High-order above-threshold ionization in stretched molecules

By numerical solution of the time-dependent Schrödinger equation, highly accurate electron spectra are calculated for strong-field ionization of a one-dimensional $\mathrm{H}_2{}^{+}$ molecular ion at large internuclear distances. Compared to the atomic spectra, we find additional cutoffs at high electron energies. The classical model for ionization (simple-man’s model) adapted to the molecular geometry predicts cutoffs up to $50U_{\mathrm{p}}$, where $U_{\mathrm{p}}$ is the ponderomotive potential. The cutoffs correspond to various scattering scenarios of the tunneled electron from the molecular sites. For certain internuclear distances the agreement between the classical predictions and the numerical spectra is unsatisfactory. We propose a modified simple-man’s model based on complex electron trajectories, allowing for electrons appearing in the continuum with nonzero initial velocity from the tunneling ionization process. Agreement of the resulting cutoffs with the numerical results is recovered.

Figure 1

(Color online) Predictions from the classical model for the maximal electron energies at the detector for the $AB$, $AAB$, and $ABA$ scattering processes as a function of the internuclear separation (in units of $R_0\equiv \pi {\alpha }_0$; see text). Left panel: monochromatic laser field. Right panel: a trapezoidal seven-cycle pulse.

Figure 2

(Color online) Above-threshold ionization spectra (see text for laser parameters). Different labels show the cutoffs predicted by the simple-man’s model. The red (dashed) arrows show the corresponding cutoffs given by the CTSM model. Left panel: $R=0.5R_0$. Right panel: $R=2.5R_0$.

Figure 3

(Color online) Above-threshold ionization spectra for $R=3.5R_0$. Different labels show the cutoffs predicted by the simple-man’s model. The red (dashed) arrows show the corresponding cutoffs given by the CTSM model.

Figure 4

Left panel $AB$: The maximal ionization rate $w$ [Eq. (5)] for a given final energy. Right panel: same as the left panel, but for the $ABA$ process. The internuclear distance is $R=3.5R_0$.

Figure 5

(Color online) The detachment position $x_1$ as a function of the electric field at time of detachment. The field is monochromatic. Two values of the Keldysh parameter are considered. The dashed and the dot-dashed curve that approach the full and the dotted curve, respectively, at maximum field show the position given by the simple expression $x_1=-I_{\mathrm{p}}∕E\left(t_1\right)$.

Figure 6

(Color online) The maximal ionization rate $w_{\mathrm{m}}$ for a given final kinetic energy, as a function of the internuclear distance. Upper panel: the $AB$ process; lower panel: the $ABA$ process.