Circuit-QED: How strong can the coupling between a Josephson junction atom and a transmission line resonator be?

After reviewing the limitation by the fine structure constant α of the dimensionless coupling constant of an hydrogenic atom with a mode of the electromagnetic field in a cavity, we show that the situation presents itself differently for an artificial Josephson atom coupled to a transmission line resonator. Whereas the coupling constant for the case where such an atom is placed inside the dielectric of the resonator is proportional to α1/2, the coupling of the Josephson atom when it is placed in series with the conducting elements of the resonator is proportional to α-1/2 and can reach values greater than 1.