  We study the interaction of a (classical) light field with the magnetic
degrees of freedom in the two-dimensional antiferromagnet Sr$_2$IrO$_4$. The
reduced space group symmetry of the crystal allows for several channels for
spin-operator bilinears to couple to the electric field. Integrating out
high-energy degrees of freedom in a Keldysh framework, we derive induced
effective fields which enter the equations of motion of the low-energy mode of
in-plane rotations which couple to the out-of-plane magnetization. Considering
a pump-probe protocol, these induced fields excite magnetization oscillations
which can subsequently probed, e.g. using Kerr rotation. We discuss how the
induced fields depend on polarization and frequency of the driving light, and
our study applies to both resonant and non-resonant regimes. Crucially, the
induced fields depend on the two-magnon density of states, thus allowing for
further insight into properties of the magnetic excitation spectrum.
Furthermore, these effects rely upon (weak) magnon-interactions, and so are
beyond a "Floquet magnon" description.
