Fig. 6: Optimal quantum-frequency estimation.

a Maximal sensitivity to amplitude changes is obtained for π-pulses at the signal nodes \(\left( {\theta = \frac{\pi }{2}} \right)\), resulting in a linear accumulation of φ. Maximal sensitivity to frequency changes is obtained for π-pulses at the signal antinodes (θ = 0). b Experimental data of the phase difference for a 1 µT change in amplitude (red) and 10 kHz change in frequency (black). Frequency estimation results in a quadratic increase of φ, whereas amplitude estimation produces a linear increase. c Measured NV population as a function of π-pulse spacing \(\tau _\pi\), given \(\theta = \frac{\pi }{2}\) (blue) and θ = 0 (green). Fits correspond to Supplementary Eq. 39. d Measured derivative of the NV population as a function of π-pulse spacing \(\tau _\pi\), given \(\theta = \frac{\pi }{2}\) (blue) and θ = 0 (green). Fits correspond to Supplementary Eq. 41. e Measured frequency uncertainty as a function of interaction time, for X (dark purple) or Y (light purple) measurement basis. Red line corresponds to Eq. (8) and pink line to Eq. (9). f Measured frequency uncertainty as a function of interaction time, for X (dark orange) or Y (light orange) measurement basis using a single SSR measurement. Red line corresponds to Eq. (8). g Frequency estimation for short coherence signals (for ensemble averaging): measured frequency uncertainty given \(\theta = \frac{{\pi }}{2}\) (blue), θ = 0 (green), and unknown initial phase (gray) for a single measurement with interaction time shorter than the signal coherence time (200 µs). For longer times, the sensitivity is obtained from averaging multiple measurements, each with an interaction time of 97 µs.