  We study analytically and numerically the properties of Jaynes-Cummings model
under monochromatic driving. The analytical results allow to understand the
regime of two branches of multi-photon excitation in the case of close
resonance between resonator and driven frequencies. The rotating wave
approximation allows to reduce the description of original driven model to an
effective Jaynes-Cummings model with strong coupling between photons and qubit.
The analytical results are in a good agreement with the numerical ones even if
there are certain deviations between the theory and numerics in the close
vicinity of the resonance. We argue that the rich properties of driven
Jaynes-Cummings model represent a new area for experimental investigations with
superconducting qubits and other systems.
