Quantum-limited estimation of the frequency shift between two interfering photons by time sampling of their quantum beats

Abstract

We present a sensing scheme for estimating the frequency difference of two non-entangled photons. The technique consists of time-resolving sampling measurements at the output of a beam splitter. With this protocol, the frequency shift between two photons can be estimated with the ultimate precision achievable in nature, overcoming the limits in precision and the range of detection of frequency-resolving detectors employed in standard direct measurements of the frequencies. The sensitivity can be increased by increasing the coherence time of the photons. We show that, already with $\sim 1000$ sampling measurements, the Cram\'{e}r-Rao bound is saturated independently of the value of the difference in frequency.