Exceptional precision of a nonlinear optical sensor at a square-root singularity

Exceptional points (EPs) -- spectral singularities of non-Hermitian linear systems -- have recently attracted great interest for sensing. While initial proposals and experiments focused on enhanced sensitivities neglecting noise, subsequent studies revealed issues with EP sensors in noisy environments. Here we propose a single-mode Kerr-nonlinear resonator for exceptional sensing in noisy environments. Based on the resonator's dynamic hysteresis, we define a signal that displays a square-root singularity akin to an EP. In contrast to EP sensors, our sensor has a signal-to-noise ratio that increases with the measurement speed, and a precision enhanced at the square-root singularity. Remarkably, averaging the signal can quickly enhance and then degrade the precision. These unconventional features open up new opportunities for fast and precise sensing beyond the constraints of linear systems. While we focus on optical sensing, our approach can be extended to other hysteretic systems.