  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.
