Laser-Driven Recollisions under the Coulomb Barrier

Photoelectron spectra obtained from the ab initio solution of the time-dependent Schrödinger equation can be in striking disagreement with predictions by the strong-field approximation (SFA), not only at low energy but also around twice the ponderomotive energy where the transition from the direct to the rescattered electrons is expected. In fact, the relative enhancement of the ionization probability compared to the SFA in this regime can be several orders of magnitude. We show for which laser and target parameters such an enhancement occurs and for which the SFA prediction is qualitatively good. The enhancement is analyzed in terms of the Coulomb-corrected action along analytic quantum orbits in the complex-time plane, taking soft recollisions under the Coulomb barrier into account. These recollisions in complex time and space prevent a separation into sub-barrier motion up to the “tunnel exit” and subsequent classical dynamics. Instead, the entire quantum path up to the detector determines the ionization probability.

Figure 1

TDSE spectra around the $2U_p$ cutoff compared to plain-SFA spectra. (a) Same $I_p=0.14$ (3.8 eV) and laser parameters as in Ref. [21], i.e., $\omega =0.0065$ (7 microns), $\widehat{E}=0.0045$ ($I=7.1\times 10^{11}\mathrm{W}/{\mathrm{cm}}^2$), ${\alpha }_C=12.4$ (asymptotic charge $Z=1$), ${\alpha }_L=8.6$, using a 6-cycle ${\sin }^2$-shaped pulse envelope. The shaded area between TDSE and SFA spectrum highlights the order-of-magnitude discrepancy. Vertical lines indicate 1, 2, and $4U_p$. The SMM prediction is plotted bold black. Panel (b) shows two TDSE PES vs scaled momentum, both for the ground state as initial state and both with ${\alpha }_C={\alpha }_L=4$ but one for $Z=1$, $\omega =0.0625$, $\widehat{E}=0.0625$ (purple), and the other for $Z=2$, $\omega =0.25$, $\widehat{E}=0.5$ (green, shifted for better comparison). The SFA result is included (cyan). Panel (c) shows two TDSE PES for similar $I_p$ realized via an excited state (purple) and an effective potential (green), respectively (see text). In both cases ${\alpha }_C=11.3$ and ${\alpha }_L=5.7$. The corresponding SFA spectrum is in striking disagreement. Panel (d) shows TDSE and SFA PES in good agreement for $H(1s)$ as in (b) but for ${\alpha }_L=6$.

Figure 2

How long and short orbits are affected by the Coulomb correction [same parameters as in Fig. 1]. Panels (a) and (b) show integration contours (cyan) in the complex-time plane for a long and a short orbit at the cutoff momentum, respectively. The color scale indicates $\mathrm{Re}\sqrt{{\mathit{r}}^2(t)}$, the colored lines constant $\mathrm{Im}\sqrt{{\mathit{r}}^2(t)}$ ($\text{red}<0$, $\text{yellow}=0$, $\text{green}>0$). White lines indicate branch cuts starting at branch points. For the short trajectory in (b) the integration contour can be chosen in the standard way because the recollision branch points and cuts are far away from the real axis, i.e., outside the complex-time domain shown in (b). For the long trajectory in (a) the integration path needs to be deformed in order to navigate through the branch cut gate. Panel (c) shows partial spectra from long and short trajectories for different angles $\vartheta$ and a plain SFA spectrum (calculated with only one trajectory per momentum to inhibit interferences). The angular dependence of the short-trajectory spectrum is negligible. Panel (d) displays the trajectory weights $\mathrm{Im}\mathrm{\Delta }{S}_{\mathrm{CC}}$, measured along the integration contours in (a) and (b). The initial value $\simeq 12$ results from the matching at $t_{0\alpha }$. Note that every kink in (d) is related to a change in the integration direction in (a),(b).