# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 4 QED Description of Time-Resolved Nonresonant X-ray Scattering
Namely, the calculation of the scattering patterns in Figure 1a considered  
\[\frac{dP^{\text{NRXS}}(\mathbf{Q})}{d\Omega}\propto\left|\int d^{3}r\rho( \mathbf{r},t_{p})e^{i\mathbf{Q}\cdot\mathbf{r}}\right|^{2}. \tag{13}\]0 Scattering patterns in the \(Q_{x}-Q_{z}\) plane (\(Q_{y}=0\)) and electronic charge distributions of the wavepacket \(\Psi(t)\). (**a**) Scattering patterns obtained with Equation (18); (**b**) scattering patterns obtained with Equation (18), but the calculation is restricted to the two electronic states involved in the dynamics; (**c**) electronic charge distributions and (**d**) scattering patterns obtained with Equation (13), at pump-probe delay times 0, \(T/4\), \(T/2\), \(3T/4\), and \(T\), where the oscillation period of the electronic wavepacket is \(T=6.25\) fs. The intensities of the patterns are shown in units of the DSP from a free electron in all cases. Figure is adopted with permission from Reference [24]. Copyright National Academy of Sciences, 2012.  
all possible final states matching the assumed photon detection width of 0.5 eV. However, the most important difference appears at delay times \(T/4\) and \(3T/4\), when the electronic charge distributions are identical, but the electronic wave packet carry a different phase. The correct scattering patterns in Figure 1a are different from each other at delay times \(T/4\) and \(3T/4\) unlike the scattering patterns at delay times \(T/4\) and \(3T/4\) in Figure 1d, which are equal to each other. Thus, the correct scattering patterns depend on the phase of the wave packet, but not on the electron density at the time of measurement as the scattering patterns in Figure 1d do. The further principal difference is that the scattering patterns in Figure 1a are not centrosymmetric (i.e., they are not equal for \(\mathbf{Q}\) and \(-\mathbf{Q}\)) at delay times \(T/4\) and \(3T/4\) breaking the [PERSON]'s law [50]. For the sake of comparison, [PERSON] et al. have shown scattering patterns calculated with Equation (18) within the QED framework, but restricting the calculation to the two electronic states involved in the dynamics [see Figure 1b]. The structure of these patterns is similar to the structure of the semiclassical patterns in Figure 1d. However, analogously to the patterns in Figure 1a, they depend on the phase of the electronic wave packet and are not centrosymmetric at delay times \(T/4\) and \(3T/4\). Thus, not only inelastic transitions to a manifold of possible final states do lead to a disagreement between the QED and semiclassical scattering patterns, but also an incorrect description of the time dependence of the DSP in Equation (13) does. Obviously, the connection to the electronic state encoded in time-resolved scattering pattern is much less straightforward as suggested by Equation (13). In order to interpret information encoded in time-resolved scattering patterns, a Fourier analysis has been proposed for time-resolved resonant scattering patterns [28]. Let us apply such an analysis to the nonresonant case. Equation (18) can be represented as  
\[\frac{dP^{\text{NRXS}}(\mathbf{Q})}{d\Omega}\propto\left|\int d^{3}r\rho( \mathbf{r},t_{p})e^{i\mathbf{Q}\cdot\mathbf{r}}\right|^{2}. \tag{13}\]1 \[\frac{dP^{\text{NRXS}}(\mathbf{Q})}{d\Omega}\propto\left|\int d^{3}r\rho( \mathbf{r},t_{p})e^{i\mathbf{Q}\cdot\mathbf{r}}\right|^{2}. \tag{13}\]2  
The function entering the expression above  
\[\frac{dP^{\text{NRXS}}(\mathbf{Q})}{d\Omega}\propto\left|\int d^{3}r\rho( \mathbf{r},t_{p})e^{i\mathbf{Q}\cdot\mathbf{r}}\right|^{2}. \tag{13}\]3  
is connected to the real part of the elements of the electron density matrix \(\hat{\rho}^{m}(t_{p})\) [cf. Equation (2)]. Therefore, its temporal dependence correlates with the electron density [cf. Equation (3)] and related quantities such as charge distributions at the time \(t_{p}\). Please notice that this temporal dependence is different to that given by the semiclassical Equation (13), which correlates with the time-dependent electron density squared.