Generalised P-representations in quantum optics

TLDR

Quantum operator representations vary by integration domain in a complex extension of canonical phase space, with the complex P‑representation defined on contours and the positive P‑representation covering the entire complex plane. The authors introduce a class of normal‑ordering representations that generalise the Glauber‑Sudarshan P‑representation via nondiagonal coherent‑state projections. These representations are constructed by employing nondiagonal coherent‑state projection operators to extend the Glauber‑Sudarshan P‑representation. The generalized representations enable practical solutions of quantum master equations, often yielding exact steady‑state solutions and positive‑semidefinite Fokker‑Planck diffusion, thereby providing genuine probability distributions useful for nonclassical statistics.

Abstract

A class of normal ordering representations of quantum operators is introduced, that generalises the Glauber-Sudarshan P-representation by using nondiagonal coherent state projection operators. These are shown to have practical application to the solution of quantum mechanical master equations. Different representations have different domains of integration, on a complex extension of the usual canonical phase-space. The 'complex P-representation' is the case in which analytic P-functions are defined and normalised on contours in the complex plane. In this case, exact steady-state solutions can often be obtained, even when this is not possible using the Glauber-Sudarshan P-representation. The 'positive P-representation' is the case in which the domain is the whole complex phase-space. In this case the P-function may always be chosen positive, and any Fokker-Planck equation arising can be chosen to have a positive-semidefinite diffusion array. Thus the 'positive P-representation' is a genuine probability distribution. The new representations are especially useful in cases of nonclassical statistics.