# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 5 QED Description of Time-Resolved Resonant X-ray Scattering
Therefore, the information about electron dynamics is contained in the absorption term of \(\hat{f}_{CF}(t_{p})\), but not in the emission term. The difference between the semiclassical approach to obtain the time-resolved DSP according to Equation (32) and the correct expression in Equation (30) has been illustrated in Reference [27] by the calculation of the DSP from an ionized Br\({}_{2}\) molecule with coherent electron dynamics in the valence shell. The two highest occupied molecular orbitals of Br\({}_{2}\) are the \(\pi g\) and \(\pi u\) orbitals of Br \(4p\) character. They assumed that a photoionizing pump pulse created an electron hole initially localized at the \(4 pz\) orbital of one of Br\({}_{2}\) atoms (at site A in Figure 2) at time \(t_{0}=0\), which is actually a coherent superposition of the electron hole being in the \(4p\pi g\) orbital and the electron hole being in the \(4p\pi u\) orbital. Such electron-hole localization in a molecule by a photoionizing pump pulse is possible as has been demonstrated in Reference [4]. Then, the electron hole, which was initially localized on one of atoms, starts oscillating coherently between the two atoms of the Br\({}_{2}\) molecule moving from site A to site B and back (see Figure 2). The electron hole dynamics is probed by an X-ray pulse with the photon energy of 13.5 keV tuned to the \(K\) edge of Br. The DSP from the Br\({}_{2}\) molecule has been calculated according to the semiclassical expression in Equation (32) and the correct expression in Equation (30), and compared in \[|\Psi_{\{n\}}^{\text{RXS}},t_{f}\rangle= -\int_{t_{0}}^{t_{f}}dt^{\prime}\,e^{i\hat{H}_{0}t^{\prime}}\hat {H}_{\text{int}}^{(1)}\,e^{-i\hat{H}_{0}t^{\prime}} \tag{27}\]0 In both cases, the signals are constant when the hole is localized on one of the atoms (at times 0 and \(T/2\)), since the scattering channel for the other atom is blocked at these moments, and there is no interference. In contrast to the semiclassical scattering patterns, the scattering patterns according to Equation (30) are different from each other at times \(T/4\) and \(3T/4\), when the electron hole densities are equal, and depend on whether the hole is moving from site A to site B or vice versa. These scattering patterns are  
\[|\Psi_{\{n\}}^{\text{RXS}},t_{f}\rangle= -\int_{t_{0}}^{t_{f}}dt^{\prime}\,e^{i\hat{H}_{0}t^{\prime}}\hat {H}_{\text{int}}^{(1)}\,e^{-i\hat{H}_{0}t^{\prime}} \tag{27}\]1 DSP from a single Br\({}_{2}\) molecule with a coherently oscillating electron hole in the \(Q_{x}-Q_{z}\) plane at \(Q_{y}=0\) according to Equation (30) (left) and Equation (32) (right) and the corresponding schematic representation of the hole density (middle) at different \(t_{p}\). The Br\({}_{2}\) molecule is aligned along the \(x\) direction parameters, the interatomic distance is \(R_{x}=2.3\) Å and the oscillation period \(T\) is 1.7 fs [60]. Probe pulse is polarized along the \(z\) direction and propagates along the \(y\)-axis with corresponding parameters \(\tau_{p}=200\) as, \(\omega_{\rm in}=13.5\) keV, \(I_{0}=10^{18}\) W/cm\({}^{2}\). \(N\) is a positive integer. The dependence on polarization is not shown. The ranges are limited by \(Q_{x}^{2}+Q_{z}^{2}\leq\omega_{\rm in}^{2}/c^{2}\). It is assumed that the detector acceptance range \(W(\omega_{\rm{\bf k}_{x}})\) allows detecting only \(1s-4p\pi g\) and \(1s-4p\pi u\) scattering events. Figure is reproduced with permission from Reference [27]. Copyright American Physical Society, 2015.  
not centrosymmetric with respect to \(\mathbf{Q}\) as in the case of NRXS (see Section 4). The QED patterns at times \(T/4\) and \(3T/4\) are not just phase-shifted with respect to the semiclassical patterns, but also have different amplitudes. For example, their intensity is nonzero at any \(\mathbf{Q}\) in contrast to the semiclassical patterns, which are zero at \(Q_{x}R_{x}=\pi/2\).