Light-Matter Interaction in dispersive Superconducting Circuit QED

It is well known that superconducting waveguides strongly attenuate the propagation of electromagnetic waves with frequencies beyond the superconducting gap. In circuit QED, the interaction between non-linear charge qubits and superconducting resonators invariably involves the qubit coupling to a large set of resonator modes. So far, strong dispersion effects near and beyond the superconducting-gap have been ignored in quantization models. Rather, it is assumed that the superconducting resonator behaves ideally across the large frequency intervals. We present a quantization approach which includes the superconducting frequency-dependent surface impedance and demonstrate that superconducting dispersion plays a role in determining the effective light-matter interaction cut-off.