  We present a detailed theoretical analysis of a multi-level quantum system
coupled to two radiation fields and subject to decoherence. We concentrate on
an effect known from quantum optics as the Autler-Townes splitting, which has
been recently demonstrated experimentally [M. A. Sillanpaa et al., Phys. Rev.
Lett. 103, 193601 (2009)] in a superconducting phase qubit. In the three-level
approximation, we derive analytical solutions and describe how they can be used
to extract the decoherence rates and to account for the measurement data.
Better agreement with the experiment can be obtained by extending this model to
five levels. Finally, we investigate the stationary states created in the
experiment and show that their structure is close to that of dark states.
