Second-order optical susceptibility of a pure spin current

Spin currents are an essential element of spintronics. A pure spin current, formed by opposite spins moving in opposite directions with the same amplitude, bears neither net magnetization nor charge current and therefore is difficult to be detected by traditional electromagnetic induction method. While indirect measurement of a spin current via the inverse spin Hall effect or spin accumulation at stopping edges is possible, a fundamental question remains: How can a pure spin current be directly seen? By a systematic analysis of the peculiar symmetry properties of spin currents and microscopic calculations, we show that a pure spin current can have sizable second-order nonlinear optical susceptibility. This forms the basis of direct, non-destructive measurement of pure spin currents by standard nonlinear optical spectroscopy.