We address the problem of identifying the dynamical variables representing the phase of a quantum field by analyzing what is measured in two simple homodyne experiments. This leads us to identify operators CM.SM corresponding to the measured cosine and sine of the phase difference. The predictions of the theory are tested in an experiment in which the input fields are derived from a highly stable He:Ne laser. Photon-counting measurements extending from mean photon numbers 30 down to 10"" show good agreement between theory and experiment. The implications for identifying the "correct" phase operator are briefly discussed. PACS numbers: 42.50.Wm, 03.65.Bz sinlThe question of which is the proper dynamical variable corresponding to the phase of a quantum field has been the subject of discussion and controversy for a long time. The problem appeared to be solved by the introduction of two Hermitian dynamical variables analogous to the sine and cosine of the phase [1,2], but as the two variables do not commute, this has tended to be regarded as an unsatisfactory solution.There have been numerous attempts to construct other, more satisfactory phase operators [3-13], and we mention particularly the recent extensive work of Pegg and Barnett [9-11], A few experiments [14][15][16] were also reported in which phase differences and their fluctuations were measured as a function of average photon number, and attempts were made to test some of the definitions against experiments [17][18][19][20][21], but no clear conclusion emerged.We have approached the phase problem in a diff'erent, more operational way, by analyzing what is actually measured in a typical homodyne experiment to determine the phase diff'erence between two fields, both in classical and in quantum optics [22]. This leads to different dynamical variables CM^SM representing the cosine and sine of the measured phase difference for different experiments. We examine two measurement schemes for extracting the phase, which have been considered before [23][24][25][26][27][28][29][30]. For comparison we then present the results of experiments in which the phase difference between two coherent fields is measured for mean detected photon numbers im) ranging from =30 down to = 10~l This range extends more than 2 orders of magnitude beyond that of previously reported phase measurements.Classical theory.-Measurements of the phase difference between two light beams are generally based on some kind of interference or homodyne (or heterodyne) experiment like that shown in Fig. 1. Two incoming light beams are mixed by a 50:50 beam splitter BS, resulting in mixed signals of intensities hJ^ emerging from the two exit ports. The numbers of photoelectric pulses mj^m^ detected by two similar photodetectors D3,D4 of quantum efficiency a in some counting time interval T are registered. When the numbers rrij are large, they are representative of the integrated light intensity Wj 1426 © 1991 The American Physical Society ^aSl^^Ij{t')dt' 0=3,4). Within the domain of classical optics ...

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Prom a series of phase difference measurements by our scheme 2, in which the field at one interferometer input is phase shifted progressively in steps of 18°, we obtain the probability distribution of the phase difference. The results are in agreement with theoretical predictions based on our operational approach to the quantum phase.

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