Sensing nano-NMR diffusion spectra using a nitrogen vacancy center

Nano-NMR is an emerging field, striving to bring classic NMR techniques to a single molecule resolution. In recent years nitrogen vacancy centers, have been shown to be a promising experimental platform for that purpose. The main idea behind these experiments in liquids, is that the molecules create a time varying magnetic field at the NV's location, which causes the NV to dephase. The dephasing rate is related to the power spectrum of the magnetic noise, and thus encodes the NMR spectra. Understanding the diffusion spectrum is therefore key to future advancements in the field. In this work we calculate analytically the auto-correlation of the magnetic field at the NV's location. We then provide an asymptotic behavior of the power spectrum. These two results can be used to estimate the liquid's self-diffusion coefficient and the NV's distance from the diamond surface.