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Physical Review Letters

Is There a Unique Operator for the Phase Difference of Two Quantum Fields?

J. R. Torgerson, L. Mandeļ

Published May 20, 1996• 46 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.76.3939

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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 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.
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.
Recently, Torgerson and Mandel [Phys. Rev. Lett. 76, 3939 (1996)] have reported a disagreement between two schemes for measuring the phase difference of a pair of optical fields. We analyze these schemes and derive their associated phase-difference probability distributions, including both their strong and weak field limits. Our calculation confirms the main point of Torgerson and Mandel of the non-uniqueness of an operational definition of the phase distribution. We further discuss the role of postselection of data and argue that it cannot meaningfully improve the sensitivity.
We address the problem of strong local oscillator limit of the recently introduced operational approach to phase uncertainty measurements. In this limit the operator character of the local oscillator mode is neglected and its coherent amplitude tends to infinity. Thereby reasonable new operators for the sine and the cosine of the phase difference between an arbitrary signal field and the coherent local oscillator field are found which describe an already well-known physical situation connected with strong amplification of the signal field.
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.