Engenharia

PHYSICAL REVIEW A

VOLUME 46, NUMBER 5

Further investigations
Department

1

SEPTEMBER 1992

of the operationally defined quantum phase

J. W. Noh, A. Fougeres, and L. Mandel of Physics and Astronomy, University of Rochester, Rochester,

New York

14627

(Received 3 February 1992)

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,
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 over the 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 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.
~

„n,

~ ~.

PACS number(s): 42. 50.Wm, 03.65. — w I.

INTRODUCTION

The problem of identifying the dynamical variable(s) representing the phase of a quantum field has already been tackled in numerous different 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. [27] have attempted to define a probability density for the quantum phase via the phase states used by Pegg and Barnett [14], and Schleich,
Bandilla, and Paul [28] have defined a probability density
We have recently