Toolkit for semiclassical computations for strongly driven molecules: Frustrated ionization of ${\mathrm{H}}_2$ driven by elliptical laser fields

We study the formation of highly excited neutral atoms during the breakup of strongly driven molecules. Past work on this significant phenomenon has shown that during the formation of highly excited neutral atoms (${\mathrm{H}}^{*}$) during the breakup of ${\mathrm{H}}_2$ in a linear laser field the electron that escapes does so either very quickly or after remaining bound for a few periods of the laser field. Here, we address the electron-nuclear dynamics in ${\mathrm{H}}^{*}$ formation in elliptical laser fields, through Coulomb explosion. We show that with increasing ellipticity two-electron effects are effectively “switched off.” We perform these studies using a toolkit we have developed for semiclassical computations for strongly driven multicenter molecules. This toolkit includes the formulation of the probabilities of strong-field phenomena in a transparent way. This allows us to identify the shortcomings of currently used initial phase-space distributions for the electronic degrees of freedom. In addition, it includes a three-dimensional method for time propagation that fully accounts for the Coulomb singularity. This technique has been previously developed in the context of celestial mechanics and we currently adopt it to strongly driven systems. Moreover, we allow for tunneling during the time propagation. We find that this is necessary in order to accurately describe the fragmentation of strongly driven molecules.

Figure 1

Double-ionization events of strongly driven ${\mathrm{H}}_2$ for an intensity in the below-the-barrier intensity regime, $2.03\times {10}^{14}{\mathrm{W}/\mathrm{cm}}^2$ (grey solid line with full triangles), and for an intensity in the over-the-barrier intensity regime, $2.14\times {10}^{14}{\mathrm{W}/\mathrm{cm}}^2$, using method 1 (black solid line with full circles) and method 2 (black dashed line with full squares): the distribution for the initial phase ${\varphi }_0$ (a) and $P_{{\varphi }_0}^{\mathrm{proc}}$ as a function of ${\varphi }_0$ (b).

Figure 2

The final energy distribution of the ${\mathrm{H}}^{+}$ or ${\mathrm{H}}^{*}$ fragments for a laser field intensity of $1.5\times {10}^{14}{\mathrm{W}/\mathrm{cm}}^2$ for different initial momentum distributions of the nuclei: Wigner distribution (black solid line with full circles), 4.3 a.u. relative momentum of the nuclei (black dashed line with full squares), and nuclei initially at rest (grey solid line with full triangles).

Figure 3

The distribution of the quantum number $n$ for a field intensity $1.5\times {10}^{14}{\mathrm{W}/\mathrm{cm}}^2$ (a), and $2.5\times {10}^{14}{\mathrm{W}/\mathrm{cm}}^2$ (b). The black solid line with full circles corresponds to $\epsilon =0$, and the grey dashed line with full squares corresponds to $\epsilon =0.45$.

Figure 4

The final energy distribution of the ${\mathrm{H}}^{+}$ or ${\mathrm{H}}^{*}$ fragments for a field intensity $1.5\times {10}^{14}{\mathrm{W}/\mathrm{cm}}^2$ (a), and $2.5\times {10}^{14}{\mathrm{W}/\mathrm{cm}}^2$ (b). The black solid line with full circles corresponds to $\epsilon =0$, and the grey dashed line with full squares corresponds to $\epsilon =0.45$.

Figure 5

Schematic illustration of the two routes leading to formation of ${\mathrm{H}}^{*}$ for $\epsilon =0$: (a) pathway A, (b) pathway B. Shown is the time-dependent position along the laser field for electrons (black lines) and ions (gray broken lines). This figure appears in [15]; we also include it here for completeness.

Figure 6

Probabilities for the two pathways for an intensity $1.5\times {10}^{14}{\mathrm{W}/\mathrm{cm}}^2$ (a), and $2.5\times {10}^{14}{\mathrm{W}/\mathrm{cm}}^2$ (b). The black solid line with full circles is for pathway A, and the grey dashed line with full squares is for pathway B.

Figure 7

The 2D electron momentum distribution for an intensity $1.5\times {10}^{14}{\mathrm{W}/\mathrm{cm}}^2$ (a) and (b), and for $2.5\times {10}^{14}{\mathrm{W}/\mathrm{cm}}^2$ (c) and (d). Figures on the left are for $\epsilon =0$, and those on the right are for $\epsilon =0.45$. The momentum is expressed in $\sqrt{U_p}$.

Figure 8

Schematic illustration of the sites an electron can tunnel from: (a) up-field well, (b) low-field well, and (c) single well.

Figure 9

(a) The ionization rate of ${\mathrm{H}}_2$ vs the field strength for a laser field parallel to the molecular axis, calculated with Eq. (A1) (black solid), obtained from [54] (grey dashed), and obtained from [55] (full circle).