  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.
