# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 3 Interaction with a Probe Pulse within the QED Framework  
When the electronic system is probed by an electromagnetic pulse, the total Hamiltonian of the whole system, matter and light, can be written as [44]  
\[\hat{H}= \hat{H}_{0}+\hat{H}_{\rm int}\] \[= \hat{H}_{\rm m}+\sum_{\textbf{k},p}\omega_{\textbf{k}}\hat{a}_{ \textbf{k},p}^{\dagger}\hat{a}_{\textbf{k},p}+\hat{H}_{\rm int}, \tag{4}\]  
where \(\hat{a}_{\textbf{k},p}^{\dagger}\) and \(\hat{a}_{\textbf{k},p}\) are creation and annihilation operators of a photon in the \(\textbf{k}\), \(p\) mode of the radiation field with energy \(\omega_{\textbf{k}}=|\textbf{k}|c\), where \(c\) is the speed of light. \(\hat{H}_{\rm int}\) is the minimal coupling interaction Hamiltonian between the matter and the electromagnetic field in Coulomb gauge  
\[\hat{H}_{\rm int} =\hat{H}_{\rm int}^{(1)}+\hat{H}_{\rm int}^{(2)}\] \[=\frac{1}{c}\int d^{3}r\hat{\psi}^{\dagger}(\textbf{r})\left( \hat{\textbf{A}}(\textbf{r})\cdot\textbf{p}\right)\hat{\psi}(\textbf{r})+\frac {1}{2c^{2}}\int d^{3}r\hat{\psi}^{\dagger}(\textbf{r})\hat{\textbf{A}}^{2}( \textbf{r})\hat{\psi}(\textbf{r}), \tag{5}\]  
where \(\hat{\textbf{A}}\) is the vector potential operator of the electromagnetic field, \(\textbf{p}\) is the canonical momentum of an electron, \(\hat{\psi}^{\dagger}\) and \(\hat{\psi}\) are electron creation and annihilation field operators. Depending on the character and conditions of the interaction of the electronic system with the electromagnetic pulse, one of two terms in Equation (5) becomes relevant for the interaction process and the other one turns to be insignificant. TRARPES, which is governed by absorption, is determined by Hamiltonian \(\hat{H}_{\rm int}^{(1)}\). Scattering driven by the second term in Equation (5) is dominant in the case of high-energy X-raysand, thus, \(\hat{H}_{\rm int}^{(2)}\) determines NRXS. However, the scattering cross section given by \(\hat{H}_{\rm int}^{(1)}\) becomes much larger than that given by \(\hat{H}_{\rm int}^{(2)}\) in the case of a resonant X-ray pulse and, thus, \(\hat{H}_{\rm int}^{(1)}\) determines RXS. In the case of X-ray scattering, the signal is determined by the probability \(P^{\rm XS}({\bf k}_{\rm s})\) to observe a scattered photon with momentum \({\bf k}_{\rm s}\), which differs from the incoming photon momenta. Since we consider energy-unresolved scattering patterns, the signal is given by the energy average of \(P^{\rm XS}({\bf k}_{\rm s})\), i.e., by the differential scattering probability (DSP),  
\[\frac{dP^{\rm XS}}{d\Omega}=\frac{V}{(2\pi c)^{3}}\int_{0}^{\infty}d\omega_{{ \bf k}_{\rm s}}\omega_{{\bf k}_{\rm s}}^{2}\,P^{\rm XS}({\bf k}_{\rm s}), \tag{6}\]  
where \(V\) is the quantization volume. Within the density-matrix formalism, the expectation value of some observable \(\hat{O}\) in a state, represented by a density matrix \(\hat{\rho}\), is given by the relation \(\langle\hat{O}\rangle={\rm Tr}[\hat{\rho}\hat{O}]\)[45]. Consequently, the probability of X-ray scattering is connected to the operator \(\hat{O}_{{\bf k}_{\rm s}}\), which describes the observation of a photon in the scattering mode \({\bf k}_{\rm s}\) independently on its polarization,  
\[P^{\rm XS}({\bf k}_{s})={\rm Tr}\left[\hat{\rho}_{f}(t_{f})\hat{O}_{{\bf k}_{ \rm s}}\right], \tag{7}\]  
where \(\hat{\rho}_{f}(t_{f})\) is the total density matrix of the electronic system and the electromagnetic field at time \(t_{f}\) after the action of the probe pulse [44; 45; 24]. The operator \(\hat{O}_{{\bf k}_{\rm s}}\) is given by  
\[\hat{O}_{{\bf k}_{\rm s}}=\sum_{p_{\rm s}}W(\omega_{{\bf k}_{\rm s}})\hat{a}_{{ \bf k}_{\rm s},p_{\rm s}}^{\dagger}\hat{a}_{{\bf k}_{\rm s},p_{\rm s}}. \tag{8}\]  
Here, the sum is over two possible polarizations of a scattered photon. It is assumed that a photon detector has some acceptance range represented by the function \(W(\omega_{{\bf k}_{\rm s}})\).