Spin Detection via Parametric Frequency Conversion in a Membrane Resonator

Recent demonstrations of ultracoherent nanomechanical resonators introduce the prospect of developing protocols for solid-state sensing applications. Here, we propose to use two coupled ultracoherent resonator modes on a $\mathrm{Si}$${}_{3}$$\mathrm{N}$${}_{4}$ membrane for the detection of small nuclear spin ensembles. To this end, we employ parametric frequency conversion between nondegenerate modes. The nondegenerate modes result from coupled degenerate resonators, and the parametric conversion is mediated by periodic inversions of the nuclear spins in the presence of a magnetic scanning tip. We analyze potential noise sources and derive the achievable signal-to-noise ratio with typical experimental parameter values. Our proposal reconciles the geometric constraints of optomechanical systems with the requirements of scanning force microscopy and brings forth a promising platform for spin-phonon interaction and spin imaging.