Compact inductor-capacitor resonators at sub-gigahertz frequencies

Compact inductor-capacitor (LC) resonators, in contrast to coplanar waveguide (CPW) resonators, have a simple lumped-element circuit representation but usually call for sophisticated finite-element method (FEM) simulations for an accurate modelling. Here we present an simple analytical model for a family of coplanar LC resonators where the electrical properties are directly obtained from the circuit geometry with a satisfying accuracy. Our experimental results on $10$ high-internal-quality-factor resonators ($Q_{\mathrm{i}}\gtrsim 2\times {10}^5$), with frequency ranging roughly from $300\mathrm{M}\mathrm{H}\mathrm{z}$ to $1\mathrm{G}\mathrm{H}\mathrm{z}$, show an excellent consistency with both the derived analytical model and detailed FEM simulations. These results showcase the ability to design sub-gigahertz resonators with less than $2\mathrm{\%}$ deviation in the resonance frequency, which has immediate applications, for example, in the implementation of ultrasensitive cryogenic detectors. The achieved compact resonator size of the order of a square millimeter indicates a feasible way to integrate hundreds of microwave resonators on a single chip for realizing photonic lattices.