# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 4 QED Description of Time-Resolved Nonresonant X-ray Scattering
The resulting density matrix is connected to the first-order wave function  
\[\left|\Psi_{\{n\}}^{\text{NRXS}},t_{f}\right\rangle=-\,i\int_{t_{0}}^{t_{f}} dt\,e^{i\hat{H}_{0}t}\,\hat{H}_{\text{int}}^{(2)}\,e^{-i\hat{H}_{0}t}|\{n\} \rangle\,\left|\sum_{I}C_{I}\Phi_{I}\right\rangle, \tag{14}\]  
which must be substituted for \(|\Psi_{\{n\}},t_{f}\rangle\) in Equation (11). Thus, according to Equations (6) and (7), the DSP of NRXS from an electronic system evolving coherently is\[\frac{dP^{\text{NRXS}}(\mathbf{k}_{\text{s}})}{d\Omega}= \frac{\sum_{p_{\text{s}}}\left|(\mathbf{\epsilon}_{\text{in}}\cdot\mathbf{ \epsilon}_{\text{s}}^{*})\right|^{2}}{4\pi^{2}c^{3}\omega_{\text{in}}^{2}}\int_ {0}^{\infty}d\omega_{\mathbf{k}_{\text{s}}}\omega_{\mathbf{k}_{\text{s}}}W( \omega_{\mathbf{k}_{\text{s}}})\sum_{F}\int_{t_{0}}^{+\infty}dt_{1}\int_{t_{0} }^{+\infty}dt_{2} \tag{15}\] \[\times\int d^{3}r_{1}\int d^{3}r_{2}G^{(1)}(\mathbf{r}_{1},t_{1}, \mathbf{r}_{2},t_{2})e^{-i\mathbf{k}_{\text{s}}\cdot(\mathbf{r}_{1}-\mathbf{r} _{2})}M_{F}^{\text{NRXS}}(\mathbf{r}_{1},t_{1})\big{[}M_{F}^{\text{NRXS}}( \mathbf{r}_{2},t_{2})\big{]}^{\dagger}.\]  
with the function  
\[M_{F}^{\text{NRXS}}(\mathbf{r},t)= e^{i(E_{F}+\omega_{\mathbf{k}_{\text{s}}})t}\langle\Phi_{F}| \hat{\psi}^{\dagger}(\mathbf{r})\hat{\psi}(\mathbf{r})|\mathbf{\Psi}(t)\rangle. \tag{16}\]  
Here, \(G^{(1)}(\mathbf{r}_{1},t_{1},\mathbf{r}_{2},t_{2})=\text{Tr}[\hat{\beta}_{0}^ {3}\hat{\mathbf{E}}^{-}\hat{\mathbf{E}}^{+}]\) is the first-order radiation field correlation function, where \(\hat{\mathbf{E}}^{+}=(\hat{\mathbf{E}}^{-})^{\dagger}=\sum_{\mathbf{k}}\sqrt {2\pi\omega_{\mathbf{k}}/V}\hat{a}_{\mathbf{k},p}\epsilon_{p}e^{-i\omega_{ \mathbf{k}}t+i\mathbf{k}\cdot\mathbf{r}}\) with \(\epsilon_{p}\) being the unit vector corresponding to the polarization \(p\)[44, 49]. \(\omega_{\text{in}}\) is the photon energy and \(\mathbf{\epsilon}_{\text{in}}\) is the mean polarization vector of the incoming X-ray beam. The sum is over all possible final states \(F\) with the corresponding energies \(E_{F}\). The role of the function \(W(\omega_{\mathbf{k}_{\text{s}}})\) is to represent the acceptance range of the photon detector, which can, for example, resolve only certain scattered energies \(\omega_{\mathbf{k}_{\text{s}}}\). This expression considers a general case, which is independent on a probe-pulse duration and coherence properties. Equation (15) can be simplified under certain conditions. [PERSON] et al. considered a perfectly coherent Gaussian-shaped probe pulse with the amplitude of the electric field  
\[\mathcal{E}(\mathbf{r}_{0},t)=\sqrt{(8\pi/c)I_{0}(\mathbf{r}_{0})}e^{-2\ln 2 \left(\frac{t-t_{p}}{\tau_{p}}\right)^{2}}, \tag{17}\]  
where \(\mathbf{r}_{0}\) is the position of the object, \(t_{p}\) is the time of the measurement, \(\tau_{p}\) is the pulse duration (FWHM of the pulse intensity) and \(I_{0}(\mathbf{r}_{0})=cE^{2}(\mathbf{r}_{0},t=0)/(8\pi)\) is the peak intensity. It was further assumed that the probe-pulse duration is much shorter than the characteristic time scale of changes in the electron density. Hereafter, we will refer to this assumption as the frozen-density approximation. As we discuss in Section 7, this is a necessary condition for an appropriate time-resolved measurement.