Theory of energy- and time-resolved two-photon photoemission from metal surfaces - influence of pulse duration and excitation condition

The theory of energy- and time-resolved two-photon photoemission (2PPE) spe ctra of metal surfaces is presented using density matrix formulation for a three-level system consisting of an initial occupied, intermediate unoccupi ed and final photoelectron states, A perturbation expansion method is emplo yed to calculate the energy-resolved 2PPE spectrum for continuous light bea ms. We have obtained analytical expressions of the 2PPE spectrum correspond ing to a step-by-step one-photon process through the intermediate state and a direct, two-photon-ionization process via virtual transition. It is demo nstrated that the intermediate stare can also be populated via the nonreson ant virtual process. This indicates an absolute importance of "pure dephasi ng" associated with the transition between the initial and intermediate sta tes. Evolution of the 2PPE spectrum as a function of the pump photon energy is calculated to demonstrate the conditions under which the intrinsic line width (total dephasing time) can be deduced from the lineshape analysis. It is also found that the intensity ratio of the two peaks due to the initial and the intermediate states in 2PPE spectrum can be used to estimate the p ure dephasing time. Transient behavior of the excited-state population foll owing pulse excitation is calculated with a focus on how the ultrafast rela xation times of the excited states such as image-potential states of metal surfaces are deduced from the transient 2PPE response observed with a pulse laser with much longer duration, The time-resolved 2PPE spectra are calcul ated for varying detuning from the resonant excitation from the initial sta te to the intermediate state. Transient responses of the 2PPE signal due to direct ionization and step-by-step processes are also calculated to demons trate that the nonresonant former process has an influence on the analysis of the cross-correlation trace of the intermediate state, by which the popu lation relaxation time is estimated. Attempts are also made to apply the pr esent theory to a recent time-resolved 2PPE study of the relaxation dynamic s of the image-potential states as well as hot electrons in Cu(100) and Ag( 100) surfaces.