# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 5 QED Description of Time-Resolved Resonant X-ray Scattering  
RXS is an element specific technique that provides insight into charge, orbital and spin degrees of freedom [55; 56; 57; 58]. Due to the resonant nature of this process, the scattering cross section can be considerably enhanced in comparison to the nonresonant case discussed in the previous Section. In contrast to NRXS, which probes simultaneously electrons involved in the dynamics and electrons that are essentially stationary [59], RXS can directly probe (quasi-)particles involved in the dynamics. Therefore, RXS particularly suits for measurement of heavy elements, where the vast majority of electrons are stationary, and resonant energies approach several keV corresponding to Angstrom spatial resolution. RXS is a two-step process consisting of an absorption and an emission process, which involves an intermediate state in distinction to NRXS. In the first step, the absorption of a photon from an X-ray pulse induces a resonant transition of an electron from a core shell to a valence shell of an atomic system (e.g., a molecule or a crystal) being measured. Thus, after the first step, the electronic system is brought from its initial state \(I\) to an intermediate state \(J_{C}\) with an electron hole in a core shell of a scattering atom \(C\) and an additional electron in the valence shell. In the second step, an electron from either the same valence shell or some other shell fills the electron hole in the core shell leading to a spontaneous emission of a photon, which is detected. The electronic system is in a final state \(F\) afterwards. If elastic scattering has taken place, then the final state \(F\) coincides with the initial state \(I\), and the energy of the scattered photon is equal to \(\omega_{\text{in}}\). An inelastic scattering event results in the final state \(F\) being different from the initial state \(I\) and leads to scattering of a photon with the energy \(\omega_{\text{in}}-(E_{F}-E_{I})\). As in the case of NRXS, elastic and inelastic scattering events from a stationary system can be distinguished by the spectroscopy of a scattered photon, and the contribution of elastic scattering events to RXS scattering patterns are dominating. Through elastic scattering, resonant scattering patterns are determined by the electron density of the object being measured, but are indirectly connected to it via the relation [55]  
\[\frac{dP^{\text{stRXS}}(\mathbf{Q})}{d\Omega}\propto\left|\sum_{C}f_{C}e^{i \mathbf{Q}\cdot\mathbf{R}_{C}}\right|^{2}, \tag{26}\]  
where \(f_{C}\) is a scattering amplitude of an atom \(C\) located at a position \(\mathbf{R}_{C}\). Since RXS involves transitions of electrons from core shells, which are highly localized in comparison to X-ray wavelengths, the spatial distribution of the X-ray electric field is treated within the dipole approximation for each absorbing atom. A scattering amplitude \(f_{C}\) is proportional to the product of the dipole matrix elements describing the absorption process and the emission process: \(f_{C}\propto\langle\Phi_{I}|\mathbf{e}_{s}^{*}\cdot\mathbf{r}|\Phi_{J_{C}}\rangle \langle\Phi_{J_{C}}|\mathbf{e}_{\text{in}}\cdot\mathbf{r}|\Phi_{I}\rangle\). Now, let us consider the interaction of a resonant X-ray pulse with an atomic system with coherent electron dynamics in the valence shell described by the state \(|\Psi(t)\rangle=\sum_{I}C_{I}e^{-iE_{I}(t-t_{0})}|\Phi_{I}\rangle\). In the absorption step, an electron from a core shell is excited into the valence shell, whereby it destroys the coherent electron dynamics. The system is brought to an intermediate state \(J_{C}\), and the following step, emission, is determined by this intermediate state and is not connected to the state \(|\Psi(t)\rangle\). Thus, as in the case of NRXS, it is, first, very unlikely that the final state after the emission process would coincide with the superposition \(|\Psi(t)\rangle\) and, second, this situation would be spectroscopically indistinguishable.