Dependence on crystal parameters of the correlation time between signal and idler beams in parametric down conversion calculated in the Wigner representation

The theory of parametric down conversion within the framework of the Wigner representation has been treated recently in a series of papers using the s tandard model Hamiltonian. Here we take a more fundamental point of view st udying the mechanism, inside the crystal, for the production of the signal and idler beams. We begin from the evolution equations for the quantum fiel d operators, pass to the Wigner function and solve the resulting (Maxwell) equations with the use of the Green's function method. We derive the time d ependence of the coincidence detection probability as a function of the par ameters of the nonlinear crystal (in particular the length) the radius of t he pumping beam, and the bandwidth of the filters in front of thr detectors .