Generalized Dicke model: transitions $\Delta M=\pm 2$

A theory of two-atom quantum transitions with optical photon frequency doubling is developed on the basis of the third order QED effects consisting in virtual photon exchange and emission (absorption) of a double-frequency real photon. In these effects we distinguish two parts of interaction corresponding to intermediate states with positive and negative frequencies. According to this, we consider electronic and positronic polarizations of the medium that leads to the existence of two different mechanisms of frequency doubling. To describe coherent optical processes in electric dipole approximation we developed the generalized Dicke model for the system of $N$ two-level atoms with possible quantum transitions having $\Delta M=\pm 1,\pm 2$, where $M$ is population half-difference of resonant atomic levels. Formula for the intensity of quantum transitions with $\Delta M=\pm 2$ is derived for this model.