# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 4 QED Description of Time-Resolved Nonresonant X-ray Scattering
The resulting time-dependent DSP can be represented as  
\[\frac{dP^{\text{NRXS}}(\mathbf{Q})}{d\Omega}=\Gamma_{0}^{\text{NRXS}}\sum_{F }\int d^{3}r_{1}\int d^{3}r_{2}e^{i\mathbf{Q}\cdot(\mathbf{r}_{1}-\mathbf{r} _{2})}\langle\Psi(t_{p})|\hat{G}_{F}^{\text{NRXS}}(\mathbf{r}_{1},\mathbf{r}_ {2})|\Psi(t_{p})\rangle e^{-\Omega_{F}^{2}\tau_{p}^{2}/(4\ln 2)} \tag{18}\]  
with the operator  
\[\hat{G}_{F}^{\text{NRXS}}=\hat{\psi}(\mathbf{r}_{2})\hat{\psi}^{\dagger}( \mathbf{r}_{2})|\Phi_{F}\rangle\langle\Phi_{F}|\hat{\psi}(\mathbf{r}_{1})\hat{ \psi}^{\dagger}(\mathbf{r}_{1})\int_{0}^{\infty}d\omega_{\mathbf{k}_{\text{s} }}\omega_{\mathbf{k}_{\text{s}}}W(\omega_{\mathbf{k}_{\text{s}}}). \tag{19}\]  
Here, \(\mathbf{Q}=\mathbf{k}_{\text{in}}-\mathbf{k}_{\text{s}}\) is the scattering vector with \(\mathbf{k}_{\text{in}}\) being the mean wave vector of the incoming X-ray beam, \(\Gamma_{0}^{\text{NRXS}}=I_{0}\tau_{p}^{2}\sum_{p_{\text{s}}}\big{|}(\mathbf{ \epsilon}_{\text{in}}\cdot\mathbf{\epsilon}_{\text{s}}^{*})\big{|}^{2}/(4\ln 2\,\omega_{ \text{in}}^{2}c^{3})\) and \(\Omega_{F}=\omega_{\text{in}}-\omega_{\mathbf{k}_{\text{s}}}-E_{F}+\langle E\rangle\) with \(\langle E\rangle\) being the mean energy of the coherent superposition in Equation (1). Let us now compare the resulting relation for the time-resolved DSP in Equation (18) with the relation in Equation (13). Equation (13) that can be represented as \(dP^{\text{NRXS}}(\mathbf{Q})/d\Omega\propto\int d^{3}r_{1}\int d^{3}r_{2}e^{i \mathbf{Q}\cdot(\mathbf{r}_{1}-\mathbf{r}_{2})}\langle\Psi(t_{p})|\hat{\psi}^{ \dagger}(\mathbf{r}_{1})\hat{\psi}(\mathbf{r}_{1})|\Psi(t_{p})\rangle\langle \Psi(t_{p})|\hat{\psi}^{\dagger}(\mathbf{r}_{2})\hat{\psi}(\mathbf{r}_{2})|\Psi( t_{p})\rangle\) assumes that elastic scattering has taken place and the final state coincides with the initial electronic state \(|\Psi(t_{p})\rangle\). Equation (18) contains a sum over final states \(F\), which are eigenstates of the Hamiltonian of the electronic system \(\hat{H}_{\text{m}}\), but not a superposition of electronic states. Contributions due to scattering to final states \(F\) are summed incoherently and are weighted by the function \(e^{-\Omega_{F}^{2}\tau_{p}^{2}/(4\ln 2)}\) originating from the probe-pulse spectral density. Even if one could spectroscopically resolve scattering exclusively to final states, which are involved in the wave packet \(\Psi(t_{p})\), a contribution due to scattering to each eigenstate would have to be added incoherently, and it would not be possible to substitute \(\Psi(t_{p})\) for \(\Phi_{F}\) in Equation (19). As a result, the state \(\Psi(t_{p})\) enters Equation (18) twice instead of four times as in Equation (13). Thus, the assumption that time-resolved X-ray scattering is given the Fourier transform of the time-dependent electron density is incorrect and even fails to reflect the correct time-dependence of scattering patterns on the evolution of the electronic system. In order to illustrate the difference between Equations (13) and (18), [PERSON] et al. considered an electronic wave packet prepared by a pump pulse as a coherent superposition \(\Psi(t)=1/\sqrt{2}e^{-iE_{M}(t-t_{0})}\left|3d\right\rangle+1/\sqrt{2}e^{-iE_{4y }(t-t_{0})}\left|4f\right\rangle\) of the \(3d\) and \(4f\) eigenstates of atomic hydrogen with the projection of orbital angular momentum equal to zero, the polarization direction along the \(z\)-axis and the corresponding energies \(E_{3d}\) and \(E_{4f}\). The probe X-ray pulse was assumed to have duration of 1 fs, 4 keV photon energy and propagate along the \(y\) direction. They calculated scattering patterns from the electronic wave packet with Equation (13) and with Equation (18) and compared them in Figure 1.  
Figure 1a shows scattering patterns calculated with Equation (18) within the QED framework. Their structure is much more diverse than the structure of the semiclassical scattering patterns calculated with Equation (13) shown in Figure 1d due to inelastic processes that have to be taken into account in Equation (18).