Probing Electronic Fluxes via Time-Resolved X-Ray Scattering

The current flux density is a vector field that can be used to describe theoretically how electrons flow in a system out of equilibrium. In this work, we unequivocally demonstrate that the signal obtained from time-resolved x-ray scattering does not only map the time evolution of the electronic charge distribution, but also encodes information about the associated electronic current flux density. We show how the electronic current flux density qualitatively maps the distribution of electronic momenta and reveals the underlying mechanism of ultrafast charge migration processes, while also providing quantitative information about the timescales of electronic coherences.

Figure 1

Conceptual sketch of the charge migration mechanism. An $x$-polarized pump pulse induces nonstationary charge migration associated with an electronic wave packet in benzene. The electron density (blue shaded area) migrates from one side of the molecule to the other with a period $\tau =504$ as. Black arrows correspond to the electronic flux density associated with this process (for $z=1a_0$).

Figure 2

Time-resolved signals in the $Q_x-Q_y$ plane (integrated over $Q_z$) as a function of pump-probe delay times with a charge migration period of $\tau =504$ as for benzene. The time-resolved signal at time zero $t=0$ is subtracted from the signals at later delay times. (a) Scattering patterns obtained using Eq. (1), and (b) time derivative of the momentum-space electron density $\rho (\mathbf{Q})$. The intensity of the scattering patterns are presented in units of $d{P}_e/d\mathrm{\Omega }$.

Figure 3

Comparison of the time-resolved signals from Eq. (1) (blue) and the time derivative of the momentum-space electron density $\rho (\mathbf{Q})$ (orange) along $Q_x$ ($Q_y=Q_z=0$) at pump-probe delay times $t=\tau /4$ and $t=3\tau /4$.

Figure 4

Panel (a): Fourier transform of the time derivative of the momentum-space electron density $\rho (\mathbf{Q})$ (color map) and the associated current flux density (arrows) as a function of pump-probe delay times. Panel (b): Fourier transform of the time derivative of the momentum-space electron density $\rho (\mathbf{Q})$ (color map) and velocity field, $\mathbf{v}(\mathbf{r},t)=\mathbf{j}(\mathbf{r},t)/\rho (\mathbf{r},t)$, (arrows) depicted only at positions where the electron density is above $\rho (r)>{10}^{-10}{a}_0^{-3}$.