Quantum behavior of a flux qubit coupled to a resonator

A detailed theory for a system of a superconducting qubit coupled to a transmission line resonator is presented. We describe the system by solving analytically and numerically the master equation for the density matrix, which includes a dissipative Lindblad term. We calculate the transmission coefficient, which provides a way to probe the dressed states of a qubit. The theoretical results are related to experiments with intermediate coupling between a qubit and a resonator when the coupling energy is of the same order as the qubit relaxation rate.