# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 5 QED Description of Time-Resolved Resonant X-ray Scattering
This again demonstrates that the notion of 'elastic scattering' for an interaction of a light pulse with a nonstationary electronic system is unclear. Since the relation in Equation (26) relays on the assumption that elastic scattering provides a dominating contribution to a scattering pattern, it must be reconsidered for time-resolved RXS. The interaction of an ultrashort resonant X-ray pulse with an electronic system evolving coherently has been described in Reference [27] as follows. Since RXS is a two-step process governed by the Hamiltonian \(\hat{H}_{\text{int}}^{(1)}\) in Equation (5), the second-order wave function  
\[|\Psi_{\{n\}}^{\text{RXS}},t_{f}\rangle= -\int_{t_{0}}^{t_{f}}dt^{\prime}\,e^{i\hat{H}_{0}t^{\prime}}\hat {H}_{\text{int}}^{(1)}\,e^{-i\hat{H}_{0}t^{\prime}} \tag{27}\] \[\times\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\,e^{i\hat{H}_{0} t^{\prime\prime}}\hat{H}_{\text{int}}^{(1)}\,e^{-i\hat{H}_{0}t^{\prime\prime}} |\{n\}\rangle\,\left|\sum_{I}C_{I}\Phi_{I}\right\rangle,\]  
must be substituted for \(|\Psi_{\{n\}},t_{f}\rangle\) in Equation (11). Thus, according to Equations (6) and (7), the DSP of RXS from an electronic system evolving coherently is  
\[\frac{dP^{\text{RXS}}(\mathbf{Q})}{d\Omega}= \frac{1}{4\pi^{2}c^{3}\omega_{\text{in}}^{2}}\int_{0}^{\infty}d \omega_{\mathbf{k}_{s}}\omega_{\mathbf{k}_{s}}W(\omega_{\mathbf{k}_{s}})\sum _{F,S}\int_{t_{0}}^{+\infty}dt_{1}^{\prime}\int_{t_{0}}^{+\infty}dt_{2}^{\prime }\int_{t_{0}}^{t_{1}^{\prime}}dt_{1}^{\prime\prime}\int_{t_{0}}^{t_{2}^{ \prime}}dt_{2}^{\prime\prime} \tag{28}\] \[\times\int d^{3}r_{1}\int d^{3}r_{2}G^{(1)}(\mathbf{r}_{1},\mathbf{ r}_{1}^{\prime\prime},\mathbf{r}_{2},t_{2}^{\prime\prime})e^{-i\mathbf{k}_{s} \cdot(\mathbf{r}_{1}-\mathbf{r}_{2})}M_{F}^{\text{RXS}}(t_{1}^{\prime},t_{1}^{ \prime\prime},\mathbf{r}_{1})\big{[}M_{F}^{\text{RXS}}(t_{2}^{\prime},t_{2}^{ \prime\prime},\mathbf{r}_{2})\big{]}^{\dagger}\]with the function  
\[M_{F}^{\text{RXS}}(t^{\prime},t^{\prime\prime},\mathbf{r})=e^{i(E_{F}-\omega_{ \mathbf{k}_{s}})t^{\prime}}\sum_{J_{C}}e^{i(E_{J_{C}}-i\Gamma_{J_{C}}/2)(t^{ \prime\prime}-t^{\prime})}\langle\Phi_{F}|\hat{T}_{\mathbf{s}}^{\dagger}( \mathbf{r})|\Phi_{J_{C}}\rangle\langle\Phi_{J_{C}}|\hat{T}_{\text{in}}(\mathbf{ r})|\Psi(t^{\prime\prime})\rangle, \tag{29}\]  
where \(\hat{T}_{\mathbf{s}(\text{in})}(\mathbf{r})=\hat{\psi}^{\dagger}(\mathbf{r})( \boldsymbol{\epsilon}_{\mathbf{s}(\text{in})}\cdot\mathbf{p})\hat{\psi}( \mathbf{r})\) and \(\Gamma_{J_{C}}\) is the decay width of the intermediate state \(J_{C}\). This expression is general for a resonant X-ray pulse with arbitrary coherence properties, and assumes neither the frozen-density approximation nor the dipole approximation. The relation in Equation (28) has much in common with the general expression for time-resolved NRXS from a coherently evolving electronic system in Equation (15). Analogously to Equation (15), the DSP for time-resolved RXS involves the sum over final states \(F\) with corresponding transition amplitudes summed incoherently and also depends on the first-order correlation function \(G^{(1)}(\mathbf{r}_{1},t_{1}^{\prime\prime},\mathbf{r}_{2},t_{2}^{\prime \prime})\).