The problem of the threshold and growth of parametric oscillation in a lossy optical cavity is treated quantum theoretically. It is shown that during the growth of oscillation each of the generated modes is statistically as chaotic as those observed previously in the lossless case, and is so chaotic that the additional uncertainty introduced by the loss processes has no further effect on the single-mode photon distributions. However, there is considerable cross correlation between the idler and signal modes. The cross correlation in the numbers of idler and signal quanta is essentially complete for large numbers, but is shown to be reduced when the numbers are smaller, because of noise due to loss. Correlations in phase are also investigated and it is shown that, under certain statistical conditions necessary for phase to be a relevant concept, the phase correlation is also essentially complete.

• Received 25 February 1970

DOI:https://doi.org/10.1103/PhysRevA.3.180