Fano’s Propensity Rule in Angle-Resolved Attosecond Pump-Probe Photoionization

In a seminal article, Fano predicts that absorption of light occurs preferably with increase of angular momentum. We generalize Fano’s propensity rule to laser-assisted photoionization, consisting of absorption of an extreme-ultraviolet photon followed by absorption or emission of an infrared photon. The predicted asymmetry between absorption and emission leads to incomplete quantum interference in attosecond photoelectron interferometry. It explains both the angular dependence of the photoionization time delays and the delay dependence of the photoelectron angular distributions. Our theory is verified by experimental results in Ar in the 20–40 eV range.

Figure 1

Propensity rules in laser-assisted photoionization in He $1s$ (cross), Ne $2p$ (inverted triangle), Ar $3p$ (circle), and Kr $3d$ (plus). The color of the curves indicates the angular momentum of the intermediate state in (a) and the final state in (b),(c) [gray, $s$; blue, $p$; green, $d$; red, $f$; orange, $g$]. (a) Probability ratio between increasing ($\lambda \to \lambda +1$) and decreasing angular momentum ($\lambda \to \lambda -1$) in the case of absorption of a photon from the intermediate state. (b) Probability ratio between absorbing and emitting a photon in the case of increasing angular momentum. (c) Probability ratio between emitting and absorbing a photon in the case of decreasing angular momentum. (Insets) An energy versus angular momentum diagram illustrating the propensity rule in each case.

Figure 2

Delay dependence of the PAD. (a) Asymmetry parameters ${\beta }_2$ (blue) and ${\beta }_4$ (yellow) as a function of the delay for SB 20 in He. The pictures at the top show the evolution of the PADs together with polarization direction $\widehat{z}$. (b) Temporal dependence of $|A_{20}{|}^2$ (yellow) and $|A_{00}{|}^2$ (blue) normalized to the maximum of $|A_{00}{|}^2$.

Figure 3

Angular dependence of the photoionization time delay. Contributions of the absorption ($A^{(+)}$, blue) and emission ($A^{(-)}$, yellow) paths and atomic delay (${\tau }_A$, black) in SB 20 in He.

Figure 4

Experimental and theoretical results in Ar. (a) Experimental and (b) theoretical variation of ${\beta }_2$ as a function of delay for SBs 14 (blue), 20 (magenta), and 22 (yellow). Solid lines in (a) are fits to the data. (c) Experimental (circles) and theoretical (solid curves) angle dependence of the atomic delay for SBs 14, 20, and 22. The error bars correspond to the standard deviation returned from the fitting algorithm.