Bayesian frequency estimation at the fundamental quantum limit

Abstract: Searching for a weak signal at an unknown frequency is a canonical task in experiments probing fundamental physics such as gravitational-wave observatories and ultra-light dark matter haloscopes. These state-of-the-art sensors are limited by quantum noise arising from the fundamental uncertainty about the state of the device. Classically, frequency estimation suffers from a threshold effect in the signal-to-noise ratio such that weak signals are extremely hard to localise in frequency. We show that this phenomenon persists at the fundamental quantum limit but that the classical approach, a quadrature measurement, can nevertheless be beaten by a coherent protocol of projecting onto the "quantum whitened" possible quantum states. Quantum whitening is a covariant measurement, and we examine it analytically in the wide-prior limit and numerically for finite-width priors. Beyond accelerating searches for unknown frequencies, quantum whitening may be used generally to sense the parameter of a unitary encoding given no prior information about the parameter.