The Positive<it>P</it> Representation

Abstract

The positive P representation generalizes the Glauber–Sudarshan P representation. It was introduced by Drummond and Gardiner [11.1] specifically to deal with the problem of phase-space equations of motion in the Fokker– Planck form that do not possess positive semidefinite diffusion. The degenerate parametric oscillator provides one of the simplest examples. Although this representation has been used extensively by its originators and their colleagues—especially by Drummond and collaborators to treat squeezing in optical fibers [11.2, 11.3, 11.4, 11.5,11.6,11.7,11.8] and more recently quantum gases [11.9,11.10,11.11,11.12]—it is probably fair to say that it is poorly understood outside this circle of the initiated. One reason for this, certainly, is that problems with non-positive-semidefinite diffusion can often be avoided using either the Wigner or the Q representation, as we have seen in Sect. 10.1.2; why would one become entangled in the mysteries of negative diffusion if it can be avoided? Beyond this, any inclination against the representation is reinforced by the appearance that there is something unreasonable, or at least unnecessarily complicated, about it: unreasonable because the positive P representation allows real quantities to be driven by imaginary noise; unnecessarily complicated because it accomplishes this feat by representing each field mode by a pair of complex phase-space variables instead of just one—the positive P distribution is defined within a phase space that has double the usual number of dimensions.