The PESCADO Method for Autonomous Systems: An Application to Photoionization at Near-optical Wavelengths

TL;DR

The paper advances the PESCADO framework by formulating a semi-analytical route to infinite-time photoelectron spectra for hydrogen under near-optical fields, using a local CAP to absorb outgoing flux on a truncated grid. It distinguishes non-autonomous (during the pulse) and autonomous (after the pulse) regimes, deriving expressions for the ionization observable via the absorbed density operator and its spectral expansion with complex eigenvalues. Key findings include robust convergence of energy and angle-differential spectra with respect to the CAP onset Rc, and a clear detector-like interpretation of absorption-angle distributions when the CAP is near the nucleus. The approach preserves Coulomb effects and enables efficient, accurate spectra without long propagation times, with potential extensions to multi-particle systems via second quantization.

Abstract

In a recent publication, Dalen, and Selstø, Phys. Rev. A {\bf 111}, 033116 (2025), it was demonstrated how converged photo electron spectra could be determined using a complex absorbing potential on a truncated numerical domain considerably smaller than the extension of the dynamical wave function. That approach required simulation until virtually all unbound parts of the wave function was absorbed, far beyond the duration of the interaction with the external field. In this work we formulate the method in a semi-analytical manner which allows us to extrapolate to infinite times after the interaction with the external field. In addition to obtaining photoelectron spectra for hydrogen differential in energy and ejection angle, we also demonstrate how -- and when -- the absorber may be seen as a detector, distorting the angular distributions when the detector is placed in the extreme vicinity of the atom.

Figures (9)

• Figure 1: The norm of the wave function, $|\Psi(t)|^2$, as a function of time for various choices of the CAP onset $R_c$ as a function of time during the interaction with the laser pulse. In all cases, a significant part of the wave function as been absorbed before the pulse is over.
• Figure 2: All panels display the energy distribution of the photoelectron. They are obtained using different values for the CAP onset parameter $R_c$. In addition to the total energy distribution (black curve), we have also displayed contributions picked up during (blue, dashed curve) and after (red, dashed-dotted curve) the interaction with the laser pulse.
• Figure 3: The ionization probability differential in energy for various CAP onsets $R_c$, ranging from $40$ to $120$ a.u. The spectra are shown using linear axes in the upper panel and with a logarithmic $y$-axis in the lower one.
• Figure 4: The ionization probability differential in ejection angle, $\Omega_k$. To the left it is visualized as a three dimensional distribution while the right panel shows the distribution, which is independent of the azimuthal angle in the dipole approximation, in the polar angle $\theta_k$. For this particular case, the CAP onset $R_c=60$ a.u.
• Figure 5: The doubly differential ionization probability obtained using various choices for the CAP onset $R_c$. Specifically, from left to right, $R_c= 40, 60, 80$ and $100$ a.u. The $x$ and $y$-axis correspond to $\varepsilon \cos \theta_k$ and $\varepsilon \sin \theta_k$, respectively, in atomic units.