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arXiv:1008.3130 · IEEE Transactions on Applied Superconductivity

Superposition of Inductive and Capacitive Coupling in Superconducting LC Resonators

Sergey P. Gladchenko, Moe S. Khalil, Christopher J. Lobb, F. C. Wellstood, Kevin D. Osborn

Published Dec 9, 2010• 5 citations
arXivHTMLPDF

Abstract

We present an experimental investigation of lumped-element superconducting LC resonators designed to provide different types of coupling to a transmission line. We have designed four resonator geometries including dipole and quadrupole configured inductors connected in parallel with low-loss${\rm SiN}_{\rm x}$dielectric parallel-plate capacitors. The design of the resonator allows a change in the symmetry of the inductor or grounding of the capacitor to create LC resonators with: 1) inductive coupling, 2) capacitive coupling, 3) both types of coupling, or 4) greatly reduced coupling. We measured all four designs at a temperature of 30 mK at different values of power. We compare the extracted data from the four resonator types and find that both capacitive and inductive coupling can be included and that when nominally turned off, the coupling is greatly reduced from only minor changes in the circuit design. We also find a variation in the measured loss tangent of less than a few percent between the designs over a large range of internal quality factors, which are created by using${\rm SiN}_{\rm x}$as a nonlinear lossy element within the capacitor.

DOI
10.1109/tasc.2010.2089774

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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.
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In this article, we present an experimental study of microwave resonators made out of Josephson junctions. The junctions are embedded in a transmission line geometry so that they increase the inductance per length for the line. By comparing two devices with different input/output coupling strengths, we show that the coupling capacitors, however, add a significant amount to the total capacitance of the resonator. This makes the resonators with high coupling capacitance to act rather as lumped element resonators with inductance from the junctions and capacitance from the end sections. Based on a circuit analysis, we show that the input and output couplings of the resonator are limited to a maximum value of ωrZ0/4Zr, where ωr is the resonance frequency and Z0 and Zr are the characteristic impedances of the input/output lines and the resonator, respectively.
Abstract We study the response of several microwave resonators made from superconducting NbTiN thin-film meandering nanowires with large kinetic inductance, having different circuit topology and coupling to the transmission line. Reflection measurements reveal the parameters of the circuit and analysis of their temperature dependence in the range 1.7–6 K extract the superconducting energy gap and critical temperature. The lumped-element LC resonator, valid in our frequency range of interest, allows us to predict the quasiparticle (QP) contribution to internal loss, independent of circuit topology and characteristic impedance. Our analysis shows that the internal quality factor is limited not by thermal-equilibrium QP, but an additional temperature-dependent source of internal microwave loss.
We report our development on design, fabrication and characterization of various superconducting resonators for quantum device applications. The resonators are developed mainly for superconducting qubit implementation in circuit QED structure, but future application will include photon detectors such as microwave kinetic inductance detectors. First we designed and fabricated half-wavelength coplanar waveguide resonators with niobium thin film on a high resistive silicon wafer, with the fundamental frequencies in the range of 1 to 8 GHz. The loaded quality factors are controlled by the coupling capacitors, and the lower bound of the intrinsic quality factor was measured down to 7 mK in the dilution refrigerator. We have also designed quarter-wavelength resonators and lumped LC resonators coupled to a transmission line, and characterized multiple resonators in a single transmission line. For the 3D transmon qubit application, we designed and fabricated a rectangular box resonator with superconducting aluminum. The intrinsic quality factor of the box resonator exceeds three million at 7 mK. We will also present basic results on the integration of a transmon qubit in the superconducting resonator.