Superconducting microwave resonators are reliable circuits widely used for detection and as test devices for material research. A reliable determination of their external and internal quality factors is crucial for many modern applications, which either require fast measurements or operate in the single photon regime with small signal to noise ratios. Here, we use the circle fit technique with diameter correction and provide a step by step guide for implementing an algorithm for robust fitting and calibration of complex resonator scattering data in the presence of noise. The speedup and robustness of the analysis are achieved by employing an algebraic rather than an iterative fit technique for the resonance circle.
Superconducting microwave resonators are reliable circuits widely used for detection and as test devices for material research. A reliable determination of their external and internal quality factors is crucial for many modern applications, which either require fast measurements or operate in the single photon regime with small signal to noise ratios. Here, we use the circle fit technique with diameter correction and provide a step by step guide for implementing an algorithm for robust fitting and calibration of complex resonator scattering data in the presence of noise. The speedup and robustness of the analysis are achieved by employing an algebraic rather than an iterative fit technique for the resonance circle.
Superconducting micro-resonators have application in sensors and quantum computing. Measurement of the resonator internal loss in the single-photon regime is a common tool to study the origins of dissipation, noise, and decoherence in quantum circuits, as well as characterization of materials used for quantum devices. However, such measurements are challenging and time-consuming with large uncertainties due to the poor signal-to-noise ratio when measured at single-photon power levels. We developed methods to extract a subset of the resonator fitting parameters at high power and fix them in the analysis of low power data, which reduce the parameter space in the regression analysis. By comparing the analysis with and without partial extraction, we show that these methods reduce the uncertainty of the results, while improving the robustness and efficiency of the loss measurement.
Abstract 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 = L 0 ( 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.
It has always been a challenge for researchers to efficiently and accurately post process experimental data which is distorted by the noise. Superconducting microwave devices e.g. resonators, directional filters, beam-splitters etc. operate at frequency of several GHz to THz and temperatures well below critical temperature (Tc) with few exceptions like transition edge sensors where devices are operated at temperatures close to Tc. These devices are measured usually with vector network analyser in terms of scattering parameters. Two kinds of errors, systematic and drift can easily be removed from the measurements taken with VNA. However, random errors are not easy to address and remove due to their unpredictability and randomness. In this manuscript we will present an algorithm to post process experimental data to cope with measurements that have been corrupted or useful spectral response is buried in spurious signal. We have developed a robust and efficient algorithm, implemented in MATLAB, to detect peaks in spectral response, remove baseline and finally estimate parameters of two-port superconductor resonator using an Improved Nelder-Mead Method for unconstrained multidimensional least square minimization. The algorithm has been successfully tested and verified by processing spectral response of half wavelength microwave transmission-line resonator successfully isolating resonator response from noisy background. We were able to compute loaded quality factor, resonance frequency from response data with high reproducibility even from those experimental data sets where resonance spikes were hardly visible.
The control of material loss mechanisms is playing an increasingly important role for improving coherence times of superconducting quantum devices. Such material losses can be characterized through the measurement of planar superconducting resonators, which reflect losses through the resonance's quality factor Q l . The resonance quality factor consists of both internal (material) losses as well as coupling losses when resonance photons escape back into the measurement circuit. The combined losses are then described as Q l − 1 = Re { Q c − 1 } + Q i − 1 , where Q c and Q i reflect the coupling and internal quality factors of the resonator, respectively. To separate the relative contributions of Q i and Q c to Q l , diameter-correcting circle fits use algebraic or geometric means to fit the resonance signal on the complex plane. However, such circle fits can produce varied results, so to address this issue, we use a combination of simulation and experiment to determine the reliability of a fitting algorithm across a wide range of quality factor values from Q i ≪ Q c to Q c ≪ Q i . In addition, we develop a measurement protocol that can not only reduce fitting errors by factors ≳ 2 but also mitigates the influence of the measurement background on the fit results. This technique can be generalized for other resonance systems beyond superconducting resonators. Published by the American Physical Society 2024
Abstract Superconducting circuit testing and materials loss characterization requires robust and reliable methods for the extraction of internal and coupling quality factors of microwave resonators. A common method, imposed by limitations on the device design or experimental configuration, is the single-port reflection geometry, i.e. reflection-mode. However, impedance mismatches in cryogenic systems must be accounted for through calibration of the measurement chain while it is at low temperatures. In this paper, we demonstrate a data-based, single-port calibration using commercial microwave standards and a vector network analyzer with samples at millikelvin temperature in a dilution refrigerator, making this method useful for measurements of quantum phenomena. Finally, we cross reference our data-based, single-port calibration and reflection measurement with over-coupled 2D- and 3D-resonators against well established two-port techniques corroborating the validity of our method.
Superconducting resonators provide a convenient way to measure loss tangents of various dielectrics at low temperature. For the purpose of examining the microscopic loss mechanisms in dielectrics, precise measurements of the internal quality factor at different values of energy stored in the resonators are required. Here, we present a consistent method to analyze a LC superconducting resonator coupled to a transmission line. We first derive an approximate expression for the transmission S-parameter S21(ω), with ω the excitation frequency, based on a complete circuit model. In the weak coupling limit, we show that the internal quality factor is reliably determined by fitting the approximate form of S21(ω). Since the voltage V of the capacitor of the LC circuit is required to determine the energy stored in the resonator, we next calculate the relation between V and the forward propagating wave voltage Vin+, with the latter being the parameter controlled in experiments. Due to the dependence of the quality factor on voltage, V is not simply proportional to Vin+. We find a self-consistent way to determine the relation between V and Vin+, which employs only the fitting parameters for S21(ω) and a linear scaling factor. We then examine the resonator transmission in the cases of port reflection and impedance mismatch. We find that resonator transmission asymmetry is primarily due to the reflection from discontinuity in transmission lines. We show that our analysis method to extract the internal quality factor is robust in the non-ideal cases above. Finally, we show that the analysis method used for the LC resonator can be generalized to arbitrary weakly coupled lumped and distributed resonators. The generalization uses a systematic approximation on the response function based on the pole and zero which are closest to the resonance frequency. This Closest Pole and Zero Method is a valuable tool for analyzing physical measurements of high-Q resonators.