Generation of photon pairs in hyper-Raman Effects and its connection with transitions symmetries

In this paper we propose the model interaction Hamiltonian which takes in-to consideration the four photon process in interaction of dressed field with 39K atoms. Using the method of elimination of virtual states, we obtain this the effective interaction Hamiltonian which describe the simultaneously generation of photon pairs. The master equation for laser field is obtain, taking into account the good cavity limits in the process of two photon generation. The steady state solution of this equation for threshold case, which takes in to account the quantum fluctuation and photon statistics, is proposed. The cross-correlation function between Stokes and anti-Stokes photon pairs is studied as function of the evolution of the system.