  Light interacting with metals elicits an ultrafast coherent many-body
screening response on sub- to few-femtosecond time-scales, which makes its
experimental observation challenging. Here, we describe the coherent
two-dimensional (2D) multi-photon photoemission study of the Shockley surface
state (SS) of Ag(111) as a benchmark for spectroscopy of the coherent nonlinear
response of metals to an optical field in the perturbative regime. Employing
interferometrically time-resolved multi-photon photoemission spectroscopy
(ITR-mPP), we correlate the coherent polarizations and populations excited in
the sample with final photoelectron distributions where the interaction
terminates. By measuring the non-resonant 3- and 4-photon photoemission of the
SS state, as well as its replica structures in the above-threshold
photoemission (ATP), we record the coherent response of the Ag(111) surface by
2D photoemission spectroscopy and relate it to its band structure. We interpret
the mPP process by an optical Bloch equation (OBE) model, which reproduces the
main features of the surface state coherent polarization dynamics recorded in
ITR-mPP experiments: The spectroscopic components of the 2D photoelectron
spectra are shown to depend on the nonlinear orders of the coherent
photoemission process m as well as on the induced coherence n.
