Frequency Fluctuations in Tunable Superconducting Microwave Cavities

We present a model for measurements of the scattering matrix elements of tunable microwave cavities in the presence of resonant frequency fluctuations induced by fluctuations in the tuning parameter. We apply this model to the specific case of a two-sided cavity and find an analytic expression for the average scattering matrix elements. A key signature of this `fluctuating model' is a subtle deformation of the trajectories swept out by scattering matrix elements in the complex plane. We apply this model to experimental data and report a direct observation of this deformation in the data. Despite this signature, we show that the fluctuating and non-fluctuating models are qualitatively similar enough to be mistaken for one another, especially in the presence of measurement noise. However, if one applies the non-fluctuating model to data for which frequency fluctuations are significant then one will find damping rates that appear to depend on the tuning parameter, which is a common observation in tunable superconducting microwave cavities. We propose this model as both a potential explanation of and remedy to this apparent phenomenon.