Floquet surface hopping: Laser-driven dissociation and ionization dynamics of ${{\mathrm{H}}_2}^{+}$

A quantum-classical approach is developed to describe the strong-field molecular dynamics of ${{\mathrm{H}}_2}^{+}$, taking into account all degrees of freedom and simultaneously dissociation as well as ionization. The electron and nuclei are treated correlated, by propagating the nuclei stochastically on potential energy surfaces. It is demonstrated that Floquet surface hopping (FSH) is particularly well suited to describe the laser-driven dynamics. The method is tested against exact solutions of the time-dependent Schrödinger equation, where available. In addition, the FSH results are in excellent agreement with recent experimental data of the dissociation and ionization dynamics of ${{\mathrm{H}}_2}^{+}$. As an additional issue of this work, the primary importance of the focal volume average is worked out for the understanding of experimental results. It determines the gross features of the experimental spectra and provides also a natural explanation of the puzzling saturation effect in the dissociation spectra, observed experimentally. Future applications and further extensions of the method are discussed.

Figure 1

The two lowest BO surfaces ${\sigma }_g$ and ${\sigma }_u$ for ${{\mathrm{H}}_2}^{+}$, dressed with different photon numbers for a laser with $\lambda =800$ nm. The arrows, each with a length of $\hslash \omega$, mark the points of resonant one- and three-photon absorption, which are also easily identified via the crossings between different dressed state surfaces.

Figure 2

Four Floquet surfaces for ${{\mathrm{H}}_2}^{+}$, arising from the two lowest BO surfaces, calculated for $I={10}^{12}\mathrm{W}/{\mathrm{cm}}^2, \lambda =800$ nm, and $\vartheta =0$. For lucidity only the one- and three-photon crossings are included.

Figure 3

The Floquet surfaces corresponding to ${\sigma }_g+0\omega$ and ${\sigma }_u-1\omega$ as a function of $R$ and $\vartheta$ for two different laser intensities; the wavelength is $\lambda =800$ nm. The trench (dashed white line) in the second from top panel is used to explain the angular-resolved dissociation yield in Sec. 3. In contrast to Fig. 2, all crossings up to the nine-photon crossing are included, which is necessary for the large intensities used here.

Figure 4

A sketch of the nuclear degrees of freedom $R, \vartheta$, and $\phi$ when considering ${{\mathrm{H}}_2}^{+}$ in the center-of-mass frame with a fixed direction given by the amplitude $\vec{F}$ of the linear polarized laser field.

Figure 5

(Upper panel) The BO surfaces ${\sigma }_g$ and ${\sigma }_u$ (thick gray lines) and three Floquet surfaces for $\lambda =800$ nm and $I={10}^{12} \mathrm{W}/\mathrm{cm}{}^2$ (thin colored–black lines) as well as $I={10}^{13} \mathrm{W}/\mathrm{cm}{}^2$ (thick colored–black lines). The vibrational energies for $\nu =3,9,14$ are indicated for the discussion in Sec. 3. (Lower panel) The amplitude of the dipole coupling between ${\sigma }_g$ and ${\sigma }_u$ for $I={10}^{13} \mathrm{W}/\mathrm{cm}{}^2$ (dashed gray line) and the coupling between the Floquet surfaces as discussed in the text (thin lines for $I={10}^{12} \mathrm{W}/\mathrm{cm}{}^2$ and thick lines for $I={10}^{13} \mathrm{W}/\mathrm{cm}{}^2$; the coupling at the one-photon crossing is scaled by a factor of 5 for both intensities).

Figure 6

The total dissociation probabilities $P_d\left(\nu \right)$ as a function of the initial vibrational level calculated with the TDSE and the FSH method.

Figure 7

The photon-resolved dissociation probabilities $P_d(\nu ,n\omega )$ with contributions of the zero-, one-, and two-photon channels to the total dissociation probabilities, presented in Fig. 6, calculated with TDSE and FSH.

Figure 8

Photon- and angle-resolved dissociation probability density $p_d(\nu ,n\omega ,\vartheta )$ for $\nu =4$, calculated with TDSE and FSH.

Figure 9

Photon- and angle-resolved dissociation probability density $p_d(\nu ,n\omega ,\vartheta )$ for $\nu =9$, calculated with TDSE and FSH.

Figure 10

Photon- and angle-resolved dissociation probability density $p_d(\nu ,n\omega ,\vartheta )$ for $\nu =12$, calculated with TDSE and FSH.

Figure 11

Total dissociation and ionization probabilities as a function of intensity, calculated without (upper panel) and including (lower panel) the focal volume average.

Figure 12

Angle-integrated probability densities for dissociation (upper panel) and ionization (lower panel) as a function of the KER for different intensities.

Figure 13

Ionization probability density $p_i\left(R\right)$ as a function of the bond length $R$ (at the time of ionization).

Figure 14

KER-integrated angular probability densities for dissociation (upper panel) and ionization (lower panel) as a function of the intensity.

Figure 15

Photon- ($n=1,2$ and 3) and angle-resolved dissociation probability densities for different intensities. As done in the experimental analysis (Fig. 10 in Ref. [5]), the angular distributions are fitted by individual ${cos}^k\vartheta$ contributions with $k=2,10$, and 14, respectively.

Figure 16

Experimental (left; adapted from Ref. [7]) and theoretical (right) angle-resolved dissociation KER spectra for different intensities.

Figure 17

Experimental (upper panel, number of counts, adapted from Ref. [7]) and theoretical (lower panel, 1/eV) angle-resolved ionization KER spectra.

Figure 18

(Left column) The FC-averaged angular resolved dissociation KER calculated for the peak intensities given in the plots. (Right column) The FC-averaged angular resolved dissociation KER, focal volume averaged to match experimental peak intensities given in the plots (same as Fig. 16).

Figure 19

The focal volume average weights (calculated for a narrow and a wide ion beam; see inset), given in units of the maximal involved weight (belonging to the lowest intensity bin of beam 1). Equidistant bins $dF=0.001$ are used for the calculation; the bins $dI$ are illustrated by the dashed black line. For both ion beams, the gray area under the curves is normed to 1 (sic). The inset shows the intensity profile (B2) of the laser; some iso-intensity lines (green) are drawn in the pulse area.