# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 4 QED Description of Time-Resolved Nonresonant X-ray Scattering  
Nonresonant X-ray scattering (usually, the specification 'nonresonant' is omitted) is an established technique that is employed to reveal structural information about a sample. In the scattering process, an electronic system in an initial state \(I\) interacting with an X-ray pulse absorbs and emits a photon, which leaves a system either in the same state \(I\) or brings it to a different final state \(F\). The former case means that the photon has been scattered elastically and its energy is equal to the energy of the incoming X-ray beam \(\omega_{\text{in}}\). In the latter case, inelastic scattering has taken place and a photon with energy \(\omega_{\text{in}}-(E_{F}-E_{I})\) has been scattered. Thus, elastic and inelastic scattering events from a stationary system can be distinguished by the spectroscopy of a scattered photon. Contributions due to elastic scattering to a scattering pattern dominate over that due to inelastic scattering, since transition amplitudes of elastic scattering events sum up coherently. Thus, scattering patterns from a stationary object are determined by elastic scattering and, as a result, encode the electron density \(\rho(\mathbf{r})\) of the object via the relation  
\[\frac{dP^{\text{stNRXS}}(\mathbf{Q})}{d\Omega}\propto\left|\int d^{3}r\rho( \mathbf{r})e^{i\mathbf{Q}\cdot\mathbf{r}}\right|^{2}, \tag{12}\]  
where \(\mathbf{Q}\) is the scattering vector [46]. Solving the phase problem, one can reconstruct the electron density with the spatial resolution determined by the wave length of the X-ray pulse. Let us now consider the interaction of an X-ray pulse with a nonstationary electronic system in the state \(|\Psi(t)\rangle=\sum_{I}C_{I}e^{-iE_{I}(t-t_{0})}|\Phi_{I}\rangle\). In References [47; 48], where this problem has been studied, it was assumed that the time-dependent electron density of a nonstationary electron system at the time of measurement \(t_{p},\rho(\mathbf{r},t_{p})\), should be substituted for the electron density \(\rho(\mathbf{r})\) in Equation (12) to obtain time-resolved scattering patterns  
\[\frac{dP^{\text{NRXS}}(\mathbf{Q})}{d\Omega}\propto\left|\int d^{3}r\rho( \mathbf{r},t_{p})e^{i\mathbf{Q}\cdot\mathbf{r}}\right|^{2}. \tag{13}\]  
However, the connection of scattering patterns to the electron density stem from elastic scattering. If one would try to consider elastic scattering from an electronic system in the state \(|\Psi(t)\rangle=\sum_{I}C_{I}e^{-iE_{I}(t-t_{0})}|\Phi_{I}\rangle\), one would have to assume that the final state of a system \(F\) is exactly the same superposition \(|\Psi(t)\rangle\) as before the interaction with the X-ray pulse. In addition to the fact that the probability of such a situation is extremely weak, it is impossible to spectroscopically distinguish this scattering event from other transitions with final states within the eigenstates of the wave packet \(|\Psi(t)\rangle\). In view of this, the concept of 'elastic scattering' in the case of the interaction of an X-ray pulse with a nonstationary electronic system is ambiguous, and the connection of scattering patterns to the electron density must be carefully reconsidered for the time-dependent case [24]. In order to accurately describe the interaction of an X-ray pulse with an electronic system evolving coherently, a theoretical analysis employing the QED framework has been performed by [PERSON] et al. in Reference [24]. Let us briefly review the result of this work. As discussed in the previous Section, nonresonant X-ray scattering is governed by the Hamiltonian \(\hat{H}_{\text{int}}^{(2)}\) in Equation (5). In order to obtain the probability of nonresonant X-ray scattering with Equation (7), the total density matrix \(\hat{\rho}_{f}(t_{f})\) must be evaluated within the first-order time-dependent perturbation theory using \(\hat{H}_{\text{int}}^{(2)}\) as the perturbation.