# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 4 QED Description of Time-Resolved Nonresonant X-ray Scattering
The corresponding term in Equation (20) is centrosymmetric with respect to \(\mathbf{Q}\). The function  
\[\frac{dP^{\text{NRXS}}(\mathbf{Q})}{d\Omega}\propto\left|\int d^{3}r\rho( \mathbf{r},t_{p})e^{i\mathbf{Q}\cdot\mathbf{r}}\right|^{2}. \tag{13}\]4  
is connected to the imaginary parts of the electron density matrix elements and its temporal dependence correlates with the electron current density, which is given by  
\[\frac{dP^{\text{NRXS}}(\mathbf{Q})}{d\Omega}\propto\left|\int d^{3}r\rho( \mathbf{r},t_{p})e^{i\mathbf{Q}\cdot\mathbf{r}}\right|^{2}. \tag{13}\]5  
The corresponding term in Equation (20) is an odd function with respect to \(\mathbf{Q}\) and is responsible for the time-resolved scattering patterns being noncentrosymmetric. Thus, the fact that the scattering patterns in Figure 1 are not centrosymmetric reflects that the electron current is nonzero, i.e., that electrons are moving. The two contributions to the scattering patterns can be decomposed performing the Fourier transform of the DSP \(\mathcal{F}^{\text{NRXS}}(\mathbf{r},t_{p})=1/(2\pi)^{3}\int d^{3}Qe^{-i\mathbf{Q} \cdot\mathbf{r}}dP^{\text{NRXS}}(\mathbf{Q})/d\Omega\). The real part of the Fourier transform is connected to the function \(\mathcal{R}^{\text{NRXS}}\) via the relation  
\[\frac{dP^{\text{NRXS}}(\mathbf{Q})}{d\Omega}\propto\left|\int d^{3}r\rho( \mathbf{r},t_{p})e^{i\mathbf{Q}\cdot\mathbf{r}}\right|^{2}. \tag{13}\]6  
and the imaginary part is connected to the function \(\mathcal{J}^{\text{NRXS}}\) according to  
\[\frac{dP^{\text{NRXS}}(\mathbf{Q})}{d\Omega}\propto\left|\int d^{3}r\rho( \mathbf{r},t_{p})e^{i\mathbf{Q}\cdot\mathbf{r}}\right|^{2}. \tag{13}\]7  
Thus, performing the Fourier transform of the scattering patterns from \(\mathbf{Q}\) space to real space obtained at different time delays \(t_{p}-t_{0}\), one obtains a complex function, which depends on the time of measurement \(t_{p}\) and the space coordinates \(\mathbf{r}\). The real part of this function correlates with charge distributions at the time of measurement \(t_{p}\), and its imaginary part correlates with electron currents at \(t_{p}\). A more specific connection of \(\text{Re}\left[\mathcal{F}^{\text{NRXS}}(\mathbf{r},t_{p})\right]\) and \(\text{Im}\left[\mathcal{F}^{\text{NRXS}}(\mathbf{r},t_{p})\right]\) to certain time-dependent quantities has to be identified for a given electronic system. Such an analysis has been performed for the case of RXS [27; 28] as described in the next Section. Several alternative approaches to image electron dynamics by means of time-resolved NRXS have been developed through the application of the QED framework. [PERSON] et al. has proposed to image the instantaneous electron density of an electronic wave packet via ultrafast X-ray phase contrast imaging [51]. They have shown that ultrafast phase contrast imaging provides the Laplacian of the electron density, which reveals complex bonding and topology of the charge distributions in an electronic system. However, this technique is quite experimentally challenging, since it requires detector pixels that are small enough to resolve the image of small objects such as molecules. Another approach has been suggested by [PERSON] et al. in Reference [52], who have shown that the Compton-scattering cross section, in the impulse approximation, depends solely on the electron momentum distribution. Thus, time-resolved Compton scattering can be used to obtain momentum-space images of the sample to be probed. [PERSON] et al. have suggested time-resolved X-ray diffraction as a probe of molecules in the gas phase undergoing nonadiabatic avoided-crossing dynamics involving strongly coupled electrons and nuclei [53]. They have shown that it provides signatures of a created electronic coherence on top of dominant ground- and excited-state wavepacket motions. A photon-coincidence measurement based on the time- and wavevector-resolved detection of photons generated by the scattering of multiple X-ray pulses with variable delays has been proposed by [PERSON] et al. in Reference [54].