Generalized Husimi operator as the rotated and squeezed coherent state projector and its application

We propose a so-called generalized Husimi operator Delta(gh)(x, p, kappa, theta) corresponding to the generalized Husimi function and prove Delta(gh)(x, p, kappa, theta) = vertical bar x, p, kappa, theta > < x, p, kappa, theta vertical bar, i.e. Delta(gh)(x, p, kappa, theta) is actually a pure state projector. Further, vertical bar x, p, kappa, theta > turns out to be a rotated and squeezed coherent state 1/root 2R(theta) S(root kappa)D(beta)vertical bar 0 >. As an application, we propose a measured scheme on the generalized Husimi function in the context of a single trapped ion.