Transverse-electron-momentum distribution in pump-probe sequential double ionization

We study the transverse-electron-momentum distribution (TEMD) of the wave packets launched in a pump-probe sequential double ionization from the valence shell of a noble gas atom. Our calculations, based on an accurate numerical solution of the time-dependent Schrödinger equation (TDSE), reproduce a characteristic cusp of the TEMD which is attributed to the Coulomb singularity. The evolution of the TEMD with the time delay between the pump and probe pulses is shown to be similar to the prediction of the standard tunneling formula (TF), as was observed experimentally for argon by Fechner et al. [Phys. Rev. Lett. 112, 213001 (2014)]. However, TDSE calculations show a clear deviation from the TF and predict a much more complicated structure which cannot be reproduced by the target orbital momentum profile filtered by the tunneling Gaussian. The accuracy of the TF can be improved if the target momentum profile is calculated with the Coulomb waves instead of the plane waves.

Figure 1

Top panel: The TEMD $W_m\left(p_{\perp }\right)$ of the Ar $3pm=0$ (solid red line) and $m=1$ (dotted blue line) states calculated with the HF orbitals (right) and the hydrogenic orbitals [15] (left). $W_{m=1}\left(p_{\perp }\right)$ is multiplied by a factor of 10 for better visibility. Middle panel: Same for the ${\mathrm{Ar}}^{+}$ ion. The green dashed lines on the right show the calculation with the localized potential. Bottom panel: Gaussian filter functions for the field intensity of $5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ for the neutral Ar atom (solid red line) and the ${\mathrm{Ar}}^{+}$ ion (dotted blue line) in the present calculation (right) and from Ref. [15] (left).

Figure 2

Right: The TEMD $W_m\left(p_{\perp }\right)$ of the ${\mathrm{Ar}}^{+}$ ion for the $3p$ states with $m=0$ (solid red line) and $m=1$ (dotted blue line). The calculation by Fechner et al. [15] is shown with the colored dots. Left: The TEMD $W_m^c\left(p_{\perp }\right)$ calculated with the Coulomb functions instead of the Bessel functions.

Figure 3

The time-differential TEMD $\Delta W(p_{\perp },\tau )$ for the ${\mathrm{Ar}}^{+}$ ion calculated at the time delays equal to an integer $\tau =nT$ (top) and half-integer $\tau =nT/2$ (bottom) periods of the spin-orbit oscillation. The Bessel function calculations with the HF (red solid lines) and localized (green dashed lines) orbitals are compared with calculations of Fechner et al. [15] with hydrogenic orbitals (blue dotted lines). The dotted purple line shows the Coulomb function calculation with the HF orbital.

Figure 4

The time-averaged TEMD of the ${\mathrm{Ar}}^{+}$ ion. Right: TDSE calculation. Left: Experiment by Fechner et al. [15] and the TF calculation with the Coulomb function and the HF orbital (blue solid line).

Figure 5

Top panel: The time-differential TEMD $\Delta \overline{W}(p_{\perp },\tau )$ for the ${\mathrm{Ar}}^{+}$ ion calculated at the time delays equal to an integer number of periods of the spin-orbit oscillation. Incoherent density matrix (21) (red solid line); coherent density matrix (25) (green dashed line). Bottom panel: Same for time delays equal to a half-integer number of periods of the spin-orbit oscillation.

Figure 6

The time-differential TEMD $\Delta W(p_{\perp },\tau )$ for the ${\mathrm{Ar}}^{+}$ ion calculated at various time delays equal to an integer $\tau =nT$ (top panel) and a half-integer $\tau =nT/2$ (bottom panel) period of the spin-orbit oscillation. TDSE calculations (red solid line) are compared with the TF calculations with the hydrogenic orbital (blue dotted lines), the TF calculations with the Coulomb waves and the HF orbitals (green dashed lines), and the experiment of Fechner et al. [15] (error bars).

Figure 7

The time-averaged TEMD of the ${\mathrm{Ar}}^{+}$ ion for circularly polarized driving pulse. The binned TDSE calculation is shown with boxes. The TF calculation with the localized $3p$ orbital of the ${\mathrm{Ar}}^{+}$ ion is shown on the right by the blue dotted line. The Gaussian fit $exp(-p_{\perp }^2/{\sigma }^2)$ is shown by the black solid line on the left.

Figure 8

The time-differential TEMD $\Delta W(p_{\perp },\tau )$ for the ${\mathrm{Ar}}^{+}$ ion calculated at time delays equal to integer $\tau =nT$ (top) and half-integer $\tau =nT/2$ (bottom) periods of the spin-orbit oscillation. Solid line: TDSE calculation; boxes: TDSE calculation binned into $\Delta p_{\perp }=0.05$ a.u. intervals of transverse momentum for better comparison with experiment.