Time-resolved Electron Momentum Spectroscopy with Ultrashort Electron Pulses: Confined Probing and Effects of Vacuum Dispersion

Abstract

Previous theoretical studies have shown that attosecond electron dynamics can, in principle, be captured in electron momentum spectroscopy (EMS) using ultrashort electron pulses. By including further analytical considerations on the scattering probability, we here study the effect of the finite transversal extend of the projectile electron wave packet. We find that in wave packet scattering, the target momentum distribution is probed solely in a finite spatial region. This is evident from a spatially filtering Gabor transform appearing in the scattering probability, replacing the full momentum wave function appearing in the conventional plane wave treatment. In addition, by spatially shifting the target with regard to the wave packet focus, we illustrate the influence of vacuum dispersion, i.e., the spatial broadening of the wave packet as it propagates. Our findings are significant for the possibility to correctly interpret future attosecond-EMS results and the considered effects reflect fundamental aspects of wave packet scattering. The EMS setup may, therefore, constitute a useful framework for understanding scattering with finite wave packets.

Figures (5)

• Figure 1: Wave packet EMS in the symmetric non-coplanar geometry where $\theta = 45^\circ$ and only the angle $\phi$ is varied. $\bm k_0$ is the central momentum of the incident wave packet and $\bm k_a$ and $\bm k_b$ are the outgoing momenta.
• Figure 2: (a) The momentum density along the momentum space $y$-axis [see Eq. \ref{['q_formula']}] for the coherent superposition of a $3p_y$ and $4p_y$ state in atomic hydrogen for different values of $t_d$. (b)-(d) Comparison between the "exact" double differential scattering probability (DDP) from Eq. (\ref{['eq:ExactScatProb']}) and Eq. (\ref{['eq:TransitionProb']}) (full lines) and the approximate DDP of Eq. (\ref{['eq:ApproxScatProb']}) (dotted lines) for a 10 keV Gaussian wave packet for different values of $t_d$. In (c)-(e) only positive detection angles are considered due to the symmetry around $\phi=0$. In (b) and (c) the wave packet is transversally narrow $\sigma_\perp = k_0 \times 1 \ \mathrm{mrad}$ and the exact and approximate DDPs are nearly identical. The difference in the spectra originate from the overlap coefficient $B_{3p_y, 4p_y} \approx 0.999$ for the 0.1 fs pulse and $B_{3p_y, 4p_y} \approx 0.103$ for the 5 fs pulse. In (d), however, the projectile wave packet is transversely broad $\sigma_\perp = k_0 \times 5 \ \mathrm{mrad}$ and the approximate DDP of Eq. (\ref{['eq:ApproxScatProb']}) is no longer valid due to significant vacuum dispersion. There is a notable dependence on $t_d$ even though $B_{3p_y, 4p_y} \approx 0$. (e) The differences between scattering probabilities for different values of the delay parameter $t_d$ for a 0.1 fs pulse (full lines) and a 5 fs pulse (dotted lines), respectively. For equal pulse lengths these differ only by a sign, and for different pulse lengths they are nearly the same up to a scaling factor (the ratio) as illustrated by the black dotted line. This indicates that the scattering probabilities behave similarly to temporal averages of the target momentum distribution.
• Figure 3: The double differential scattering probabilities (full lines) for a target described by $t_d = 0$ and a pulse duration of $\tau = 100 \ \mathrm{as}$ for different transversal widths $\sigma_\perp$. The dotted lines, which nearly perfectly overlap with the full lines, show the momentum distributions $\rho_G$ isolated with the cylindrical Gabor transform with an equivalent width. The momentum densities have been scaled to match the scattering probabilities. Thus in this case, the Gabor transform describes the shape of the EMS spectrum.
• Figure 4: The EMS double differential scattering probability (DDP) of 100 as wave packet for different shifts $t_p$ of the wave packet focus. The target wave function is categorized by the delay parameter $t_d = T/4$. A notable difference between positive and negative shifts is observed. The dashed black lines represent results for a focused wave packet for $t_d = 0,\ T/4, \ T/2$ respectively (as given in Fig. \ref{['fig:ApproxScaProb']} (b)) and serve as a reference.
• Figure 5: The propagation of a Gaussian wave packet across the origin. The wave packet focus is shifted by $t_p = T/4$ and is shown in (e). As the wave packet crosses the target atom (black dot), which is defined by $t_d = T/4$, located at the origin the projectile electron density changes. When propagating away from the origin, shown in (d), the electron density is slightly higher compared to when it moved towards the origin (b). The target momentum density (shown on the insert) is therefore probed more at $t=T/40$ than $t=-T/40$ effecting the shape of the resulting EMS spectrum. Note, the dispersion of the wave packet has been exaggerated for illustration and does not resemble the actual dispersion for the parameters used in Fig. \ref{['fig:OutOfFocus']}.