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Physical Review A

Comment on ‘‘Further investigations of the operationally defined quantum phase’’

Z. Hradil, J. Bajer

Published Aug 1, 1993• 6 citations
Publisher

Abstract

The normalization of the phase experiment of the scheme 1 in the paper of Noh, Foug`eres, and Mandel [Phys. Rev. A 46, 2840 (1992)] is inappropriate. The correct treatment of this measurement tends to the same result as in their scheme 2.
DOI
10.1103/physreva.48.1717

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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.
Interpretational difficulties of a quantum measurement [J. W. Noh, A. Foug\`eres, and L. Mandel, Phys. Rev. A 45, 424 (1992)] are pointed out. The nonunitarity of the respective exponential phase operator reflects an enhanced noise, which can be reduced by discarding ambiguous output data. Consequently, dispersion of such measurements does not represent a meaningful performance measure relevant to the input state. Moreover, the proposed model of quantum phase measurements provides a discrete spectrum of the quantum phase operator, whereas the phase-shift parameter is continuous.
In response to Hradil's Comment [preceding Comment, Phys. Rev. A 47, 4532 (1993)], we point out that his prescription for putting both the cosine and sine equal to zero when the data yield n3{\mathit{n}}_{3}n3​=n4{\mathit{n}}_{4}n4​ and n5{\mathit{n}}_{5}n5​=n6{\mathit{n}}_{6}n6​ can lead to meaningless results. Moreover, Hradil's Eq. (5) for obtaining D2{\mathit{D}}^{2}D2 from the measured variance leads to disagreement with experiment. Finally, it is possible to obtain a continuous spectrum of phases with our technique.
It is shown that there exist two kinds of operational (or measured) phase operators, those of the Hradil type [Phys. Rev. A 47 (1993), 4532], defined on the entire Hilbert space, and those of the Noh-Fougères-Mandel type [Phys. Rev. A 45 (1992), 424], defined only on a subspace. As a consequence, strictly speaking, the Noh-Fougères-Mandel measured phase cannot be defined in the case of, for example, coherent states, squeezed states and the vacuum. The difference between these operators is made clear by using the concept of the Bargmann distance between a quantum state and its “reduced” state. Since Noh et al. did not use the correct variances of the Susskind-Glogower and Pegg-Barnett phase operators when they compared their data with the results predicted through use of these operators, the correct expressions in that case are presented. Comparing the measured operators with the Hermitian phase operator in the extended Susskind-Glogower formalism, we show that the quantum phase given by the Hermitian phase operator is more appropriate than the measured phases. The variance of the Hermitian operator in the vacuum is consistent with the complementarity of the photon number and phase, which implies that the vacuum has a uniform phase distribution. Hradil's measured operator does not have such a property. Although the Noh-Fougères-Mandel operator has such a property in a particular limit, this operator does not represent, in principle, a meaningful measure of the phase measurement in a typical quantum region, because the Bargmann distance becomes large in such a region.
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.