Superconducting Ring Resonators: Modelling, Simulation, and Experimental Characterisation

TL;DR

We address how superconducting ring resonators can host two nearly degenerate, orthogonal modes with high quality factors suitable for quantum technologies. We develop a perturbative transmission-line model, complemented by a coupled-mode/flow-graph analysis, to predict frequency shifts, mode splitting, and rotation arising from line inhomogeneities, and validate these predictions with four-port simulations and microwave experiments. The experiments on Al and Nb rings in microstrip and CPW geometries confirm the predicted phenomena, including sharp frequency splitting, mode rotation, and flux-trapping–related variability, establishing the practical sensitivity and controllability of these devices. The work highlights the potential of high-$Q$ superconducting ring resonators for passive components, parametric amplifiers, and flux-tunable quantum circuits, while outlining fabrication tolerance challenges and directions for future integration.

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.

Figures (8)

• Figure 1: Schematic diagram of a four-port superconducting ring resonator. A circular superconducting transmission line of radius $r$ is capacitively coupled to external transmission lines at four evenly spaced points. Two lowest-order orthogonal electrical modes share a common ring system.
• Figure 2: Flow graph representation of (a) a two-port waveguide, and (b) a three-port coupler.
• Figure 3: Flow graph of four-port ring resonator.
• Figure 4: Flow graph simulations of ring resonators with identical coupling capacitances at all external ports $C_c = 20.8\,fF$. S42 and S31 correspond to the transmission between two orthogonal pairs of opposite ports, while S41 corresponds to the transmission between adjacent ports. (a) No perturbation on the ring transmission lines. (b) Quadrupole impedance perturbation on the ring transmission lines: lines 1 and 3 have $Z_{t}=50.05\,\Omega$ and lines 2 and 4 have $Z_{t}=49.95\,\Omega$. (c) Weaker quadrupole impedance perturbation on the ring transmission lines: lines 1 and 3 have $Z_{t}=50.01\,\Omega$ and lines 2 and 4 have $Z_{t}=49.99\,\Omega$.
• Figure 5: Flow graph simulations of ring resonators with coupling capacitances of $C_c = 20.8\,fF$ at port 2 and 4, and coupling capacitances of $C_c = 10.8\,fF$ at port 1 and 3. S42 and S31 correspond to the transmission between two orthogonal pairs of opposite ports, while S41 corresponds to the transmission between adjacent ports. (a) No perturbation on the ring transmission lines. (b) Dipole impedance perturbation on the ring transmission lines: lines 1 and 2 have $Z_{t}=49.95\,\Omega$ and lines 3 and 4 have $Z_{t}=50.05\,\Omega$. (c) Quadrupole impedance perturbation on the ring transmission lines: lines 1 and 3 have $Z_{t}=50.05\,\Omega$ and lines 2 and 4 have $Z_{t}=49.95\,\Omega$.