  A theoretical frame for two-photon photoemission is derived from the general
theory of pump-probe photoemission, assuming that not only the probe but also
the pump pulse is sufficiently weak. This allows us to use a perturbative
approach to compute the lesser Green function within the Keldysh formalism.
Two-photon photoemission spectroscopy is a widely used analytical tool to study
non-equilibrium phenomena in solid materials. Our theoretical approach aims at
a material-specific, realistic and quantitative description of the
time-dependent spectrum based on a picture of effectively independent electrons
as described by the local-density approximation in band-structure theory. To
this end we follow Pendry's one-step theory of the photoemission process as
close as possible and heavily make use of concepts of multiple-scattering
theory, such as the representation of the final state by a time-reversed
low-energy electron diffraction state. The formalism is fully relativistic and
allows for a quantitative calculation of the time-dependent photocurrent for
moderately correlated systems like simple metals or more complex compounds like
topological insulators. An application to the Ag(100) surface is discussed in
detail.
