# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 8 Discussion  
We reviewed the theoretical framework based on QED developed for an accurate description of an interaction between a coherently evolving electronic system and an ultrashort light probe pulse. We concentrated on its application to three techniques, namely, time-resolved NRXS, time-resolved RXS and TRARPES that can be employed for a measurement of electron dynamics in real space and real time [24; 26; 27; 28; 32; 51; 52; 53; 54; 68]. It turned out that the corresponding theories and their results have much in common. The outcomes of the theoretical analyses appeared to be quite counter-intuitive in all considered cases. Time- and momentum-resolved signals from a nonstationary electronic system obtained by means of the considered time-resolved techniques encode different information from their stationary analogues. X-ray scattering, which is determined by the electron density of a stationary object being measured, is not determined by the time-dependent electron density in a time-resolved measurement. The connection of scattering patterns to the electron density relay on elastic scattering dominating over inelastic scattering. In the case of an interaction of a light pulse with a nonstationary electronic system, the concept of elastic scattering is ambiguous, since it is extremely improbable that a final state after the interaction would be the same nonstationary electronic state. Moreover, such a situation would not be spectroscopically distinguishable from other possible inelastic scattering events. As a result, time-resolved scattering patterns encode spatial and temporal correlations substantially deviating from quantities encoded in stationary scattering patterns. A similar consequence of a time-resolved measurement on photoelectron spectroscopy has been demonstrated. Contrary to an intuitive assumption that electron dynamics can be measured by means of time-dependent chemical shifts (i.e., by following temporal changes of electronic binding energies), time-resolved photoelectron spectra consist of a series of photoelectron peaks centered on time-independent positions. Electron dynamics is indeed encoded in time-dependent amplitudes of these peaks, which exhibit prominent temporal dependence in photoelectron angular distributions at fixed photoelectron energies. In all cases considered, time- and momentum-resolved signals, i.e., scattering patterns in the case of NRXS and RXS, and photoelectron angular distributions in the case of TRARPES, do not follow instantaneous electron density. They depend on the phase of the electronic wave packet being measured and are not centrosymmetric with respect to the momentum (except for certain time points). It turns out that they have a quite similar dependence on the electronic state at the time of measurement \(\Psi(t_{p})\) and the spectral density of a probe pulse. In all cases, they are connected to a function, which can be represented as \(\langle\Psi(t_{p})|\hat{H}(\mathbf{r}_{1},\mathbf{r}_{2})|\Psi(t_{p})\rangle\), where an operator \(\hat{H}(\mathbf{r}_{1},\mathbf{r}_{2})\) includes a sum over final states of corresponding transition amplitudes weighted by a function determined by the spectral density of a probe pulse. The exact form of the operator \(\hat{H}(\mathbf{r}_{1},\mathbf{r}_{2})\) depends on the technique by which the signal was obtained. There are two contributions to the time- and momentum-resolved signals. The first contribution is centrosymmetric with respect to the momentum and is determined by a function connected to the real parts of electron density matrix elements. This means that the temporal evolution of this function correlates with the time-dependent electron density and charge distributions. The second contribution is an odd function with respect to the momentum and is determined by a function connected to the imaginary parts of electron density matrix elements. Its temporal dependence correlates with electron currents. Thus, the fact that the signals are not centrosymmetric reflects that electrons are moving.