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Physical Review A

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

J. W. Noh, A. Fougères, L. Mandeļ

Published Aug 1, 1993• 6 citations
Publisher

Abstract

It is pointed out in response to Hradil and Bajer [preceding Comment, Phys. Rev. A 48, 1717 (1993)] that C${\mathrm{^}}_{\mathit{M}}$ and S${\mathrm{^}}_{\mathit{M}}$ cannot both be measured together by scheme 1, whereas they can by scheme 2. There is no phase angle whose cosine and sine both vanish, and putting C+iS equal to zero for certain measurement outcomes breaks the connection with phase.

DOI
10.1103/physreva.48.1719

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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 ${\mathit{n}}_{3}$=${\mathit{n}}_{4}$ and ${\mathit{n}}_{5}$=${\mathit{n}}_{6}$ can lead to meaningless results. Moreover, Hradil's Eq. (5) for obtaining ${\mathit{D}}^{2}$ from the measured variance leads to disagreement with experiment. Finally, it is possible to obtain a continuous spectrum of phases with our technique.

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Physical Review A•Jan 1, 1992•237 citations
It is pointed out in response to Hradil and Bajer [preceding Comment, Phys. Rev. A 48, 1717 (1993)] that C{\mathrm{^}}_{\mathit{M}} and S{\mathrm{^}}_{\mathit{M}} cannot both be measured together by scheme 1, whereas they can by scheme 2. There is no phase angle whose cosine and sine both vanish, and putting C+iS equal to zero for certain measurement outcomes breaks the connection with phase.
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.
We have previously developed a formalism for identifying quantum operators corresponding to the sine and cosine of the phase differences between two quantum fields.1 Our approach is based on what is actually measured in an experiment, and it leads to different operators for different measurement schemes. Predictions based on our theory differ from those based on the Susskind-Glogower and the Pegg-Barnett operators. We now present the results of new experiments designed to test the validity of our approach, including measurements of the higher moments of the sine and cosine for different quantum states. The results of all measurements confirm our formalism in every case.
We examine the problem of determining the phase difference between two optical fields, first for classical and later for quantum fields, by reference to two simple measurement schemes that yield the sine and/or cosine of the phase difference between classical fields. We show that certain difficulties exist even within the framework of semiclassical radiation theory when the field is very weak, and particularly when amplitude and phase fluctuations are correlated. We find that a clear distinction has to be made between the measured values of the sine or cosine and the values that can be inferred from a series of repeated measurements. A corresponding distinction can be made also for a quantum field, although the interpretation is not the same. The dynamical variables chosen to represent the cosine and sine that emerge from the discussion of the measurement schemes commute when the sine and cosine are obtained together, but not when the measurement yields one or the other. These sine and cosine operators have well-defined values only when there is a large dispersion of the photon number. We arrive at expressions for the moments of the measured and of the inferred sines and cosines that differ from most previous treatments. The expressions are applied to optical fields in several different quantum states. Only for the Fock state and for the so-called phase state, which was treated recently at some length by Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)], do the measured and the inferred moments coincide. Our analysis of the problem of phase measurement leads to the conclusion that the appropriate dynamical variables for the measured sine and cosine depend on the measurement scheme, and that different schemes correspond to different operators.