Theoretical description of field-assisted postcollision interaction in Auger decay of atoms

In a recent publication [Schütte et al., Phys. Rev. Lett. (to be published)] it was demonstrated, both experimentally and theoretically, that Auger electrons are subject to an energetic chirp if a three-body-interaction dubbed postcollision interaction (PCI) is involved. Here, we extend previous theoretical work and give a detailed analysis of field-assisted PCI based on numerical solutions of the time-dependent Schrödinger equation, extensive Monte Carlo–averaged molecular dynamics simulations, and analytical theory. The dependence on various streaking and excitation conditions is investigated, and we discuss how these findings may help to improve XUV pulse characterization as well as understanding of ultrafast atomic processes.

Figure 1

Auger line shapes obtained by solving Eqs. (1) and (2) for a series of time delays $t_X$. Shown is the case including PCI (blue solid lines, blue area) and the case neglecting PCI (red dashed lines), that is, ${\widehat{H}}_2={\widehat{H}}_1$ in Eqs. (1) and (2) for a $16.9$-fs single-cycle (${\varphi }_L=\pi /2$) streaking pulse with a frequency of ${\omega }_L=33$ THz and $U_p=98.8$ meV. The $2.83$-fs XUV pulse has a photon energy of $91$eV. The Auger decay constant was set to ${\Gamma }_A=950$ meV at an Auger energy of $E_A=34.27$ eV, thus resembling THz streaking of the Xe NOO transition scaled by a factor of $\gamma =10$ (see text for details). Note that the lines neglecting PCI have been shifted toward higher energy by $0.5$ eV for better comparison.

Figure 2

FWHM of Auger lines (right axis) obtained by solving Eqs. (1) and (2) for different scaling factors $\gamma$ [cf. Eq. (7)] of the Xe NOO transition in a $3.3$-THz streaking field keeping $U_p=98.8$ meV constant. The natural linewidth ${\Gamma }_A$ has been subtracted for each set of parameters for better comparison. The maximum of the line [proportional to $-A\left(t\right)$, gray line with solid circles labeled by $E_{\max }$, left axis] and the case neglecting PCI (black dashed line) are shown for $\gamma =10$. For $\gamma =10$, parameters are the same as in Fig. 1.

Figure 3

FWHM of Auger lines (right axis) for different photon energies ${\omega }_X$ of the XUV pulse of 28 fs duration. All other parameters are the same as in Fig. 2 for the case $\gamma =10$. The energy of the maximum of the Auger line and the case neglecting PCI (black dashed line) are given for ${\omega }_X=91$ eV. The time dependence of the electrical field $E\left(t\right)$ is sketched by the gray dotted line.

Figure 4

FWHM of Auger line (right axis) for different XUV pulse durations ${\tau }_X^{\mathrm{FW}}$. The maximum of the Auger line [proportional to $-A\left(t\right)$, solid circles] is shown for 28 fs. The time dependence of the electrical field $E\left(t\right)$ is sketched by the gray dotted line. Parameters are the same as in Fig. 2 for $\gamma =10$.

Figure 5

The same as Fig. 4 but for different ponderomotive potentials $U_p$ of the streaking field at a fixed XUV pulse duration of 28 fs ($\gamma =10$). The graph of the maximum of the line corresponds to $U_p=98.8$ meV.

Figure 6

Coincidence energy spectra for Auger and photoelectrons for selected time delays $t_X$ calculated by solving Eqs. (1) and (2). The integrated photoelectron (Auger electron) distribution is plotted with red dashed (blue solid) lines. Parameters are the same as in Fig. 1 (Xe NOO with $\gamma =10$).

Figure 7

Temporal parameters and electron distributions: The XUV pulse is centered at $t_X$ with a FWHM duration of ${\tau }_X$. At $t_{P_i}$ during the pulse the photoelectron is excited, which triggers Auger decay at a time instant $t_{A_i}$. The measurement is performed long after the pulses and the decay are over, at time $t_d$ ($t_d\to \infty$).

Figure 8

Auger line shapes of the Kr MNN transition for a set of time delays $t_X$ in a 1-THz single-cycle streaking field with $U_p=80$ meV for a XUV pulse duration of 28 fs at a photon energy of 97 eV. Results are obtained by MC averaging of MD trajectories. Shown is the case including photoelectron–Auger-electron interactions (blue solid lines) and neglecting $e$-$e$ interactions (red dashed lines). Note that the latter lines are shifted by 100 meV toward higher energy for clarity.

Figure 9

FWHM of the Auger lines (right axis) of the Xe NOO and Kr MNN transitions calculated utilizing MC MD simulations. For Xe (Kr) the photon energy of 91 eV (97 eV) leads to a photoelectron kinetic energy of 24 eV ($2.6$ eV). All other parameters are the same as in Fig. 8. The position of the maximum of the line, $E_{\max }$, for both cases is shown in gray. The case neglecting $e$-$e$ interactions for Kr is given by the black dotted line.

Figure 10

Line shapes of the Xe NOO transition ($\gamma =10$) at three different time delays $t_X$, corresponding to the FWHM maxima (see curve in Fig. 3 for 91 eV) and $t_X=0$, for streaking with $U_p=98.8$ meV. Results from TDSE simulations (top row) and MC-averaged MD simulations (bottom row) are given. The cases including PCI (blue solid lines) and neglecting PCI (red dashed lines) are compared for falling (left), rising (center), and zero (right) slope of $A\left(t\right)$.

Figure 11

Analytical model for the Auger line shape neglecting PCI [Eq. (26)] compared to corresponding MD simulations (excluding PCI and sampling of $p_{A_i}$) at zero transitions of the vector potential for increasing (left) and decreasing (right) slope of $A$. Shown is Xe $NOO$ in a 1-THz streaking field with a duration of 1 ps and a ponderomotive potential of 100 meV. The insets show the same data semilogarithmically.

Figure 12

Auger line shape for minima (left panel, ${\ddot{A}}_P>0$) and maxima (right panel, ${\ddot{A}}_P<0$) of $A$. Displayed is Eq. (29) compared to MD simulations. Parameters are the same as for Fig. 11.

Figure 13

Scheme of simplified 1D propagation with PCI effects: In phase 1 ($t<t^{*}$), the Auger electron (solid circles) catches up with the slow photoelectron (open circles) and overtakes it at $t=t^{*}$ (phase 2); $p_A>p_P$ is assumed. The propagation toward the detector in phase 3 ($t>t^{*}$) is similar to phase 1. Figure after Ref. [20].

Figure 14

Field-assisted PCI without streaking contribution at zero transitions of $A$. Shown is Eq. (39) for $\dot{A}>0$ (red dashed line) and $\dot{A}<0$ (blue dotted line) in comparison to the field-free case [Eq. (41)]. Data is for NOO transition of Xe in a 1-THz streaking field with $U_p=100$ meV for $\delta r^{*}=1$.

Figure 15

Auger line shapes at zero transition of $A$ including streaking and PCI contributions. The analytical line shapes [Eqs. (45) and (46)] (bold, red lines) are compared to MD simulations (filled area). The case of ${\dot{A}}_P>0$ (${\dot{A}}_P<0$) is shown in the left (right) panel. Parameters are the same as for Fig. 14.

Figure 16

Time-to-energy mapping of the Auger electrons in the streaking field for ${\dot{A}}_P>0$ (left) and ${\dot{A}}_P<0$ (right). The mappings without PCI [Eq. (25), blue dashed lines], without streaking [Eq. (38), green dotted lines], and including PCI and streaking [Eq. (37) red solid lines] are shown. Different signs of ${\dot{A}}_P$ lead to a drastic change of the mapping: For ${\dot{A}}_P>0$ the whole energy space is accessible in a bijective way, whereas in contrast for ${\dot{A}}_P<0$ a forbidden region below ${\epsilon }_0$ occurs and “early” and “late” Auger electrons are mapped to the same energy.

Figure 17

Auger line shapes for zero transitions of $A$ neglecting PCI contributions for a finite XUV pulse duration of 20 fs FWHM. All parameters refer to THz streaking of the Xe NOO transition. The convoluted line shape [Eq. (51)] (bold red line) is plotted against MD simulations (blue area). The case of infinitesimal XUV pulse duration [Eq. (26)] is shown also for comparison (blue dashed lines).

Figure 18

The same as Fig. 17 but for the case including PCI. The result of a numerical convolution [Eq. (52)] (red bold lines) is compared to MD simulations with sharp momenta (blue area) and with proper sampling over initial momenta $p_{P_i}$ and $p_{A_i}$ (blue dashed lines). Additionally, 3D MD simulations (cf. Sec. 3) are shown (orange dotted lines).

Figure 19

Energy (left axis) and FWHM (right axis) of the Auger electron (dotted) and the photoelectron distribution (PCI, solid; without PCI, dashed). Shown is the case of Xe NOO scaled by $\gamma =10$ obtained within TDSE simulations. Parameters are the same as for Fig. 4.

Figure 20

Auger line shape of the Kr MNN transition in a $3.3$-THz streaking field at fixed ponderomotive potential of 100 meV for different XUV pulse durations with a photon energy of 97 eV. Shown are solutions of the analytical model [Eq. (52)].

Figure 21

The same as Fig. 20 but for different ponderomotive potential of the streaking field at fixed XUV pulse duration of 28 fs.

Figure 22

Position of the Auger line's maximum ${\epsilon }_m^{\pm }$, the corresponding width $w^{\pm }$ and the asymmetry parameter $\xi$ [cf. Eq. (53)] vs XUV pump pulse duration. Data is for Kr MNN decay at a ponderomotive potential of 100 meV of the streaking pulse. The forbidden region for ${\dot{A}}_p<0$ at infinitesimal short XUV pulse duration is marked by ${\epsilon }_0$. The superscript “+” (“$-$”) refers to ${\dot{A}}_p>0$ (${\dot{A}}_p<0$).

Figure 23

The same as Fig. 22 but for varied ponderomotive potential $U_p$ of the streaking field at a fixed pump pulse duration of 28 fs.

Figure 24

Asymmetry parameter $\xi$ for different ponderomotive potentials of the streaking pulse as a function of the XUV pulse duration. Parameters are the same as for Fig. 22.