Quantum theory of optical phase correlations

We extend the theory of the Hermitian optical phase operator to analyze the quantum phase properties of pairs of electromagnetic field modes. The operators representing the sum and difference of the two single-mode phases are simply the sum and difference of the two single-mode phase operators. The eigenvalue spectra of the sum and difference operators have widths of 4\ensuremath{\pi}, but phases differing by 2\ensuremath{\pi} are physically indistinguishable. This means that the phase sum and difference probability distributions must be cast into a 2\ensuremath{\pi} range. We obtain mod(2\ensuremath{\pi}) probability distributions for the phase sum and difference that unambiguously reveal the signatures of randomness, phase correlations, and phase locking. We use our approach to investigate the phase sum and difference properties for uncorrelated modes in random and partial phase states and the phase-locked properties of the two-mode squeezed vacuum states. We reveal the fundamental property of two-mode squeezed states that the phase sum is locked to the argument of the squeezing parameter. The variance of the phase sum depends dilogarithmically on 1+tanhr, where r is the magnitude of the squeezing parameter, vanishing in the large squeezing limit.