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Optics Letters

Measurements of the probability distribution of the phase difference between two quantum fields

A. Fougères, Carlos Henrique Monken, L. Mandeļ

Published Nov 1, 1994• 11 citations
Publisher

Abstract

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.

DOI
10.1364/ol.19.001771

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Physical Review Letters•May 20, 1996•46 citations
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.
Although the operational approach to the quantum-phase problem developed earlier by us [Phys. Rev. A 45, 424 (1992)] has now been confirmed experimentally for several different states of the electromagnetic field, so far no measurements have been reported with partially coherent light beams. This situation is remedied in the present work, where we report the results of measurements on light obeying thermal statistics with a variable degree of mutual coherence. Once again, we find agreement between experiment and our theoretical approach.
The problem of identifying the dynamic variable representing the phase difference between quantized electromagnetic fields is approached via the measurement process. By comparison with classical optics, certain operators corresponding to the measured sine and cosine of the phase difference are identified, but different measurement schemes lead naturally to different operators. First and second moments of the sine and cosine operators are calculated for a two-mode field in a coherent state. The theory is then tested in a photon counting experiment, for mean detected photon numbers varying from 30 to 10-2 which extends the range of previous measurements 300- fold. Good agreement is found between theory and experiment.
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.