We describe a formalism suitable for studying the ultrafast dynamics and nonadiabatic effects associated with propagation of a single electron injected into an empty band. Within the band the electron is coupled to vibrational or electronic excitations that can be modeled by bosons. The formalism is based on the application of cumulant expansion to calculations of diagonal single particle propagators that are used in the interpretations of time resolved measurements of the surface electronic structure. Second and fourth order cumulants which arise from linear coupling to bosonic excitations and give leading contributions to the renormalization of propagators are explicitly calculated in the real time domain and their properties analyzed. This approach enables the assessment of transient effects and energy transfer associated with nonadiabatic response of the system to promotion of electrons into unoccupied bands, as well as of higher order corrections to the lifetimes and energy shifts of the initial electronic states that in the adiabatic regime are obtained from Fermi’s golden rule approach or its improvements such as the $\mathit{GW}$ approximation. In the form presented the formalism is particularly suitable for studying the non-Markovian evolution and ultrafast decoherence of electronic states encountered in electron spectroscopies of quasi-two-dimensional bands on metal surfaces whose descriptions are inaccessible to the approaches based on the adiabatic hypothesis. The fast convergence of the results obtained by this procedure is demonstrated for a simple model system relevant to surface problems. On the basis of this and some general properties of cumulants it is argued that in the majority of surface problems involving electron-boson interactions the ultrafast dynamics of quasiparticles is accurately described by the second order cumulant, which can be calculated with the effort not exceeding those encountered in the standard $\mathit{GW}$ approximation calculations.

• Received 11 April 2005

DOI:https://doi.org/10.1103/PhysRevB.72.165406