Estimation of nuclear polarization via discrete measurement of nitrogen-vacancy-center spin evolution

We propose a method for the estimation of the initial polarization of spinful nuclei of the $^{13}\mathrm{C}$ isotope in diamond via a measurement of the evolution of the coherence of a nitrogen-vacancy-center spin qubit. Existing polarization measurement methods are difficult to implement experimentally because they require direct interference in the environment of the qubit. Here, in order to obtain the information, it is necessary to measure the qubit coherence at certain points of time, which are unambiguously determined by the applied magnetic field. For sufficiently high magnetic fields, the minimum value of the measured coherence constitutes an upper bound on the product of the initial polarizations of each environmental spin. The most significant advantage of the method, which allows to infer initial values of nuclear polarizations without any direct access to the environment, lies in its simplicity and the small amount of experimental resources that it requires. We exemplify the operation of the scheme on a realistic, randomly generated environment of eight nuclear spins, obtaining a reasonably accurate estimation of the initial polarization.