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.
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