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Physical Review Letters

Measurements of the probability distribution of the operationally defined quantum phase difference

J. W. Noh, A. Fougères, L. Mandeļ

Published Oct 18, 1993• 96 citations
Publisher

Abstract

From a series of phase difference measurements by our scheme 2, in which the field at one interferometer input is phase shifted progressively in steps of 18\ifmmode^\circ\else\textdegree\fi{}, we obtain the probability distribution of the phase difference. The results are in agreement with theoretical predictions based on our operational approach to the quantum phase.

DOI
10.1103/physrevlett.71.2579

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From a series of phase difference measurements by our scheme 2, in which the field at one interferometer input is phase shifted progressively in steps of 18\ifmmode^\circ\else\textdegree\fi{}, we obtain the probability distribution of the phase difference. The results are in agreement with theoretical predictions based on our operational approach to the quantum phase.
The recent operational approach to the phase operator of a quantized electromagnetic field is applied to the problem of determining the phase difference between two weak coherent fields in two different ways. The phase difference is determined directly, and indirectly by measuring each field in turn against the field of a strong local oscillator. We show by measurements with an eight-port interferometer that these two different schemes lead to quite different outcomes for the probability distribution of the phase difference, in agreement with our theory, but in disagreement with most other theoretical approaches.
We have measured the probability distribution of the phase difference Phi(2) - Phi(1) between two signal fields produced by parametric downconversion from two cascaded nonlinear crystals for several different values of the degree of mutual coherence. The experimental results are compared with theory, based on our operational approach to phase operators for quantum fields, and good agreement is obtained. These results are of interest because they constitute what we believe is the first experimental test of our phase theory for nonclassical states of light.
Our operational approach to the problem of identifying dynamical variables to represent the phase, or the sine and cosine of the phase, of a quantum field is reviewed. We start from classical optics and examine what is usually measured to determine the phase difference between two fields. We point out that difficulties in phase measurements exist even in the classical domain, and these difficulties persist when the classical relations are used as guides in the construction of phase operators. We find that different measurement procedures lead naturally to different phase operators, so that there appears to be no one "correct" phase operator. We carry out a series of experiments to measure the phase difference between two fields in various quantum states, and then compare the results with the predictions of our formalism and some other formalisms. We obtain good agreement between experiment and our theoretical approach in every case, so far without exception.
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 use the field-strength eigenstates, that is, the quadrature eigenstates rotated by an angle ccphi, to define a phase distribution of a single mode of the radiation field. A measurement procedure lies at the heart of this operational phase distribution: A balanced homodyne-detection scheme measures, in principle, the field-strength probability curve in its dependence on ccphi. The probability of finding a zero electric field plotted versus ccphi constitutes the proposed distribution. For a wide class of quantum states, this curve is in good agreement with the abstract phase probability curve obtained from phase states, but it is free of the familiar problems accompanying the notion of a Hermitian phase operator. The phase distribution of a quantum state has been achieved without using phase states; this can be summarized as ``phase without phase.''