Normal modes of a superconducting transmission-line resonator with embedded lumped element circuit components

We present a method to identify the coupled, normal modes of a superconducting transmission line with an embedded lumped element circuit. We evaluate the effective transmission-line nonlinearities in the case of Kerr-like Josephson interactions in the circuit and in the case where the embedded circuit constitutes a qubit degree of freedom, which is Rabi coupled to the field in the transmission line. Our theory quantitatively accounts for the very high and positive Kerr nonlinearities observed in a recent experiment [M. Rehak, P. Neilinger, M. Grajcar, G. Oelsner, U. Hubner, E. Il'ichev, and H.-G. Meyer, Appl. Phys. Lett. 104, 162604 (2014)], and we can evaluate the accomplishments of modified versions of the experimental circuit.