Collective operator method for the resonance fluorescence in photonic band gap

Summary form only given. In this work we suggest and discuss the method of collective atomic-field operators for the resonance fluorescence of two-level atom with the transition frequency near the band gap edge. The method is based on the formal diagonalization of the interaction Hamiltonian in the absence of driving. In the absence of driving the method also allows to obtain the exact solution for the case when there are two photons in the field. We show that due to closeness to the gap edge the collective operators are weakly coupled. This allows to develop an effective approximation applicable when the problem is strongly non-Markovian, i.e. when in the absence of driving the effect of "freezing" of upper-level atomic population takes place. Then the dynamics of all but one collective operators becomes similar to the bosonic one. Only the dynamics of this one collective operators is nonlinear. We show that in this case a perturbation theory could be developed using as a small parameter a respective coupling strength between this one collective operator and others. In fact, the method of collective operators gives possibility to develop a perturbation theory "tailored" especially for the essentially non-Markovian problem. We derive a condition of applicability for this theory. Essentially, it is the condition of weak excitation of all collective atom-field "modes" corresponding to the collective operators with dynamics similar to the bosonic one. For the moderate intensities of the driving field even an approximation of independent collective operators gives good results. In the limit of weak driving field a simple analytic approximation is developed. As comparison shows, this approximation allows to describe a basic features of atomic population dynamics even for the cases when the driving field could not any more be considered weak.