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Physical Review A

Photon counting, quantum phase, and phase-space distributions

Matthias Freyberger, Wolfgang P. Schleich

Published Jan 1, 1993• 107 citations
Publisher

Abstract

We analyze a recent proposal for a new phase operator along with the experiment [J. W. Noh, A. Foug\`eres, and L. Mandel, Phys. Rev. Lett. 67, 1426 (1991)] using phase-space functions. We show that the measured dispersion in phase results from a phase distribution given by the appropriate Wigner function integrated over radius. We relate this experiment to a different phase operator [H. Paul, Fortschr. Phys. 22, 657 (1974)] and the integrated Q function.

DOI
10.1103/physreva.47.r30

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We analyze a recent proposal for a new phase operator along with the experiment [J. W. Noh, A. Foug`eres, and L. Mandel, Phys. Rev. Lett. 67, 1426 (1991)] using phase-space functions. We show that the measured dispersion in phase results from a phase distribution given by the appropriate Wigner function integrated over radius. We relate this experiment to a different phase operator [H. Paul, Fortschr. Phys. 22, 657 (1974)] and the integrated Q function.
Generalizing a recent theoretical result obtained by Freyberger and Schleich [Phys. Rev. A 47, R30 (1993)], we show that the phase-measurement scheme proposed and realized by Noh, Foug\`eres, and Mandel [Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992)] amounts to measuring the Q function for the light under investigation, provided the reference beam (used for homodyne detection) is a very strong coherent field so that it can be described classically. The desired phase distribution follows from the Q function by averaging over the field amplitude. Since an analysis of an earlier proposal by Bandilla and Paul [Ann. Phys. (Leipzig) 23, 323 (1969)] to measure phase distributions via amplification led to just the same result, a perfect physical equivalence of those two approaches has thus been established.
We address the problem of identifying an operational prescription for quantum phase measurements. As is known, different experiments can lead to different measured phase operators. However, we show that ambiguities of interpretation can arise even if a single experiment, such as that of Noh, Foug\`eres, and Mandel [Phys. Rev. Lett. 67, 1426 (1991)], is chosen as defining a phase measurement. We show by reference to a simple but fundamental example that it is not possible to deduce a unique phase difference from the measurements.
Various quasiprobability distributions have been developed in the past using the Hilbert space of the single-mode light field. The development of a quasiprobability distribution associated with a phase operator has previously been impossible because of the absence of a unique Hermitian phase operator defined on the Hilbert space. Recently, however, Pegg and Barnett [Europhys. Lett. 6, 483 (1988); Phys. Rev. A 39, 1665 (1989)] and Barnett and Pegg [J. Mod. Optics 36, 7 (1989)] introduced a new formalism that does allow the construction of a Hermitian phase operator and associated phase eigenstates. In this paper we develop a quasiprobability distribution associated with the number and phase operators of the single-mode light field in the new formalism. The new distribution, which we call the number-phase Wigner function, has properties analogous to the Wigner function. We also derive the number-phase Wigner representation of number states, phase states, general physical states, coherent states, and the squeezed vacuum. We find this new representation has features that are related to the number and phase properties of states. For example, the number-phase Wigner representation of a number state is nonzero only on a circle, while the representation of a phase state is only nonzero along a radial line.