  Ultrafast characterization and control of many-body interactions and
elementary excitations are critical to understanding and manipulating emergent
phenomena in strongly correlated systems. In particular, spin interaction plays
an important role in unconventional superconductivity, but efficient tools for
probing spin dynamics especially out of equilibrium, are still lacking. To
address this question, we develop a theory for nonresonant time-resolved Raman
scattering, which can be a generic and powerful tool for nonequilibrium
studies. We also use exact diagonalization to simulate the pump-probe dynamics
of correlated electrons in the square-lattice single-band Hubbard model.
Different ultrafast processes are shown to exist in the time-resolved Raman
spectra and dominate under different pump conditions. For high-frequency and
off-resonance pumps, we show that the Floquet theory works well in capturing
the softening of bimagnon excitation. By comparing the Stokes/anti-Stokes
spectra, we also show that effective heating dominates at small pump fluences,
while a coherent many-body effect starts to take over at larger pump amplitudes
and frequencies on resonance to the Mott gap. Time-resolved Raman scattering
thereby provides the platform to explore different ultrafast processes and
design material properties out of equilibrium.
