Wigner Time Delay in Photoionization: A 1D Model Study

TL;DR

This work demonstrates that photoionization can exhibit a measurable Wigner-Smith time delay, contrary to claims that such delays are inaccessible in photoemission. By analyzing a 1D short-range symmetric potential with time-dependent perturbation theory and a dipole interaction, the authors show the emitted wavepacket carries a delay equal to the energy derivative of the appropriate scattering phase shift, mediated by the dipole matrix element under the ingoing boundary condition. They provide analytic results for narrow-bandwidth scenarios and a general convolution-based formula for arbitrary bandwidths, supported by numerical simulations across multiple resonance regimes. The findings establish a coherent link between photoionization delays and scattering phase shifts, with the notable result that the observed delay corresponds to half of the standard scattering delay, and offer insight into how measurements of attosecond delays relate to phase-shift dynamics in short-range models (with implications for Coulombic cases).

Abstract

In scattering theory, the Wigner-Smith time delay, calculated through a phaseshift derivative or its multichannel generalization, has been demonstrated to measure the amount of delay or advance experienced by colliding particles during their interaction with the scattering potential. Fetic, Becker, and Milosevic argue that this concept cannot be extended to include photoionization, viewed as a half-scattering experiment. Their argument is based on the lack of information about scattering phaseshifts in the part of the wavefunction (satisfying the ingoing-wave boundary condition) going to the detector. This article aims to test this claim by examining a photodetachment process in a simple 1D model with a short-range symmetrical potential. Using time-dependent perturbation theory with a dipole interaction, the relevant wavepacket of the outgoing particle is analyzed and compared to the free wavepacket as a reference. Our findings confirm that, indeed, a time delay arises in the liberated fragmentation wavepacket, which is expressed as an energy derivative of the scattering phaseshift. Our study highlights that the source of the phaseshift content in the wavepacket arriving at the detector is the dipole matrix element, which is a direct consequence of imposing the ingoing-wave boundary condition. We illustrate our results through numerical simulations of both the non-free and free wavepackets. The amount of the observed time delay is found to be half of that appearing in a typical scattering experiment.

Figures (6)

• Figure 1: The potential energy $V_0(x)$ in Eq. (\ref{['potential_eq']}) is plotted versus $x$. The Dirac-delta well is represented by a downward arrow at the origin.
• Figure 2: The even- and odd-parity phaseshifts (divided by $\pi$) are plotted versus energy for $a=4, \ b=6, \ V_0=1$ and $\alpha=1$. Each rapid rise of $\delta/\pi$ by approximately unity corresponds to a scattering resonance.
• Figure 3: The dipole matrix element square root is plotted versus final state energy for $a=4, \ b=6, \ V_0=1$ and $\alpha=1$. Two peaks can be seen at the same resonant energies seen in the odd-parity phaseshift in Fig. \ref{['phaseshift']}
• Figure 4: (First Case) (a): the squared norm of the wavepackets for $V_0=0$ (blue dashed curve) and $V_0 \neq 0$ (red solid curve) are plotted versus distance at a large positive time. The two wavepackets have been rescaled to have the same normalization. The inset exhibits the Gaussian envelope $g(E)^2$, arbitrarily scaled, (blue dashed curve) over-plotted with the odd phaseshift energy derivative $\delta'_o(E)$ (red solid curve). (b): $\langle x \rangle$ for $V_0=0$ (blue dashed curve) and $V_0 \neq 0$ (red solid curve) is plotted versus time. The inset focuses on the asymptotics and shows their straight line fit equations. The parameters values are: $a=4,\ b=6,\ E_0=0.83695$, $s=500$, and $\alpha=1$.
• Figure 5: (Second Case) (a): the squared norm of the wavepackets for $V_0=0$ (blue dashed curve) and $V_0 \neq 0$ (red solid curve) are plotted versus distance at a large positive time. The two wavepackets have been rescaled to have the same normalization. The inset exhibits the Gaussian envelope $g(E)^2$, arbitrarily scaled, (blue dashed curve) over-plotted with the odd phaseshift energy derivative $\delta'_o(E)$ (red solid curve). (b): $\langle x \rangle$ for $V_0=0$ (blue dashed curve) and $V_0 \neq 0$ (red solid curve) is plotted versus time. The inset focuses on the asymptotics and shows their straight line fit equations. The parameters values are: $a=4,\ b=6,\ E_0=0.83695$, $s=20$, and $\alpha=1$.