Approximate controllability of the Jaynes-Cummings dynamics

We investigate the controllability of the Jaynes-Cummings dynamics in the resonant and nearly resonant regime. We analyze two different types of control operators acting on the bosonic part, corresponding - in the application to cavity QED - to an external electric and magnetic field, respectively. We prove approximate controllability for these models, for all values of the coupling constant g except those in a countable set S which is explicitly characterized in the statement. The proof relies on a spectral analysis which yields the non-resonance of the spectrum for every real g which is not in S.