  Raman spectroscopy uses light scattering to extract information on low-energy
excitations of solids. The Raman process is described by diagrams which are
fourth order in the light-matter interaction, and in particular the resonant
contribution, which involves four different space-time arguments, is difficult
to evaluate. If one instead simulates explicitly the incoming (classical) light
pulse, the Raman signal is given by the outgoing photon flux and can be
determined from a two-point correlation function. Such a formalism can be used
to compute the time-resolved Raman spectrum of non-equilibrium systems, as well
as nonlinear signals which are higher order in the incoming field, such as
hyper Raman scattering. Here we explain how to implement this time-dependent
formalism within the dynamical mean field theory framework. The method is
illustrated with applications to the Holstein-Hubbard model in the strong
electron-phonon coupling regime. We demonstrate hyper Raman scattering in
measurements with strong probe fields and frequency mixing signals in the
presence of a pump field, and simulate the evolution of Stokes and anti-Stokes
features after photo-excitations of metallic and Mott insulating systems.
