  Optical pump-probe experiments carried out in the time domain reveal both the
intrinsic low energy dynamics and its connections to higher energy excitations
in correlated electron systems. In this work, we propose two microscopic
mechanisms for the optical generation of coherent magnetic modes in van der
Waals magnets, and derive the corresponding effective light-spin interactions:
either through pumping atomic orbital excitations resonantly or via a
light-induced Floquet spin Hamiltonian, the ground state of the system is
driven out of equilibrium. The subsequent long-time relaxational dynamics can
then be probed using, e.g. the magneto-optical Kerr effect or transient grating
spectroscopy. As an example, we apply our framework to NiPS$_3$, which is
magnetically ordered in the bulk, and is conjectured to realize the XY model in
the monolayer limit. Our theory makes explicit how the material's low-energy
response depends sensitively on the microscopic details of the light-spin
coupling as well as pump fluence, frequency and polarization. For the case of
bulk NiPS$_3$, we find quantitative agreement with recent experiments by
Afanasiev et al. [Science Advances 7, abf3096 (2021)]. We further propose
pump-probe experiments for monolayer NiPS$_3$ and detail how anomalous
relaxational behaviour may reveal excitations of a (proximate) BKT phase in a
proposed effective XY model.
