This manuscript reviews our recently developed theory, the dynamical projective operatorial approach (DPOA), for studying pump–probe setups in ultra-fast regimes. After reviewing the general formulation of the DPOA, we focus on its lattice version and provide a formalism that is particularly suitable for several pumped semiconductors. Within the DPOA, we also compute the TR-ARPES signal through out-of-equilibrium Green's functions and establish an out-of-equilibrium counterpart of the fluctuation–dissipation theorem. Moreover, we generalize the linear response theory to pumped systems and address, within the DPOA, the differential transient optical properties, providing an overall robust theoretical and computational framework for studying pump–probe setups. Considering a minimal model for a semiconductor, we illustrate the capabilities of the DPOA and discuss several features emerging in this case study that are relevant to real materials. [ABSTRACT FROM AUTHOR]

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