Title: Generalized Linear Response Theory for Pumped Systems and its Application to Transient Optical Properties within DPOA

Abstract: We derive the two-time linear response theory for out-of-equilibrium pumped systems, generic pump-probe delays and probe frequencies. Such a theory enormously simplifies the numerical calculations, for instance, of the optical conductivity with respect to the actual procedure, which requires computing the effect of the probe pulse for each time delay with respect to the pump pulse\textcolor{red}{.} The theory is given for a generic observable and pumped Hamiltonian and then specialized for a system with a quadratic Hamiltonian and its transient optical properties, exploiting the Dynamical Projective Operatorial Approach (DPOA). The theory is complemented by a set of crucial numerical guidelines that help perform actual calculations in a computationally affordable way. The optical response (differential transient reflectivity and absorption) of a prototypical three-band (core, valence, and conduction) model in the XUV regime is analyzed in detail to illustrate the theory and its application. Using some generalizations of the density of states, we provide a systematic approach to exploring the optical properties in terms of the system band structure features and the pump parameters. Such an analysis can be extremely helpful in understanding the actual results of experimental optical measurements. Moreover, we study the effects of inter-band and intra-band transitions, the local dipole coupling, and single and multi-photon processes. The latter is further investigated by varying the central frequency of the pump pulse to have different regions of the first Brillouin zone in resonance with it. We also study the effect of varying the pump pulse intensity. Finally, we study and analyze the transient optical properties in the probe pulse regime of IR and visible.