The Degenerate Parametric OscillatorII: Phase-Space Analysisinthe Small-Noise Limit

Abstract

We turn now to an analysis of the degenerate parametric oscillator within the phase-space representations. Overall, the program is similar to the one carried out for the laser in Chap. 8, but with one fundamentally new element encountered here. At the start of the exercise, our premise is that the phase-space approach will lead to an acceptable Fokker–Planck equation. This premise will be seen to be incorrect, and two new topics emerge from the problems encountered. First, a new representation is introduced, the so-called positive P representation, formulated as a generalization of the Glauber–Sudarshan P representation to rescue the overall strategy of the quantum–classical correspondence in the small-noise limit; we make use of the positive P representation here, though its systematic development is postponed until Chap. 11. Second, the issue of quantum fluctuations outside the small-noise limit becomes a concern, since for large noise the positive P representation encounters difficulties of its own. These difficulties are explored in some detail in Chap. 12.