Abstract
We discuss how reactive and dissipative nonlinearities affect the intrinsic response of superconducting thin-film resonators. We explain how most, if not all, of the complex phenomena commonly seen can be described by a model in which the underlying resonance is a single-pole Lorentzian, but whose centre frequency and quality factor change as external parameters, such as readout power and frequency, are varied. What is seen during a vector-network-analyser measurement is series of samples taken from an ideal Lorentzian that is shifting and spreading as the readout frequency is changed. According to this model, it is perfectly proper to refer to, and measure, the resonant frequency and quality factor of the underlying resonance, even though the swept-frequency curves appear highly distorted and hysteretic. In those cases where the resonance curve is highly distorted, the specific shape of the trajectory in the Argand plane gives valuable insights into the second-order physical processes present. We discuss the formulation and consequences of this approach in the case of nonlinear kinetic inductance, two-level-system loss, quasiparticle generation, and a generic model based on a power-law form. The generic model captures the key features of specific dissipative nonlinearities, but additionally leads to insights into how general dissipative processes create characteristic forms in the Argand plane. We provide detailed formulations in each case, and indicate how they lead to the wide variety of phenomena commonly seen in experimental data. We also explain how the properties of the underlying resonance can be extracted from this data. Overall, our paper provides a self-contained compendium of behaviour that will help practitioners interpret and determine important parameters from distorted swept-frequency measurements.
Highlights
Superconducting thin-film microwave resonators are being developed for a wide range of applications
Thin-film superconducting resonators are a natural system for exploring chip-based Quantum Electrodynamics (QED) [15, 16], and are being realised in exotic combinations, such as superconducting electromagnetic resonators coupled to micromechanical cantilevers for studying quantum-statistical processes [17]
We show that a considerable amount of physical information is contained in the behaviour of the quality factor, not just in the resonant frequency, as external parameters, such as the readout power, are changed
Summary
Superconducting thin-film microwave resonators are being developed for a wide range of applications. A crucial point is that when superconducting resonators are measured, they often do not behave in a simple linear way having a near-perfect Lorentzian response, but instead show transmission and reflection coefficients that display peculiar shapes in the complex plane. Their behaviour changes as the readout power is increased, and often the resonance curves switch hysteretically between two stable states as the readout frequency is swept up and down. We describe a range of methods for extracting physical information from distorted resonance curves, which can be used for optimising performance, and for predicting operational aspects of behaviour such as optimal readout power, small signal nonlinearity, and noise
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