1. The two-time correlator of the collective polarization/inversion operator does not scale with the square of the atom numbers. It scales with N instead, since we assume there is no correlation between two different atoms.
2. When applying the Wierner-Khinckin theorem with operators, do not make the Fourier transform of an operator A^\dagger(w) =[A(-w)]^*. Keep all the operators as they are but perform the identity for their prefactors if needed.
