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Physica Scripta

Measuring the Quantum Phase of the E.M. Field by Interference as ?0

J. R. Torgerson, L. Mandeļ

Published Jan 1, 1998• 2 citations
Publisher

Abstract

We show that the measured probability distribution of the phase difference between two electromagnetic fields can be sharpened progressively as data involving more and more detected photons are discarded, and that the phase difference can be well-defined even when the mean photon number n → 0. In our operational phase theory the variance of the directly measured phase difference has no lower bound other than zero, and this is consistent with the experimental results.

DOI
10.1238/physica.topical.076a00110

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We show that the measured probability distribution of the phase difference between two electromagnetic fields can be sharpened progressively as data involving more and more detected photons are discarded, and that the phase difference can be well-defined even when the mean photon number n → 0. In our operational phase theory the variance of the directly measured phase difference has no lower bound other than zero, and this is consistent with the experimental results.
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 formalism that we have previously developed for the phase difference between two quantized electromagnetic fields, which is intimately connected with the measurement process, is explored further, both theoretically and experimentally. We calculate the higher moments of the measured cosine and sine operators for certain two-mode Fock states \ensuremath{\Vert}n1{\mathit{n}}_{1}n1​,n2{\mathit{n}}_{2}n2​〉, and show how the measurement itself exerts a bias on the outcome. We find that the corresponding phase difference becomes uniform over the interval 0 to 2\ensuremath{\pi} only in the limit n1{\mathit{n}}_{1}n1​,n2{\mathit{n}}_{2}n2​\ensuremath{\rightarrow}\ensuremath{\infty}. On the other hand, the phase difference associated with the product of a coherent state \ensuremath{\Vert}v〉 with a Fock state can be random for large \ensuremath{\Vert}v\ensuremath{\Vert}, because of the availability of an infinite number of photons. Several of our theoretical predictions are compared with predictions based on the Susskind-Glogower and the Pegg-Barnett operators, and they are also tested by experiment. We find that the experimental results confirm our theory in every case, and this includes tests of the higher moments of the measured cosine operator.
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.
The uncertainty principle applied to electromagnetic fields implies that the field phase and photon number cannot simultaneously have arbitrarily narrow distributions. Here we describe an experiment which determines, for the first time, the variances of phase and photon number for a field containing small numbers of photons. The measurements allow a test of the uncertainty relation for the phase and photon number of a mode of the electromagnetic field when the field mode is in a very weak coherent state. The phase of the field is described by the Pegg-Barnett phase operator formalism. [1] The measurements are accomplished by experimentally determining the quantum state of the field by using optical homodyne tomography. [2] This method of state characterization requires that an ensemble of many measurements be performed. [3] In this case, the relevant interpretation of the uncertainty relation is that the state of the field does not have well defined phase and photon number.