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Progress of Theoretical Physics

Extended Pegg-Barnett Phase Operator

K. Kakazu

Published Oct 1, 2001• 1 citation
Publisher

Abstract

Vorontsov and Rembovsky pointed out recently a problem that unphysical states appear after an approximate quantum measurement of the Pegg-Barnett phase [Phys. Lett. A 254 (1999), 7; 262 (1999), 486]. To give a solution to this problem, we introduce a quantum phase in the extended number-state space and show that there is no Vorontsov-Rembovsky problem in the extended space. In the infinite-dimensional limit, the present extended formalism gives the same results in almost all cases as the Pegg-Barnett formalism. Also, it is shown that unphysical states can be suppressed by using the projection operator to the physical number space.

DOI
10.1143/ptp.106.721

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Vorontsov and Rembovsky pointed out recently a problem that unphysical states appear after an approximate quantum measurement of the Pegg-Barnett phase [Phys. Lett. A 254 (1999), 7; 262 (1999), 486]. To give a solution to this problem, we introduce a quantum phase in the extended number-state space and show that there is no Vorontsov-Rembovsky problem in the extended space. In the infinite-dimensional limit, the present extended formalism gives the same results in almost all cases as the Pegg-Barnett formalism. Also, it is shown that unphysical states can be suppressed by using the projection operator to the physical number space.
Considering an approximate quantum phase measurement, we show that the extended Pegg-Barnett formalism with dimension (2 s + 1) is not plagued by Vorontsov and Rembovsky problem whereby unphysical states can appear after the phase measurement. The approximate phase measurement in this formalism gives, in the large s limit, the same result as that in the Susskind-Glogower formalism. The difficulty of defining a Hermitian phase operator, caused by the periodicity of the phase angle, is overcome by introducing an extended number-state space and by making use of periodic generalized functions (distributions) and their Fourier transforms. This space is the large s limiting case of the extended Pegg-Barnett formalism. Similarly, with the help of periodic generalized functions, Hermitian phase operators can be defined in other infinite-dimensional extended theories. It is also shown that approximate phase measurements give the same results in all these extended formalisms.
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.
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.
Following the discussion - in state-space language - presented in a preceding paper, we work on the passage from the phase-space description of a degree of freedom described by a finite number of states (without classical counterpart) to one described by an infinite (and continuously labelled) number of states. With this it is possible to relate an original Schwinger idea to the Pegg-Barnett approach to the phase problem. In phase-space language, this discussion shows that one can obtain the Weyl-Wigner formalism, for both Cartesian and angular coordinates, as limiting elements of the discrete phase-space formalism.