# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 6 QED Description of Time-Resolved Photoelectron Probability
Analogously to the nonresonant and the resonant scattering patterns presented in Sections 4 and 5, correspondingly, the angular distributions do not coincide at times \(t_{p}=T/4=2.55\) fs and \(t_{p}=3T/4=5.1\) fs, when the electron densities are equal, but the phases of the electronic wave packet are different (see Figure 6). It was suggested to analyze the angular distributions at a fixed photoelectron energy \(\varepsilon_{e}\) by performing the Fourier transform according to the following equation  
\[|\Psi_{\{n\}}^{\text{PE}},t_{f}\rangle=-\,i\int_{t_{0}}^{t_{f}}dt\,e^{i\hat{H}_ {0}t}\hat{H}_{\text{int}}^{(1)}\,e^{-i\hat{H}_{0}t}|\{n\}\rangle\left|\sum_{I}C _{I}\Phi_{I}\right\rangle \tag{39}\]1  
\[|\Psi_{\{n\}}^{\text{PE}},t_{f}\rangle=-\,i\int_{t_{0}}^{t_{f}}dt\,e^{i\hat{H}_ {0}t}\hat{H}_{\text{int}}^{(1)}\,e^{-i\hat{H}_{0}t}|\{n\}\rangle\left|\sum_{I}C _{I}\Phi_{I}\right\rangle \tag{39}\]2 (**a**) Isosurfaces of the amplitudes of the HOMO and HOMO-1 orbitals of indole calculated with the software package the MOLCAS package [67]; (**b**) Schematic representation of the electronic state of indole after the pump pulse arriving at time \(t_{0}\), which is the superposition of two states with an electron hole in HOMO and HOMO-1 orbitals; (**c**) Electron hole density at time \(0\), \(T/4\), \(T/2\), and \(3T/4\). The orbitals and electron hole densities are visualized using the VESTA software [66]. Figure is reproduced with permission from Reference [32]. Copyright American Physical Society, 2016. \[|\Psi_{\{n\}}^{\text{PE}},t_{f}\rangle=-\,i\int_{t_{0}}^{t_{f}}dt\,e^{i\hat{H}_ {0}t}\hat{H}_{\text{int}}^{(1)}\,e^{-i\hat{H}_{0}t}|\{n\}\rangle\left|\sum_{I}C _{I}\Phi_{I}\right\rangle \tag{39}\]3 Angle-resolved photoelectron spectra generated by the probe pulse arriving at time \(t_{p}\). Each \(q_{x}-q_{z}\) plane at the corresponding photoelectron energy \(\varepsilon_{e}\) is a projection of the semisphere with radius \(|\mathbf{q}|=\sqrt{2}\varepsilon_{e}\) at \(q_{y}>0\), where the color of the \(\mathbf{q}\) point on the semisphere corresponds to the probability of detecting an electron with momentum \(\mathbf{q}\). Figure is reproduced with permission from Reference [32]. Copyright American Physical Society, 2016.  
where \(\delta(|\mathbf{Q_{el}}|-Q_{0})\) is the Dirac delta function. This equation describes the Fourier transform of the spherical surfaces in \(\mathbf{Q_{el}}\) space of a fixed radius \(Q_{0}=\sqrt{2\varepsilon_{e}}\), the projections of which are shown in \[|\Psi_{\{n\}}^{\text{PE}},t_{f}\rangle=-\,i\int_{t_{0}}^{t_{f}}dt\,e^{i\hat{H}_ {0}t}\hat{H}_{\text{int}}^{(1)}\,e^{-i\hat{H}_{0}t}|\{n\}\rangle\left|\sum_{I}C _{I}\Phi_{I}\right\rangle \tag{39}\]4 It was found that the Fourier transform is connected to the time-dependent Dyson orbitals \(\phi_{F}^{D}(\mathbf{r},t_{p})=\langle\Phi_{F}^{\mathrm{N_{el}-1}}|\hat{\psi}( \mathbf{r})|\Psi(t_{p})\rangle\) via  
\[|\Psi_{\{n\}}^{\text{PE}},t_{f}\rangle=-\,i\int_{t_{0}}^{t_{f}}dt\,e^{i\hat{H}_ {0}t}\hat{H}_{\text{int}}^{(1)}\,e^{-i\hat{H}_{0}t}|\{n\}\rangle\left|\sum_{I}C _{I}\Phi_{I}\right\rangle \tag{39}\]5  
where \(A[\phi_{F}^{D}(t_{p})]=\int d^{3}r^{\prime}\phi_{F}^{D}(\mathbf{r}^{\prime}- \mathbf{r},t_{p})\phi_{F}^{D}(\mathbf{r}^{\prime},t_{p})\) is the autocorrelation function of \(\phi_{F}^{D}(\mathbf{r},t_{p})\), \((A[\phi_{F}^{D}(t_{p})]*s)(\mathbf{r})=\int d^{3}r^{\prime}A(\mathbf{r}^{ \prime})s(\mathbf{r}-\mathbf{r}^{\prime})\) denotes the convolution of the autocorrelation function \(A[\phi_{F}^{D}(t_{p})]\) with the function \(s(\mathbf{r})=\mathrm{sinc}(Q_{0}|\mathbf{r}|)\). Thus, the Fourier transform of the time- and angle-resolved photoelectron probability at a fixed photoelectron energy \(\varepsilon_{e}\) is determined by a linear combination of autocorrelation functions of Dyson orbitals \(A[\phi_{F}^{D}(t_{p})]\) with the coefficients given by the exponential factor \(e^{-(\Gamma_{F}-\varepsilon_{e})^{2}\tau_{p}^{2}/(4\ln 2)}\).