# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 5 QED Description of Time-Resolved Resonant X-ray Scattering
Applying the dipole approximation for each absorbing atom, and assuming a coherent resonant X-ray pulse with the amplitude of the electric field defined in Equation (17) and duration that is much shorter than the characteristic time scale of the changes in the electron density, the DSP of RXS can be represented as  
\[\frac{dP^{\text{RXS}}(\mathbf{Q})}{d\Omega}=\text{P}_{0}^{\text{RXS}}\sum_{C_ {1},C_{2}}e^{i\mathbf{Q}\cdot(\mathbf{R}_{C_{1}}-\mathbf{R}_{C_{2}})}\sum_{F} \langle\Psi(t_{p})|\hat{\mathcal{C}}_{F,C_{1},C_{2}}^{\text{RXS}}|\Psi(t_{p}) \rangle e^{-\Omega_{F}^{2}\tau_{p}^{2}/(4\ln 2)}, \tag{30}\]  
with the operator  
\[\hat{\mathcal{C}}_{F,C_{1},C_{2}}^{\text{RXS}} =\sum_{\mathbf{s}=1}^{2}\int_{0}^{\infty}d\omega_{\mathbf{k}_{s} }\omega_{\mathbf{k}_{s}}W(\omega_{\mathbf{k}_{s}})\hat{\mathcal{S}}_{C_{2}}^{ \dagger}|\[PERSON]_{F}\rangle\langle\[PERSON]_{F}|\hat{S}_{C_{1}}, \tag{31}\] \[\langle\[PERSON]_{F}|\hat{S}_{C} =\sum_{J_{C}}\frac{\Delta\omega_{J_{C}F}\langle\[PERSON]_{F}| \boldsymbol{\epsilon}_{\mathbf{s}}^{*}\cdot\hat{\mathbf{r}}|\[PERSON]_{J_{C}} \rangle\langle\[PERSON]_{J_{C}}|\boldsymbol{\epsilon}_{\mathbf{in}}\cdot\hat{ \mathbf{r}}}{\omega_{\mathbf{k}_{s}}-E_{J_{C}}+E_{F}+i\Gamma_{J_{C}}/2},\]  
where \(\text{P}_{0}^{\text{RXS}}=\tau_{p}^{2}l_{0}/(4\ln 2c^{4})\) and \(\hat{\mathbf{r}}=\int d^{3}r\hat{\psi}^{\dagger}(\mathbf{r})\,\mathbf{r}\, \psi(\mathbf{r})\). The expression in Equation (30) is also very similar to the expression for the DSP of NRXS in Equation (18); but, instead of the space integrals in Equation (18), it involves the sum over scattering atoms \(C_{1}\) and \(C_{2}\), and their positions \(\mathbf{R}_{C_{1}}\) and \(\mathbf{R}_{C_{2}}\) due to the dipole approximation. In particular, it has a similar temporal dependence on the electronic state at the time of measurement \(\Psi(t_{p})\). Let us compare Equation (30) with the expression for the DSP from a stationary system in Equation (26), which can be represented as \(dP^{\text{stRXS}}/d\Omega\propto\sum_{C_{1},C_{2}}e^{i\mathbf{Q}\cdot( \mathbf{R}_{C_{1}}-\mathbf{R}_{C_{2}})}f_{C_{1}}f_{C_{2}}^{*}\). The DSP according to Equation (30) could be represented as \(dP^{\text{RXS}}/d\Omega\propto\sum_{C_{1},C_{2}}e^{i\mathbf{Q}\cdot(\mathbf{R} _{C_{1}}-\mathbf{R}_{C_{2}})}\sum_{F}\widetilde{f}_{C_{1}F}(t_{p})\widetilde{f }_{C_{2}F}^{*}(t_{p})\), where the scattering amplitudes \(\widetilde{f}_{CF}(t_{p})\) are proportional to \(\langle\[PERSON]_{F}|\boldsymbol{\epsilon}_{\mathbf{s}}^{*}\cdot\hat{\mathbf{r}}| \[PERSON]_{J_{C}}\rangle\langle\[PERSON]_{J_{C}}|\boldsymbol{\epsilon}_{\mathbf{in}} \cdot\hat{\mathbf{r}}|\Psi(t_{p})\rangle\). Obviously, it is not possible to obtain the correct expression for the time-resolved DSP by simply substituting the scattering amplitudes \(f_{C}\propto\langle\[PERSON]_{I}|\boldsymbol{\epsilon}_{\mathbf{s}}^{*}\cdot\hat{ \mathbf{r}}|\Phi_{J_{C}}\rangle\langle\Phi_{J_{C}}|\boldsymbol{\epsilon}_{ \mathbf{in}}\cdot\hat{\mathbf{r}}|\Phi_{I}\rangle\) for their time-dependent analogues \(f_{C}(t_{p})\propto\langle\Psi(t_{p})|\boldsymbol{\epsilon}_{\mathbf{s}}^{*} \cdot\hat{\mathbf{r}}|\[PERSON]_{J_{C}}\rangle\langle\Phi_{J_{C}}|\boldsymbol{ \epsilon}_{\mathbf{in}}\cdot\hat{\mathbf{r}}|\Psi(t_{p})\rangle\) in Equation (26) as  
\[\frac{dP^{\text{RXS}}}{d\Omega}\propto\left|\sum_{C}e^{i\mathbf{Q}\cdot \mathbf{R}}f_{C}(t_{p})\right|^{2}. \tag{32}\]  
This assumption suggests that the absorption process first brings the coherently evolving electronic system into an intermediate state \(J_{C}\), and then the emission process brings the system from the intermediate state \(J_{C}\) into the same coherent superposition \(\Psi(t_{p})\) as before the absorption. Equation (30) indeed takes into account that the absorption process bringing the system into the intermediate state \(J_{C}\) destroys electron dynamics, and final states after the emission process are eigenstates of the Hamiltonian \(\hat{H}_{\text{m}}\).