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

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... oemission 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.