# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 5 QED Description of Time-Resolved Resonant X-ray Scattering
It has been shown in Reference [27] that the time evolution of the interatomic electron current \(j_{C_{1}C_{2}}(t_{p})\) is substantially reproduced by the function \(\mathcal{J}_{C_{1}C_{2}}^{\text{RSS}}(t_{p})\) and can be reconstructed from the Fourier transform of the DSP as shown below. Performing the Fourier transform of the DSP in Equation (33), \(\mathcal{F}^{\text{RXS}}(\mathbf{r},t_{p})=1/(2\pi)^{3}\int d^{3}Qe^{-i \mathbf{Q}\cdot\mathbf{r}}d\mathcal{P}^{\text{RXS}}(\mathbf{Q})/d\Omega\), its centrosymmetric and nonetrosymmetric contributions can be decomposed. The real part of the Fourier transform is connected to the functions \(\mathcal{R}_{C_{1},C_{2}}^{\text{RXS}}\) via the relation  
\[|\Psi_{\{n\}}^{\text{RXS}},t_{f}\rangle= -\int_{t_{0}}^{t_{f}}dt^{\prime}\,e^{i\hat{H}_{0}t^{\prime}}\hat {H}_{\text{int}}^{(1)}\,e^{-i\hat{H}_{0}t^{\prime}} \tag{27}\]8  
and its imaginary part is connected to \(\mathcal{J}_{C_{1},C_{2}}^{\text{RXS}}\) via  
\[|\Psi_{\{n\}}^{\text{RXS}},t_{f}\rangle= -\int_{t_{0}}^{t_{f}}dt^{\prime}\,e^{i\hat{H}_{0}t^{\prime}}\hat {H}_{\text{int}}^{(1)}\,e^{-i\hat{H}_{0}t^{\prime}} \tag{27}\]9  
Thus, the Fourier transform of a time-resolved resonant scattering pattern is a sum of delta peaks at the interatomic distances \(\mathbf{R}_{C_{1}}-\mathbf{R}_{C_{2}}\) weighted by complex amplitudes, which depend on the time of measurement \(t_{p}\). In particular, the imaginary part of the amplitude of a peak at \(\mathbf{R}_{C_{1}}-\mathbf{R}_{C_{2}}\) is given by \(\mathcal{J}_{C_{1},C_{2}}^{\text{RXS}}(t_{p})\) and provides the electron current between atoms \(C_{1}\) and \(C_{2}\). If there are several atomic pairs at the same interatomic distance, when the imaginary part corresponds to the sum of the interatomic currents between the corresponding pairs. The procedure to reconstruct interatomic electron currents in crystals by means of ultrafast RXS has been illustrated by the calculation of time-resolved scattering patterns from KBr and Ge crystals with coherent electron dynamics in the valence bands by [PERSON] and [PERSON] in Reference [28]. Here, we will review their calculation of coherent electron dynamics in KBr. It is an ionic crystal, where the 4\(s\) electrons of K atoms are transferred to Br atoms [65]. The \(p\)-character electrons centered on the Br atoms form the outermost valence band of KBr. It was assumed that a photoionizing pump pulse induced coherent electron dynamics in this band by creating \(p\)-type electron holes centered on Br atoms at time \(t_{0}\). A probe pulse propagating along the \(x\) direction of the duration of 200 as and the photon energy of \(\approx\) 13 keV tuned to the \(K\) edge of Br polarized along the \(y\)-axis parallel to one of the vectors connecting two Br atoms (see Figure 3) was assumed. Each coherently evolving electron hole is delocalized and distributed over many Br atoms in some region (see Figure 3). It was assumed that the concentration of the electron holes is sufficiently low to  
\[\times\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\,e^{i\hat{H}_{0} t^{\prime\prime}}\hat{H}_{\text{int}}^{(1)}\,e^{-i\hat{H}_{0}t^{\prime\prime}} |\{n\}\rangle\,\left|\sum_{I}C_{I}\Phi_{I}\right\rangle,\]0 Snapshot of the electron-hole density at time \(t_{p}\) in a fragment of (KBr)\({}_{108}\) cluster visualized using VESTA software [66]. The orange arrows represent the electron-hole currents \(j_{C_{1}C_{2}}(t_{p})\) between the Br atoms. The blue-framed arrow represents a current parallel to the probe-pulse polarization. Figure is reproduced with permission from Reference [28]. Copyright American Physical Society, 2015.  
consider these regions isolated and the holes noninteracting. A region, where a single electron hole is distributed, was simulated by a (KBr)\({}_{108}\) cluster. Since the delocalized electron hole is coherently evolving, the charge is redistributing and flowing from one atom to another resulting in nonzero interatomic electron current.