Forward-Backward Squeezing Propagator

Abstract

I show that a usual propagator cannot be defined for the pseudo-diffusion equation of the Q-functions. Instead, a forward-backward propagator is defined, which motivated a generalization of Cahill-Glauber interpolating operator. Our generalized operator ${\bf Q}(p,q;σ_p^{-1},σ_q)$ depends on two squeezing parameters $σ_p$ and $σ_q$, and is shown to obey a generalized pseudo-diffusion equation or a diffusion equation, depending on the curve $(σ_p(μ),σ_q(μ))$ along which one moves in the $(σ_p,σ_q)$ plane. An algorithm is also given for squeezing Q functions directly, using one-dimensional diffusion propagators.