Imaging electron dynamics with time- and angle-resolved photoelectron spectroscopy

We theoretically study how time- and angle-resolved photoemission spectroscopy can be applied for imaging coherent electron dynamics in molecules. We consider a process in which a pump pulse triggers coherent electronic dynamics in a molecule by creating a valence electron hole. An ultrashort extreme ultraviolet probe pulse creates a second electron hole in the molecule. Information about the electron dynamics is accessed by analyzing angular distributions of photoemission probabilities at a fixed photoelectron energy. We demonstrate that a rigorous theoretical analysis, which takes into account the indistinguishability of transitions induced by the ultrashort, broadband probe pulse and electron hole correlation effects, is necessary for the interpretation of time- and angle-resolved photoelectron spectra. We show how a Fourier analysis of time- and angle-resolved photoelectron spectra from a molecule can be applied to follow its electron dynamics by considering photoelectron distributions from an indole molecular cation with coherent electron dynamics.

Figure 1

(a) Isosurfaces of the amplitudes of the HOMO and HOMO-1 orbitals of indole calculated with the software package molcas [34]. (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 $3T/4$, where $T=2\pi /(E_2-E_1)$. Electron hole densities coincide at $T/4$ and $3T/4$. The orbitals and electron hole densities are visualized using the vesta software [37].

Figure 2

(a) Angle-averaged photoelectron spectra at different probe-pulse arrival times. (b) Energies of possible final dicationic states of indole.

Figure 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 $\left|\mathbf{q}\right|=\sqrt{2{\varepsilon }_e}$ 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 4

The differences between the angle-resolved probability distributions at times $t_p=0$ and $5.1$ fs, $P\left(0\text{fs}\right)-P\left(5.1\text{fs}\right)$, and at times $t_p=2.55$ and $7.65$ fs, $P\left(2.55\text{fs}\right)-P\left(7.65\text{fs}\right)$. The differences are shown for energies $\varepsilon =85$ and 86 eV.

Figure 5

The real (a) and imaginary (b) parts of the Fourier transforms of the angle-resolved spectra in Fig. 3 according to Eq. (19) at different times $t_p$.

Figure 6

The real (a) and imaginary (b) parts of $\mathcal{A}(\text{84}\text{eV},t_p), \mathcal{A}(\text{85}\text{eV},t_p)$, and $\mathcal{A}(\text{86}\text{eV},t_p)$ at different times $t_p$.

Figure 7

Time-resolved photoelectron spectra in the case of three eigenstates involved in the electron dynamics of indole (a) computed with Eq. (6) and (b) computed within the frozen-density approximation according to Eq. (7).

Figure 8

Angle-resolved photoelectron spectra calculated taking into account the orthogonalization correction to the plane-wave approximation. Each $q_x{q}_z$ plane at the corresponding photoelectron energy ${\varepsilon }_e$ is a projection of a semisphere with radius $\left|\mathbf{q}\right|=\sqrt{2{\varepsilon }_e}$ 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 $\mathbf{q}$.