  We present a detailed theoretical analysis for a system of a superconducting
flux qubit coupled to a transmission line resonator. The master equation,
accounting incoherent processes for a weakly populated resonator, is
analytically solved. An electromagnetic wave transmission coefficient through
the system, which provides a tool for probing dressed states of the qubit, is
derived. We also consider a general case for the resonator with more than one
photon population and compare the results with an experiment on the
qubit-resonator system in the intermediate coupling regime, when the coupling
energy is comparable with the qubit relaxation rate.
