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Higher order spin noise spectroscopy of atomic spins in fluctuating external fields

We discuss the effect of external noisy magnetic fields on mesoscopic spin fluctuations that can be probed in semiconductors and atomic vapors by means of optical spin noise spectroscopy (SNS). We show that conventional arguments of the law of large numbers do not apply to spin correlations induced by external fields, namely, the magnitude of the 4th order spin cumulant grows as $\sim N^2$ with the number $N$ of observed spins, i.e. it is not suppressed in comparison to the 2nd order cumulant. This allows us to design a simple experiment to measure the 4th order cumulant of spin fluctuations in an atomic system near thermodynamic equilibrium and develop quantitative theory that explains all observations.