# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 6 QED Description of Time-Resolved Photoelectron Probability  
TRARPES, i.e., time- and energy-resolved molecular-frame photoelectron angular distributions, has been proposed for imaging coherent electron dynamics in molecules [29; 30; 31; 32]. An advantage of this technique over X-ray scattering is that it allows achieving Angstrom spatial resolution with light pulses of a much lower photon energy. An analysis of time- and angle-resolved photoelectron distributions obtained by ultrashort XUV probe pulses inducing single-photon ionization of a coherently evolving electron system within the QED framework has been performed in Reference [32]. This study provided a correct way to take into account the consequence of the broad probe-pulse bandwidth on a signal, which has not been taken into account in earlier studies of time-resolved photoelectron probability [29], but is critical for a correct interpretation of time-resolved photoelectron spectra. This work is reviewed in this Section. Figure 5: (**a**) The imaginary part of the Fourier transform from **Q** space to real space of the scattering pattern in Figure 4b; (**b**) Solid violet line: time evolution of the amplitude of the circled peak in panel (**a**). Orange dashed line: the sum of the currents between pairs of atoms connected by the vector \((0,R_{\text{Br-Br}},0)\), where \(R_{\text{Br-Br}}\) is the interatomic distance between nearest-neighbour Br atoms in KBr. Figure is reproduced with permission from Reference [28]. Copyright American Physical Society, 2015. In order to describe the absorption process, the total density matrix must be evaluated within the first-order time-dependent perturbation theory using the Hamiltonian \(\hat{H}_{\text{int}}^{(1)}\). Thus, in order to obtain the photoelectron probability, the first-order wave function  
\[|\Psi_{\{n\}}^{\text{PE}},t_{f}\rangle=-\,i\int_{t_{0}}^{t_{f}}dt\,e^{i\hat{H}_ {0}t}\hat{H}_{\text{int}}^{(1)}\,e^{-i\hat{H}_{0}t}|\{n\}\rangle\left|\sum_{I}C _{I}\Phi_{I}\right\rangle \tag{39}\]  
must be substituted for \(|\Psi_{\{n\}},t_{f}\rangle\) in Equation (11). Thus, according to Equation (9), the photoelectron probability of a coherently evolving electronic system consisting of \(N_{\text{el}}\) electrons is  
\[P^{\text{PE}}(\mathbf{Q}_{\text{el}})=\frac{1}{\omega_{\text{in }}^{2}}\sum_{F,\sigma}\int_{t_{0}}^{t_{f}}dt_{1}\int_{t_{0}}^{t_{f}}dt_{2}\int d ^{3}r_{1}\int d^{3}r_{2}\,\phi_{\sigma e}^{\dagger}(\mathbf{Q}_{\text{el}}, \mathbf{r}_{1})\phi_{\sigma e}(\mathbf{Q}_{\text{el}},\mathbf{r}_{2}) \tag{40}\] \[\times\,G^{(1)}(\mathbf{r}_{1},t_{1},\mathbf{r}_{2},t_{2})M_{F}^ {\text{PE}}(t_{1},\mathbf{r}_{1})\big{[}M_{F}^{\text{PE}}(t_{2},\mathbf{r}_{2 })\big{]}^{\dagger}\]  
with the function  
\[M_{F}^{\text{PE}}(t,\mathbf{r})= e^{i(E_{F}^{N_{\text{el}}-1}+\epsilon_{e})t}\langle\Phi_{F}^{N_{ \text{el}}-1}|(\epsilon_{\text{in}}\cdot\mathbf{p})\hat{\psi}(\mathbf{r})| \Psi(t)\rangle. \tag{41}\]  
Here, \(\phi_{\sigma e}(\mathbf{Q}_{\text{el}},\mathbf{r})\) is the wave function of a photoelectron with spin \(\sigma\) and momentum \(\mathbf{Q}_{\text{el}}\), \(\epsilon_{e}=|\mathbf{Q}_{\text{el}}|^{2}/2\) is the photoelectron energy, \(|\Phi_{F}^{N_{\text{el}}-1}\rangle\) is a final state of the electronic system with \(N_{\text{el}}-1\) electrons and energy \(E_{F}^{N_{\text{el}}-1}\), which by assumption does not interact with the emitted photoelectron. Equation (40) does not include any further assumptions concerning probe-pulse parameters, such as coherence properties or duration.