# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 6 QED Description of Time-Resolved Photoelectron Probability
Applying the dipole and the frozen-density approximation, assuming a coherent XUV pulse with the amplitude of the electric field defined in Equation (17), and applying the plane-wave approximation to the photoelectron wave function, the photoelectron probability can be represented as  
\[P^{\text{PE}}(\mathbf{Q}_{\text{el}})= P_{0}^{\text{PE}}|\epsilon_{\text{in}}\cdot\mathbf{Q}_{\text{el}}|^{2} \sum_{F,\sigma}\int d^{3}r_{1}\int d^{3}r_{2}e^{i\mathbf{Q}_{\text{el}}\cdot( \mathbf{r}_{2}-\mathbf{r}_{1})}\langle\Psi(t_{p})|\hat{G}_{F}^{\text{PE}}( \mathbf{r}_{1},\mathbf{r}_{2})|\Psi(t_{p})\rangle e^{-(\Omega_{F}^{\text{el}}- \epsilon_{e})^{2}\pi_{p}^{2}/(4\ln 2)} \tag{42}\]  
with the operator  
\[\hat{G}_{F}^{\text{PE}}(\mathbf{r}_{1},\mathbf{r}_{2})=\hat{\psi}^{\dagger}( \mathbf{r}_{2})|\Phi_{F}^{N_{\text{el}}}\rangle\langle\Phi_{F}^{N_{\text{el}} }|\hat{\psi}^{\dagger}(\mathbf{r}_{1}), \tag{43}\]  
\(P_{0}^{\text{PE}}=I_{0}\tau_{p}^{2}/(8\pi\ln 2\omega_{\text{in}}^{2}c)\) and \(\Omega_{F}^{\text{el}}=\omega_{\text{in}}-E_{F}^{N_{\text{el}}-1}+\langle E\rangle\). The general expression Equation (40) for the time- and angle-resolved photoelectron probability has much in common with the general expressions for the DSPs of time-resolved NRXS [Equation (15)] and time-resolved RXS [Equation (28)], as well as the expression in Equation (42) is analogous to the corresponding expressions in Equations (18) and (30). The general Equations (15), (28) and (40) depend on the first-order radiation field correlation function \(G^{(1)}(\mathbf{r}_{1},t_{1},\mathbf{r}_{2},t_{2})\) multiplied by a function dependent on \(\Psi(t_{1})\) and its conjugate dependent on \(\Psi(t_{2})\). Equations (18), (30) and (42) describe analogous dependence of the corresponding time-resolved signals on the evolution of the electronic state \(\Psi(t_{p})\) and the spectral density of an ultrashort probe pulse. All these relations involve an incoherent sum over final states. The only prominent difference of the relations for the photoelectron probability from the relations for X-ray scattering is that the former comprise the photoelectron wave function \(\phi_{\sigma e}(\mathbf{Q}_{\text{el}},\mathbf{r})\) instead of the wave function of a scattered photon \(e^{i\mathbf{k}_{\text{e}}\cdot\mathbf{r}}\). In addition, analogously to time-resolved NRXS and time-resolved RXS, there are two contributions to the photoelectron probability in Equation (42), which can be represented as  
\[P^{\text{PE}}(\mathbf{Q}_{\text{el}})= |\mathbf{\epsilon}_{\text{in}}\cdot\mathbf{Q}_{\text{el}}|^{2}\int d^{3 }r_{1}\int d^{3}r_{2}\cos[\mathbf{Q}_{\text{el}}\cdot(\mathbf{r}_{2}-\mathbf{ r}_{1})]\mathcal{R}^{\text{PE}}(t_{p},\mathbf{\epsilon}_{e},\mathbf{r}_{1},\mathbf{r}_{2}) \tag{44}\] \[-|\mathbf{\epsilon}_{\text{in}}\cdot\mathbf{Q}_{\text{el}}|^{2}\int d ^{3}r_{1}\int d^{3}r_{2}\sin[\mathbf{Q}_{\text{el}}\cdot(\mathbf{r}_{2}- \mathbf{r}_{1})]\mathcal{J}^{\text{PE}}(t_{p},\mathbf{\epsilon}_{e},\mathbf{r}_{1},\mathbf{r}_{2}).\]  
The first contribution is centrosymmetric with respect to \(\mathbf{Q}_{\text{el}}\) and is determined by the function  
\[\mathcal{R}^{\text{PE}}(t_{p},\mathbf{\epsilon}_{e},\mathbf{r}_{1},\mathbf{r}_{2} )=P_{0}^{\text{PE}}\operatorname{Re}\left[\langle\Psi(t_{p})|\sum_{F}\hat{G}_ {F}^{\text{PE}}(\mathbf{r}_{1},\mathbf{r}_{2})|\Psi(t_{p})\rangle\right]e^{- (\Omega_{F}^{\text{el}}-\epsilon_{e})^{2}\tau_{p}^{2}/(4\ln 2)}, \tag{45}\]  
which is connected to the real parts of electron density matrix elements. The second contribution is an odd function with respect to \(\mathbf{Q}_{\text{el}}\) and is determined by  
\[|\Psi_{\{n\}}^{\text{PE}},t_{f}\rangle=-\,i\int_{t_{0}}^{t_{f}}dt\,e^{i\hat{H}_ {0}t}\hat{H}_{\text{int}}^{(1)}\,e^{-i\hat{H}_{0}t}|\{n\}\rangle\left|\sum_{I}C _{I}\Phi_{I}\right\rangle \tag{39}\]0  
which is dependent on the imaginary parts of electron density matrix elements.