We present an experimental and theoretical analysis of the self-and cross-Kerr effect of extended plasma resonances in Josephson junction chains. We calculate the Kerr coefficients by deriving and diagonalizing the Hamiltonian of a linear circuit model for the chain and then adding the Josephson non-linearity as a perturbation. The calculated Kerr-coefficients are compared with the measurement data of a chain of 200 junctions. The Kerr effect manifests itself as a frequency shift that depends linearly on the number of photons in a resonant mode. By changing the input power on a low signal level, we are able to measure this shift. The photon number is calibrated using the self-Kerr shift calculated from the sample parameters. We then compare the measured cross-Kerr shift with the theoretical prediction, using the calibrated photon number.

We present a theoretical treatment for the dissipative two-resonator circuit quantum electrodynamics setup referred to as quantum switch. There, switchable coupling between two superconducting resonators is mediated by a superconducting qubit operating in the dispersive regime, where the qubit transition frequency is far detuned from those of the resonators. We derive an effective Hamiltonian for the quantum switch beyond the rotating wave approximation and provide a detailed study of the dissipative dynamics. As a central finding, we derive analytically how the qubit affects the quantum switch even if the qubit has no dynamics, and we estimate the strength of this influence. The analytical results are corroborated by numerical calculations, where coherent oscillations between the resonators, the decay of coherent and Fock states, and the decay of resonator-resonator entanglement are studied. Finally, we suggest an experimental protocol for extracting the damping constants of qubit and resonators by measuring the quadratures of the resonator fields.

We have measured the current-voltage characteristics of a Josephson junction with tunable Josephson energy E J embedded in an inductive environment provided by a chain of SQUIDs. Such an environment induces localization of the charge on the junction, which results in an enhancement of the zero-bias resistance of the circuit. We explain this result quantitatively in terms of the Bloch band dynamics of the localized charge. This dynamics is governed by charge diffusion in the lowest Bloch band of the Josephson junction as well as by Landau-Zener transitions out of the lowest band into the higher bands. In addition, the frequencies corresponding to the self-resonant modes of the SQUID array exceed the Josephson energy E J of the tunable junction, which results in a renormalization of E J , and, as a consequence, an increase of the effective bandwidth of the lowest Bloch band.