The cavity mediated spin current between two ferrite samples has been reported by Bai et. al. [Phys. Rev. Lett. 118, 217201 (2017)]. This experiment was done in the linear regime of the interaction in the presence of external drive. In the current paper we develop a theory for the spin current in the nonlinear domain where the external drive is strong so that one needs to include the Kerr nonlinearity of the ferrite materials. In this manner the nonlinear polaritons are created and one can reach both bistable and multistable behavior of the spin current. The system is driven into a far from equilibrium steady state which is determined by the details of driving field and various interactions. We present a variety of steady state results for the spin current. A spectroscopic detection of the nonlinear spin current is developed, revealing the key properties of the nonlinear polaritons. The transmission of a weak probe is used to obtain quantitative information on the multistable behavior of the spin current. The results and methods that we present are quite generic and can be used in many other contexts where cavities are used to transfer information from one system to another, e.g., two different molecular systems.
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