# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 5 QED Description of Time-Resolved Resonant X-ray Scattering
A scattering pattern at a time \(t_{p}\) is shown in \[\times\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\,e^{i\hat{H}_{0} t^{\prime\prime}}\hat{H}_{\text{int}}^{(1)}\,e^{-i\hat{H}_{0}t^{\prime\prime}} |\{n\}\rangle\,\left|\sum_{I}C_{I}\Phi_{I}\right\rangle,\]1 Since scattered polarizations are different at each \(\mathbf{k}_{\mathrm{s}}\) vector, an additional \(\mathbf{Q}\)-dependence appears in the DSP and leads to a nonperiodic scattering pattern in Figure 4a. This dependence is eliminated in the scattering pattern in Figure 4b, which is determined solely by the trigonometric functions in Equation (33). Figure 5a shows the imaginary part of the Fourier transform from \(\mathbf{Q}\) space to real space of the scattering pattern in Figure 4b extrapolated to a region of infinite \(Q_{y}\) and \(Q_{z}\). In accordance with Equation (38), it consists of delta peaks at points corresponding to vectors connecting pairs of Br atoms (see Figure 3). The amplitudes of peaks at points \((R_{y},R_{z})\) are opposite to the amplitudes of peaks at \((-R_{y},-R_{z})\). Performing an analysis of the electronic structure of KBr, the authors found a way to resolve interatomic currents between Br atoms in a direction parallel to the probe-pulse polarization. They suggested to measure scattering patterns at different time delays \(t_{p}-t_{0}\), perform the Fourier transform of these scattering patterns from \(\mathbf{Q}\) space to real space and follow the amplitude of the peak outlined in the circle in Figure 5a in the imaginary part of the Fourier transform. The evolution of the amplitude of this peak depending on the time of measurement \(t_{p}\) is shown in Figure 5b with the solid violet line. It precisely reproduces the time evolution of the computed sum of the currents between all pairs of nearest-neighbor Br atoms connected by the vector \((0,R_{\mathrm{Br-Br}},0)\) shown in the orange dashed line in Figure 5b, where \(R_{\mathrm{Br-Br}}\) is the distance between the nearest-neighbor Br atoms. Thus, it has been shown in Reference [28] that if a proper polarization of the incoming probe pulse has been chosen for measurement of scattering patterns, the time evolution of the amplitude of a certain delta peak in their Fourier transform follows the time evolution of the sum of the interatomic currents between atoms connected by the vector corresponding to this peak. The required polarization of the probe pulse can be determined by an analysis of the electron structure of the crystal. \[\times\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\,e^{i\hat{H}_{0} t^{\prime\prime}}\hat{H}_{\text{int}}^{(1)}\,e^{-i\hat{H}_{0}t^{\prime\prime}} |\{n\}\rangle\,\left|\sum_{I}C_{I}\Phi_{I}\right\rangle,\]2 Scattering patterns at the probe-pulse intensity \(I_{0}\) from the ionized KBr cluster obtained by a \(y\)-polarized X-ray pulse. (**a**) No polarization filter is applied in the measurement of a scattered photon; (**b**) A polarization filter transmitting \(y\)-polarized scattered photons is applied and the dependence on scattered polarizations is eliminated. The spectral window function \(\mathsf{W}(\omega_{\mathbf{k}_{\mathrm{s}}})\) is centered at \(\omega_{\mathrm{in}}\) and suppresses all photons emitted by electrons lying deeper than the outermost valence band of KBr. Figure is reproduce with permission from Reference [28]. Copyright American Physical Society, 2015. To sum up, time-resolved RXS is a robust technique that is in many ways advantageous for probing electron dynamics in molecules and crystals. On top of the usual structural information about an atomic system, it can provide interatomic electron current in it. Feasibility analyses of the experiments to measure electron dynamics in Br\({}_{2}\)[27], and KBr and Ge crystals [28] by means of ultrafast RXS showed that they are achievable at X-ray free electron laser facilities. However, the main obstacle of such experiments is the need to produce hard X-ray pulses of subfemtosecond duration for an interatomic spatial resolution and sufficient temporal resolution.