Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Measurement of phase differences between two partially coherent fields

A. Fougères, J. W. Noh, Timothy P. Grayson, L. Mandeļ

Published Jan 1, 1994• 23 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.49.530

Related papers

Phase measurements of partially coherent light beams by photon counting

A. Fougères, J. W. Noh, L. Mandeļ

The technique developed previously for studying phase differences between quantum fields by photon counting1 has been applied to two partially coherent light beams with controllable degree of coherence.

Annual Meeting Optical Society of America•Jan 1, 1993•0 citations

New investigations of the phase distribution with variable coherence

A. Fougères, J. R. Torgerson, L. Mandeļ

Abstract New measurement of the probability distribution of the optical phase difference between two light beams have been carried out for different degrees of mutal coherence, when the average number of detected photons is of unity. The measurement time was made much longer than the coherence time. Good agreement with our operational theory for the quantum phase is again obtained.

Optics Communications•Feb 1, 1994•9 citations

Phase measurement of a quantum field

J. W. Noh, A. Fougeres, L. Mandeļ

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.

Annual Meeting Optical Society of America•Jan 1, 1991•0 citations

Further investigations of the operationally defined quantum phase

J. W. Noh, A. Fougères, L. Mandeļ

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}${\mathit{n}}_{1}$,${\mathit{n}}_{2}$〉, 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 ${\mathit{n}}_{1}$,${\mathit{n}}_{2}$\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.

Physical Review A•Sep 1, 1992•117 citations

Phase measurement andQfunction

Ulf Leonhardt, Henryk Paul

Generalizing a recent theoretical result obtained by Freyberger and Schleich [Phys. Rev. A 47, R30 (1993)], we show that the phase-measurement scheme proposed and realized by Noh, Foug\`eres, and Mandel [Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992)] amounts to measuring the Q function for the light under investigation, provided the reference beam (used for homodyne detection) is a very strong coherent field so that it can be described classically. The desired phase distribution follows from the Q function by averaging over the field amplitude. Since an analysis of an earlier proposal by Bandilla and Paul [Ann. Phys. (Leipzig) 23, 323 (1969)] to measure phase distributions via amplification led to just the same result, a perfect physical equivalence of those two approaches has thus been established.

Physical Review A•Apr 1, 1993•131 citations

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

A. Fougères, Carlos Henrique Monken, L. Mandeļ

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.

Optics Letters•Nov 1, 1994•11 citations
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 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.
Generalizing a recent theoretical result obtained by Freyberger and Schleich [Phys. Rev. A 47, R30 (1993)], we show that the phase-measurement scheme proposed and realized by Noh, Foug\`eres, and Mandel [Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992)] amounts to measuring the Q function for the light under investigation, provided the reference beam (used for homodyne detection) is a very strong coherent field so that it can be described classically. The desired phase distribution follows from the Q function by averaging over the field amplitude. Since an analysis of an earlier proposal by Bandilla and Paul [Ann. Phys. (Leipzig) 23, 323 (1969)] to measure phase distributions via amplification led to just the same result, a perfect physical equivalence of those two approaches has thus been established.
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.