  In this work we propose an accurate first-principle approach to calculate the
transient photo--absorption spectrum measured in Pump\&\,Probe experiments. We
formulate a condition of {\em adiabaticity} and thoroughly analyze the
simplifications brought about by the fulfillment of this condition in the
non--equilibrium Green's function (NEGF) framework. Starting from the
Kadanoff-Baym equations we derive a non--equilibrium Bethe--Salpeter equation
(BSE) for the response function that can be implemented in most of the already
existing {\em ab--initio} codes. In addition, the {\em adiabatic} approximation
is benchmarked against full NEGF simulations in simple model hamiltonians, even
under extreme, nonadiabatic conditions where it is expected to fail. We find
that the non--equilibrium BSE is very robust and captures important spectral
features in a wide range of experimental configurations.
