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Non-equilibrium Bethe-Salpeter equation for transient photo-absorption spectroscopy

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.