pegg–Barnett phase operator, is contained in the Hamiltonian of circular waveguide arrays. Our results may find applications in the development of new finite-dimensional photonic systems with interesting phase-dependent properties.
A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ and so the correponding limit points are ordinary operators on Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.
A similarity between the photon annihilation and creation operators and the Susskind-Glogower exponential phase operators motivates the introduction and study of quasidistributions of the cosine and sine of phase. These quasidistributions are related to the standard and antistandard orderings of trigonometric phase operators and to the normal, antinormal, and symmetrical orderings of the exponential phase operators. The symmetrical ordering is connected to an optical ideal tomography of the appropriate quasidistribution from the rotated Susskind-Glogower cosine operator.