Role of Resonant $\mathbf{k}$-Points in the Transient Optical Response of Pumped Germanium
Pump-induced transient optical properties combine contributions from electronic states throughout the Brillouin zone, but the relative relevance of off-resonant and l-photon resonant crystal momenta has remained unexplored. We address this issue in pumped germanium by resolving the transient absorptive response into momentum-space classes defined by the presence or absence of 1-, 2-, and 3-photon resonances with respect to the pump. Using the Dynamical Projective Operatorial Approach together with the related generalized linear response theory, we compute the differential imaginary part of the dielectric function and evaluate the contributions of each resonance class. Resonant regions account for nearly the entire optical response, whereas points outside the identified resonance sets contribute only negligibly. Nevertheless, the 2-photon-resonant set, although containing more than 98% of the residual (post-pump) excitation population, does not reproduce the full transient spectrum. Conversely, resonance classes with very small residual populations generate non-negligible contributions to the transient optical properties. This mismatch shows that the transient optical weight is not determined solely by the real-charge dynamics (which results in post-pulse residual excitation population) and is consistent with substantial virtual pump-induced contributions, whose dominant optical weight nevertheless arises from the resonant regions of momentum space. The class-resolved phase of the dominant 2$\omega_{\mathrm{pu}}$ oscillations further shows that, whenever a class contributes appreciably, the phase of its oscillatory component follows that of the corresponding full signal. The resulting decomposition provides a momentum-resolved connection among multi-photon resonances and transient optical observables in a realistic material.