dynamics occurring in the material under investigation. Few-femtosecond IR-XUV/SXR attosecond pulse pump-probe experiments As an example, in a typical pump-probe experimental apparatus, an attosecond (XUV-SXR) pulse and an intense ( W/cm2) low-frequency infrared pulse with a time duration of few to tens femtoseconds are collinearly focused on the studied sample. At this point, by varying the delay of the attosecond pulse, which could be pump/probe depending on the experiment, with respect to the IR pulse (probe/pump), the desired physical observable is recorded. The subsequent challenge is to interpret the collected data and retrieve fundamental information on the hidden dynamics and quantum processes occurring in the sample. This can be achieved with advanced theoretical tools and numerical calculations. By exploiting this experimental scheme, several kinds of dynamics can be explored in atoms, molecules and solids; typically light-induced dynamics and out-of-equilibrium excited states within attosecond time-resolution. Quantum mechanics foundations Attosecond physics typically deals with non-relativistic bounded particles and employs electromagnetic fields with a moderately high intensity ( W/cm2). This fact allows to set up a discussion in a non-relativistic and semi-classical quantum mechanics environment for light-matter interaction. Atoms Resolution of time dependent Schrödinger equation in an electromagnetic field The time evolution of a single electronic wave function in an atom, is described by the Schrödinger equation (in atomic units): where the light-matter interaction Hamiltonian, , can be expressed in the length gauge, within the dipole approximation, as: where is the Coulomb potential of the atomic species considered; are the momentum and position operator, respectively; and is the total electric field evaluated in the neighbor of the atom. The formal solution of the Schrödinger equation is given by the propagator formalism: where , is the electron wave function at time . This exact solution cannot be used for almost any practical purpose. However, it can be proved, using Dyson's equations that the previous solution can also be written as: where, is the bounded Hamiltonian and is the interaction Hamiltonian. The formal solution of Eq. , which previously was simply written as Eq. , can now be regarded in Eq. as a superposition of different quantum paths (or quantum trajectory), each one of them with a peculiar interaction time with the electric field. In other words, each quantum path is characterized by three steps: An initial evolution without the electromagnetic field. This is described by the left-hand side term in the integral. Then, a