Controlling Ultrafast Excitations in Germanium: The Role of Pump-Pulse Parameters and Multi-Photon Resonances

We employ the Dynamical Projective Operatorial Approach (DPOA) to investigate the ultrafast optical excitations of germanium under intense, ultrashort pump pulses. The method has very low resource demand relative to many other available approaches and enables detailed calculation of the residual electron and hole populations induced by the pump pulse. It provides direct access to the energy distribution of excited carriers and to the total energy transferred to the system. By decomposing the response into contributions from different multi-photon resonant processes, we systematically study the dependence of excited-carrier density and absorbed energy on key pump-pulse parameters: duration, amplitude, and photon energy. Our results reveal a complex interplay between these parameters, governed by resonant Rabi-like dynamics and competition between different multi-photon absorption channels. For the studied germanium setup, we find that two-photon processes are generally dominant, while one- and three-photon channels become significant under specific conditions of pump-pulse frequency, duration, and intensity. This comprehensive analysis offers practical insights for optimizing ultrafast optical control in semiconductors by targeting specific multi-photon pathways.