Zappe Photon Upconverters for Quantum Measurements of Low-Frequency Electrical Resonators, Part I: Theory*

We present the Zappe Photon Upconverter (ZPU), a Josephson-junction-based readout amplifier for performing quantum measurements on lumped-element LC resonators (100 Hz-300 MHz). The ZPU exploits the symmetries of a three-junction Zappe interferometer embedded inside a microwave resonator. The Zappe element acts as a flux-variable inductor, cleanly implementing a three-wave-mixing interaction between the low-frequency resonator at the flux input and the high-frequency (~6 GHz) microwave resonator. The interaction is analogous to the interaction between resonators in electromechanical and optomechanical systems. We establish protocols for flux sensing at the Standard Quantum Limit (SQL), as well as backaction evasion techniques that permit measurements below the SQL. We discuss applications of ZPUs, in particular axion dark matter searches, where these devices can enable scan rate enhancements of orders of magnitude.