Circuit model. (a) Distribution of the magnetic flux field along the -resonator for the fundamental resonator mode , without (red) and with (blue) a Jospehson junction. The additional Josephson inductance changes the boundary condition such that neither the current nor the voltage is zero at position . The resulting increase in the effective wavelength is indicated by the dashed blue line. (b) Transmission line resonator of length d with a Josephson junction at the grounded end. (c) Lumped element representation with indicated discretized magnetic flux field as used in Eq. (21).