Electromagnetic response of disordered superconducting cavities

We present the results for the resonant frequency shift and quality factor of disordered niobium (Nb) superconducting radio-frequency cavities driven out of equilibrium by the resonant microwave field. The theory is based on the non-equilibrium theory of superconductivity for the current response to the electromagnetic field at the vacuum–metal interface. We are able to accurately predict the observed frequency shifts with a precision of order fractions of kHz over the full temperature range 0 < T ≤ T c , including the negative frequency shift anomalies that are observed very near T c . The origin of these anomalies is shown to be the competition between the normal metal skin depth and the London penetration depth, which diverges as T→Tc− . An analytical approximation to the full current response, valid for | T − T c |≪ T c , accounts for the negative frequency shift near T c . The non-monotonic dependence of the quality factor on the quasiparticle scattering rate is related to the pair-breaking effect of disorder on the superfluid fraction and, thus, the London penetration depth.