A method to measure hyperfine interaction beyond standard statistical limit

We propose a method to measure the hyperfine interaction between a single electron spin and a nuclear spin and apply it to the trapped electron and the $^{15}$N nuclear spin in a diamond nitrogen-vacancy (NV) center. The electron spin is prepared in a pure quantum state and the nuclear spin acquires an unknown partial polarization as a consequence of the preparation of the electron state. The proposed quantum measurement protocol is independent of the incoherence of the initial nuclear spin state. The model utilizes the time $\tau$ of a sequence of quantum operations as well as the number $N$ of repeated sequences to increase the accuracy and the precision of the estimation. The deleterious effect of the electron spin decoherence on the precision during time $\tau$ is included in the simulation. While in the statistical limit the standard deviation (measure of imprecision) of the estimation is proportional to $1/\sqrt{N}$, the quantum operations in time $\tau$ enables the imprecision to decrease faster as $1/\tau$ instead of the equivalent statistical limit of $1/\sqrt{\tau}$. Thus, the net imprecision of the estimation dips below the statistical limit. The robustness test of our simulation shows that experimental implementation of such a precision measurement is possible.