Further investigations of the operationally defined quantum phase

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 ‖${\mathit{n}}_1$,${\mathit{n}}_2$〉, 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 2π only in the limit ${\mathit{n}}_1$,${\mathit{n}}_2$→∞. On the other hand, the phase difference associated with the product 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.