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.
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.
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.
We have designed, fabricated, and tested a frequency-tunable high-Q superconducting resonator made from a niobium titanium nitride film. The frequency tunability is achieved by injecting a DC through a current-directing circuit into the nonlinear inductor whose kinetic inductance is current-dependent. We have demonstrated continuous tuning of the resonance frequency in a 180 MHz frequency range around 4.5 GHz while maintaining the high internal quality factor Qi > 180 000. This device may serve as a tunable filter and find applications in superconducting quantum computing and measurement. It also provides a useful tool to study the nonlinear response of a superconductor. In addition, it may be developed into techniques for measurement of the complex impedance of a superconductor at its transition temperature and for readout of transition-edge sensors.
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.
We present a tunable coupler scheme that allows us to tune the coupling strength between a feedline and a superconducting resonator in situ over a wide range. In this scheme, we shunt the feedline with a 50-Ω lumped-element nonlinear transmission line made from a 20 nm NbTiN film. By injecting a DC current, the nonlinear kinetic inductance changes and the effective impedance shunting the resonator periodically varies from a short to an open, which tunes the coupling strength and coupling quality factor Qc. We have demonstrated Qc tuning over a factor of 40, between Qc∼5.5×104 and Qc∼2.3×106, for a 4.5 GHz resonator by applying a DC current less than 3.3 mA. Our tunable coupler scheme is easy to implement and may find broad applications in superconducting detector and quantum computing/information experiments.
Abstract We originally develop a network analysis method to study the non-ideal superconducting resonator circuits. We model the resonator device by a 3-block serial network, consisting of input feedline, resonator-coupler and output feedline. We show that the resonance curve will be modified if non-negligible discontinuity (reflection) exists at input or output side of the resonator. The extracted resonance parameters (such as coupling quality factor Q c ) from only fitting the transmission deviate from their true values as originally defined in matched impedance conditions, and the deviation depends on a single parameter which includes all the non-ideal effects from the circuits. We prove that the internal quality factor Q i can be accurately extracted from the modified transmission curve by using a very simple graphical method. Our methods and results are quantitatively verified by both simulations and experiments on superconducting microstrip resonators. Comparing to the usual circuit analysis, our network analysis method is more intuitive and easier to understand. It can serve as a general tool to study the non-ideal superconducting resonator circuits.
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.