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arXiv:1312.3145

Algorithm for spectral response analysis of superconducting microwave transmission-line resonator

Hanif, Muhammad

Published Dec 11, 2013• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1312.3145

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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.
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.
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.
Characterizing superconducting microwave resonators with highly dissipative elements is a technical challenge, but a requirement for implementing and understanding the operation of hybrid quantum devices involving dissipative elements, e.g., for thermal engineering and detection. We present experiments on λ/4 superconducting niobium coplanar waveguide resonators, terminating at the antinode by a dissipative copper microstrip via aluminum leads, such that the resonator response is difficult to measure in a typical microwave environment. By measuring the transmission both above and below the superconducting transition of aluminum, we are able to isolate the resonance. We then experimentally verify this method with copper microstrips of increasing thicknesses, from 50 nm to 150 nm, and measure quality factors in the range of 10–67 in a consistent way.
We develop a microwave frequency counter based on the complex conductivity modulation of a DC-biased superconducting transmission line while maintaining the bias below the critical current. A resonator-free platform is proposed for microwave frequency estimation. It utilizes a bias-controlled phase delay in a superconducting coplanar waveguide structure, which imprints frequency-dependent signatures on the transmitted microwave tone. Instead of relying on intentionally designed resonators, our method leverages the complex transmission spectra arising from numerous weak resonance structures due to subtle fabrication imperfections. However, even in the absence of such features, the method remains effective because the DC bias induces frequency-dependent phase shifts that modulate the microwave signal’s propagation velocity. Mixed signals are externally generated via in-phase-quadrature mixing with a reference signal, which manifests as interference-like patterns enabling high-resolution spectral identification. We measure the frequency-dependent changes in the interference-like patterns under this bias to catalog characteristic transmission loss profiles across a wide frequency range (2–8 GHz). These profiles serve as supervised training data for a machine learning algorithm that identifies unknown input microwave frequencies with high-resolution and accuracy. Our method achieves a frequency resolution (F/ΔF) of over 200 000 and a relative error in identification of below 0.01%. This scalable approach to distributed microwave sensing may be useful for quantum measurement and signal classification and is particularly well-suited for dark matter searches, such as hidden-photon detection, where broadband frequency identification without prior tuning is essential. The measured frequency range is primarily limited by the bandwidth of external microwave components, such as low-noise amplifiers and bias-tees. The superconducting device itself may be used up to its superconducting gap frequency (∼280 GHz for TiN at 3.8 K).
A method is proposed and experimentally explored for in-situ calibration of complex transmission data for superconducting microwave resonators. This cryogenic calibration method accounts for the instrumental transmission response between the vector network analyzer reference plane and the device calibration plane. Once calibrated, the observed resonator response is analyzed in detail by two approaches. The first, a phenomenological model based on physically realizable rational functions, enables the extraction of multiple resonance frequencies and widths for coupled resonators without explicit specification of the circuit network. In the second, an ABCD-matrix representation for the distributed transmission line circuit is used to model the observed response from the characteristic impedance and propagation constant. When used in conjunction with electromagnetic simulations, the kinetic inductance fraction can be determined with this method with an accuracy of 2%. Datasets for superconducting microstrip and coplanar-waveguide resonator devices were investigated and a recovery within 1% of the observed complex transmission amplitude was achieved with both analysis approaches. The experimental configuration used in microwave characterization of the devices and self-consistent constraints for the electromagnetic constitutive relations for parameter extraction are also presented.
Abstract We present a theoretical and experimental study of superconducting ring resonators as an initial step toward their implementation in superconducting electronics and quantum technologies, with promising applications including superconducting parametric amplifiers with pump-signal isolation, flux-controlled quantum circuits, ultra-sensitive measurements in quantum sensing, and THz instrumentations. These devices have the potentially valuable property of supporting two orthogonal electromagnetic modes that couple to a common Cooper pair, quasiparticle, and phonon system. We present here a comprehensive theoretical and experimental analysis of the superconducting ring resonator system. We have developed superconducting ring resonator models that describe the key features of microwave behaviour to first order, providing insights into how transmission line inhomogeneities give rise to frequency splitting and mode rotation. Furthermore, we constructed signal flow graphs for a four-port ring resonator to numerically validate the behaviour predicted by our theoretical analysis. Superconducting ring resonators were fabricated in both coplanar waveguide and microstrip geometries using Al and Nb thin films. Microwave characterisation of these devices demonstrates close agreement with theoretical predictions. Our study reveals that frequency splitting and mode rotation are prevalent in ring systems with coupled degenerate modes, and these phenomena become distinctly resolved in high quality factor superconducting ring resonators.