Parametric down and up conversion in the Wigner representation of quantum optics

We study spontaneous parametric processes starting from the quantized held equations in the Heisenberg picture using the common approximation of treat ing the laser pump as classical. After a (standard) linearization procedure we pass to the Wigner representation and the process becomes formally equi valent to the parametric interaction of two incoming zero-point modes with a pumping laser wave coupled by an optically nonlinear medium. Then the tre atment of parametric down conversion and parametric up conversion are descr ibed by the Maxwell equations and look like a parametric amplification of t he vacuum field. The fourth-order correlation function between the signal a nd idler fields is obtained in a form that makes for easy comparison with e xperimental results.