Correlation function of spin noise due to atomic diffusion

We use paramagnetic Faraday rotation to study spin noise spectrum from unpolarized Rb vapor in a tightly focused probe beam in the presence of N2 buffer gas. We derive an analytical form for the diffusion component of the spin noise time-correlation function in a Gaussian probe beam. We also obtain analytical forms for the frequency spectrum of the spin noise in the limit of a tightly focused or a collimated Gaussian beam in the presence of diffusion. In particular, we find that in a tightly focused probe beam the spectral line shape can be independent of the buffer gas pressure. Experimentally, we find good agreement between the calculated and the measured spin noise spectra for N2 gas pressures ranging from 56 to 820 Torr.