Fig. 2: Statistics of sequential measurements on a sensor spin.

a Protocol of sequential measurement. The sensor spin and the ancilla are initialized by an optical pump (green block being a pulse of 532 nm laser) and SWAP gates (repeated twice for higher fidelity). Then the sensor spin is rotated by a \(\frac{\pi }{2}\) pulse (blue block), controlled by a dynamical decoupling sequence, and rotated again by a \(\frac{\pi }{2}\) pulse (with a readout angle θ from the first \(\frac{\pi }{2}\) pulse so that \({\hat{\sigma }}_{\theta }\) is measured). The NV electron spin state is then stored in the ¹⁴N spin by a SWAP gate and the ¹⁴N spin state is repeatedly read out through the electron spin via a CNOT gate and photon counts. b Second moment S_ij of sequential measurement of a sensor spin coupled to a ¹³C nuclear spin. c Fourier transform of the second moment for a nuclear spin (upper) and a random-phased AC field (lower). The extra small peak at lower frequency in lower graph of c is not from the AC field, as checked by the dependence of its amplitude on the measurement direction e_θ (see Fig. S5 in Supplementary Note 9).