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We provide a general method to find the Hamiltonian of a linear circuit inthe presence of a nonlinearity. Focussing on the case of a Josephson junctionembedded in a transmission-line resonator, we solve for the normal modes of thesystem by taking into account exactly the effect of the quadratic (i.e.inductive) part of the Josephson potential. The nonlinearity is then found tolead to self and cross-Kerr effect, as well as beam-splitter type interactionsbetween modes. By adjusting the parameters of the circuit, the Kerr coefficientK can be made to reach values that are weak (K < \kappa), strong (K > \kappa)or even very strong (K >> \kappa) with respect to the photon-loss rate \kappa.In the latter case, the resonator+junction circuit corresponds to an in-lineversion of the transmon. By replacing the single junction by a SQUID, the Kerrcoefficient can be tuned in-situ, allowing for example the fast generation ofSchr\"odinger cat states of microwave light. Finally, we explore the maximalstrength of qubit-resonator coupling that can be reached in this setting.

Josephson junction-embedded transmission-line resonators: from Kerr medium to in-line transmon