Dispersion relations for light propagation through a two-level system

Interaction of radiation with two-level system is considered using the Dicke hamiltonian. By making use of the Schwinger representation, the hamiltonian is written in a trilinear form. The hamiltonian is made bilinear by replacing one set of boson operators by C-numbers. A Bogoliubov transformation is then used to reduce the hamiltonian to diagonal form. The density matrix separates into a product of density matrices for separate modes. The energies of the photon-like modes are found to follow dispersion relations which depend on the given set of initial conditions.