  We study electric polarization and nonlinear optical effects in spin systems
with broken inversion symmetry. We apply strong coupling expansion to the
underlying electronic Hamiltonians, and systematically derive expressions for
electric polarization in spin systems that are represented in terms of spin
operators. The magnon representation of the obtained electric polarization
operator allows us to compute linear and nonlinear optical responses by the
standard diagrammatic method. We apply our formalism to Heisenberg model with
alternating coupling constants and $J_1$-$J_2$ model with inversion symmetry
breaking. We demonstrate that these inversion broken spin systems support dc
current flow upon magnon excitations which arises from the shift current
mechanism.
