# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 1 Introduction  
Real-time imaging of electron dynamics is one of the most important and challenging tasks for modern ultrafast science [1; 2; 3; 4; 5; 6; 7; 8]. Valence electron dynamics occurs on a subfemtosecond to few-femtosecond timescale (1 fs = \(10^{-15}\) s) and its real-time measurement requires attosecond temporal resolution (1 as = \(10^{-18}\) s). At the same time, Angstrom spatial resolution is required to access inter-atomic distances in molecular structures and solids. Such ultrahigh spatial and temporal resolutions could be achieved using X-ray free-electron lasers, which can generate hard-X-ray pulses with Angstrom wavelengths [9; 10; 11; 12; 13]. Free-electron laser sources are capable of producing ultrashort X-ray pulses of femtosecond [14; 15] and even subfemtosecond duration [16; 17; 18; 19; 20; 21; 22; 23]. A remarkable achievement was demonstrated recently by [PERSON] et al., who succeeded to generate single spikes of hard X-rays that are only 200 attoseconds long at Linac Coherent Light Source [23]. (Sub-)femtosecond hard X-ray pulses can be applied for time-resolved imaging of electron dynamics either by means of nonresonant X-ray scattering (NRXS) [24; 25; 26] or by means of resonant X-ray scattering (RXS) [27; 28], which are different processes determined by distinct terms of the light-matter interaction Hamiltonian. Alternatively to using photons for imaging electron dynamics with X-ray scattering, one can consider employing photoelectrons dislodged by ultrashort light pulses for this goal. Time- and angle-resolved photoelectron spectroscopy (TRARPES), i.e., time- and energy-resolvedmolecular-frame photoelectron angular distributions, using ultrashort extreme ultraviolet (XUV) probe pulses inducing single-photon ionization has been proposed for imaging coherent electron dynamics in molecules [29; 30; 31; 32]. High-energy photoelectrons generated by XUV pulses, on the one hand, provide Angstrom spatial resolution and, on the other hand, allow for a less sophisticated interpretation than that in the case of low-energy photoelectrons, where multiple scattering effects can be substantial [33]. The goal of this article is to show that imaging a nonstationary electronic system by ultrashort light pulses is nontrivial not only from an experimental but also from a theoretical perspective. We aim to demonstrate that a simple extrapolation of an observable of a stationary measurement to a time-dependent quantity to obtain a time-resolved signal does not work and can lead to erroneous results. Therefore, a careful analysis of time-dependent processes governing a nonstationary measurement is necessary for a correct interpretation of its outcome. For a measurement of a stationary electronic system by means of X-ray scattering, its electron density is the central quantity that determines a scattering pattern. Thus, one may naively expect that time-resolved scattering patterns from a nonstationary electronic system depend on its time-dependent electron density. However, it has been demonstrated that this assumption is not correct and time-resolved scattering patterns are not connected to a time-dependent electron density due to a considerable contribution of inelastic (Compton-type) processes that cannot be physically avoided [24; 26; 27; 28; 34]. Only taking into account all transitions that can be induced by a broadband, ultrashort probe pulse, one can obtain correct time-resolved scattering patterns. Such an analysis can be accurately performed within the quantum electrodynamics (QED) framework. Analogously, one may assume that electron dynamics can be measured by means of time-dependent chemical shifts using time-resolved photoelectron spectroscopy. Since electron binding energies, which determine photoelectron spectra, depend on the local chemical environment, one would think that photoelectron peaks would shift following a time-dependent electron density.