Crossover from weak to strong coupling regime in dispersive circuit QED

We study the decoherence of a superconducting qubit due to the dispersive coupling to a damped harmonic oscillator. We go beyond the weak qubit-oscillator coupling, which we associate with a phase Purcell effect, and enter into a strong coupling regime, with qualitatively different behavior of the dephasing rate. We identify and give a physicaly intuitive discussion of both decoherence mechanisms. Our results can be applied, with small adaptations, to a large variety of other physical systems, e. g. trapped ions and cavity QED, boosting theoretical and experimental decoherence studies.