PHYSICAL REVIEW A
VOLUME 46, NUMBER 5
of the operationally defined quantum phase
J. W. Noh, A. Fougeres, and L. Mandel
of Physics and Astronomy, University of Rochester, Rochester,
(Received 3 February 1992)
The formalism that we have previously developed for the phase difference between twoquantized electromagnetic fields, which is intimately connected with the measurement process, is explored further,
We calculate the higher moments of the measured cosine and
both theoretically and experimentally.
sine operators for certain two-mode Fock states n
), and show how the measurement itself exerts a
bias on the outcome. We find that the corresponding phase difference becomes uniform overthe interval
On the other hand, the phase difference associated with the product
0 to 2~ only in the limit n &, n&
of a coherent state ~v ) with a Fock state can be random for large ~v~, 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 arealso 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.
PACS number(s): 42. 50.Wm, 03.65. —
The problem of identifying the dynamical variable(s)
representing the phase of a quantum field has already
been tackled in numerousdifferent ways over the years,
without consensus being achieved, and it has led to as
26]. Instead of introducing a
many different answers [1 —
phase operator, Agarwal et al.  have attempted to
define a probability density for the quantum phase via the
phase states used by Pegg and Barnett , and Schleich,
Bandilla, and Paul  have defined a probability density
We have recentlyapbased on the Q distribution.
proached the problem operationally, starting with an
analysis of what is usually measured in classical optics
when the phase difference is to be determined, and then
the formalism into the quantum
[29,30]. This leads to different operators CM, S~ for the
measured cosine and sine of the phase difference for
schemes, and tothe conclusion
that there may not be one universal dynamical variable
independent of the measurement scheme. As in semiclasin the
sical optics, one encounters
definitions of the measured operators CM, SM when the
photon numbers are very small, which is ultimately
resolved by making a deliberate choice. As in semiclassical optics one is led to make a distinction between themeasured and the inferred phases, corresponding to the
distinction between what can be deduced from a single
and from an ensemble of measurements.
We find that C~, SM commute when the measurement
yields the cosine and sine of the phase difference simu1taneously, but not when separate measurements are required.
We have previously applied the formalism to several
different quantumstates , and compared the predictions of our theory with those based on the Susskind and
Glogower  and those based on the Pegg and Barnett
 operators. Significant differences were encountered.
We have also previously performed photon-counting
measurements with coherent input fields , and found
excellent agreement with our theory over a range that extends more than 2 ordersof magnitude beyond previous
In the following, after a review of the formalism, we
extend both the theory and the measurements.
different experimental schemes are used, which have been
treated before [32 —
41], leading to different C'M, Sxt operators. We also measure some of the higher moments of
CM for comparison with theory. The predictions of our
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