Cross-correlation spin noise spectroscopy of interacting multi-component spin systems

Interacting multi-component spin systems are ubiquitous in semiconductor spintronics; $e.g$. carrier-mediated ferromagnetism in magnetic semiconductors, or electronic spin coupling to nuclear spin baths. Traditionally, inter-species spin interactions are studied by experimental methods that are necessarily perturbative: $e.g.$, by intentionally polarizing or depolarizing one spin species and detecting the response of the other(s). Here, we show that multi-probe spin noise spectroscopy can reveal interspecies spin-spin interactions -- under conditions of strict thermal equilibrium -- by cross-correlating the stochastic fluctuation signals exhibited by each of the constituent spin species. We develop a theory for such noise cross-correlations in thermal equilibrium. As a proof of principle, we compare the results with an experimental study of a well-understood interacting spin system -- a mixture of warm Rb and Cs vapors -- by applying a new type of \textit{two-color} spin noise spectroscopy [1,2]. Noise correlations directly reveal the presence of inter-species spin exchange interactions. Such non-invasive and noise-based techniques should be generally applicable to any multi-component spin system in which the fluctuations of the constituent components are detectable.\\[4pt] [1] D. Roy, L. Yang, S.A. Crooker, N.A. Sinitsyn, arXiv:1408.5399 (2014).\\[0pt] [2] L. Yang \textit{et al}., Nat. Comm. \textbf{5}:4949 (2014).