# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 6 QED Description of Time-Resolved Photoelectron Probability
The time- and angle-resolved photoelectron probability is a sum of an even function with respect to \(\mathbf{Q}_{\text{el}}\) determined by \(\mathcal{R}^{\text{PE}}(t_{p},\mathbf{r}_{1},\mathbf{r}_{2})\) and an odd function with respect to \(\mathbf{Q}_{\text{el}}\) determined by \(\mathcal{J}^{\text{PE}}(t_{p},\mathbf{r}_{1},\mathbf{r}_{2})\) not only in the case of the photoelectron wave function being a plane wave, but in the more general case of \(\phi_{ee}(\mathbf{Q}_{\text{el}},\mathbf{r})\) being a Hermitian function with respect to both \(\mathbf{Q}_{\text{el}}\) and \(\mathbf{r}\). By an analogy to the concept of a chemical shift in stationary photoelectron spectroscopy, one may assume that electron dynamics can be measured by means of time-dependent chemical shifts, i.e., that photoelectron peaks would shift following the time-dependent electron density in a time-resolved measurement. However, according to Equation (42), time-dependent photoelectron spectra at each emission angle consist of a series of photoelectron peaks centered at energies \(\Omega_{F}\) corresponding to a transition to a final state \(F\), whereby the position of the peaks are time independent. Therefore, quite counter-intuitively, electron dynamics cannot be followed by means of time-dependent chemical shifts. The quantity that depends on the electronic state at the time of measurement is the strength of the photoelectron peaks. Equation (42) was applied for the calculation of time- and angle-resolved photoelectron spectra of an indole molecule with coherent electron dynamics in valence orbitals. It was assumed that a broadband photoionizing pump pulse launched coherent electron dynamics in indole by creating an electron hole in a superposition of the HOMO (highest occupied molecular orbital) and HOMO-1 orbitals at time \(t_{0}\) (see Figure 6). Then, the electronic state of the indole molecular cation after the interaction with the pump pulse evolves in time as \(|\Psi(t)\rangle=C_{1}e^{-iE_{1}(t-t_{0})}|\Phi_{H}^{\text{ion}}\rangle+C_{2}e ^{-iE_{2}(t-t_{0})}|\Phi_{H-1}^{\text{ion}}\rangle\), where \(|\Phi_{H}^{\text{ion}}\rangle\) is an electronic state with an electron hole in the HOMO, \(|\Phi_{H-1}^{\text{ion}}\rangle\) is an electronic state with an electron hole in the HOMO-1, and \(C_{1}\) and \(C_{2}\) are time-independent complex coefficients, which are determined by the pump process. After the pump process, the electron density starts oscillating with the period \(T=2\pi/(E_{2}-E_{1})\approx 10.2\) fs. Since there are two electronic states involved in the dynamics, there are two time points during the density oscillation period, at which the time-dependent electron densities are identical. The zero time point was adjusted such that the time-dependent electron densities coincide at times \(t=T/4\) and \(t=3T/4\). It was suggested to probe the electron dynamics by an XUV pulse, which creates a second electron hole in the indole molecule by a single-photon ionization at time \(t_{p}\), and analyze angular distributions of photoemission probabilities at a fixed photoelectron energy depending on the time delay \(t_{p}-t_{0}\). A \(y\)-polarized probe pulse of 1 fs duration, 100 eV photon energy, and \(10^{12}\) W/cm\({}^{2}\) intensity was considered. An application of the XUV probe pulse provided photoelectron angular distributions at photoelectron energies of \(\sim 80\) eV-90 eV, which allowed for Angstrom spatial resolution. In addition, an analysis of such distributions was simplified, since the orthogonalization correction to the plane-wave approximation was suppressed at these energies in the case of the indole molecule. Figure 7 shows calculated angular distributions of photoemission probabilities at different photoelectron energies depending on time \(t_{p}\). The distributions at energies \(\varepsilon_{e}=85\) eV and \(86\) eV show quite strong dependence on the electron dynamics.