Number-phase properties of correlated chaotic fields

We study the arrangement of two down-conversion crystals with aligned and partially aligned idler beams in the quasimonochromatic approximation. Treating the signal fields as correlated chaotic modes, we arrive at a detailed description and illustration of their mathematically well defined state in terms of number and phase properties. We show it to be a mixed partial phase-difference state. We derive the closed formulae for the phase-difference distributions in this state derived from the phase-space distributions related to the normal, symmetrical and antinormal orderings of field operators and show that such distributions depend exclusively on measures of correlation which are variants of the modulus of the familiar degree of coherence. Quantum correlation influences fluctuations of photon-number sum, photon-number difference and quantum phase difference. Starting from a quasidistribution which carries the same information as the statistical operator, we arrive at the joint distributions of number sum and phase difference and the quasidistribution of number difference and phase difference.