Cross-correlation spin noise spectroscopy of heterogeneous interacting spin systems

We develop a minimally invasive approach for characterization of inter-species interactions in heterogeneous spin systems by detecting spin fluctuations alone. We consider two spin ensembles in thermal equilibrium that interact via binary exchange coupling and determine cross-correlations between the intrinsic spin fluctuations exhibited by the two spin species. Our theoretical predictions are experimentally confirmed using ``two-color'' optical spin noise spectroscopy on a mixture of interacting Rb and Cs alkali vapors. The results allow us to explore the rates of spin exchange and total spin relaxation under conditions of strict thermodynamic equilibrium.