New method for fitting complex resonance curve to study nonlinear superconducting resonators

We present a new fitting method that can robustly and accurately fit the complex transmission curve of a superconducting resonator in the nonlinear regime. This method takes into account the varying internal current in the resonator at different frequencies and the nonlinear dependence of the inductance on the internal current L = L0(1 + I-2/I-2*), where I* is a characteristic current. We demonstrate using this method to retrieve important resonator parameters, such as the quality factor and the resonance frequency, from resonators driven below, near, and above bifurcation. We further use this method to retrieve I* for lumped-element TiN resonators with various inductor designs. By fitting the resonance frequency shift at different readout powers of each resonator, we can determine the characteristic current I*, which is found to be linearly related to the cross-sectional area of the narrow inductor strip. Our method has wide applications in superconducting detector, superconducting qubit and parametric amplifier data analysis where the resonator is driven in the nonlinear regime.