Nuclear-motion effects in attosecond transient-absorption spectroscopy of molecules

We investigate the characteristic effects of nuclear motion on attosecond transient-absorption spectra in molecules by calculating the spectrum for different model systems. Two models of the hydrogen molecular ion are considered: one where the internuclear separation is fixed, and one where the nuclei are free to vibrate. The spectra for the fixed nuclei model are similar to atomic spectra reported elsewhere, while the spectra obtained in the model including nuclear motion are very different and dominated by extremely broad absorption features. These broad absorption features are analyzed and their relation to molecular dissociation investigated. The study of the hydrogen molecular ion validates an approach based on the Born-Oppenheimer approximation and a finite electronic basis. This latter approach is then used to study the three-dimensional hydrogen molecule including nuclear vibration. The spectrum obtained from ${\mathrm{H}}_2$ is compared to the result of a fixed-nuclei calculation. In the attosecond transient-absorption spectra of ${\mathrm{H}}_2$ including nuclear motion we find a rich absorption structure corresponding to population of different vibrational states in the molecule, while the fixed-nuclei spectra again are very similar to atomic spectra. We find that light-induced structures at well-defined energies reported in atomic systems are also present in our fixed nuclei molecular spectra, but suppressed in the ${{\text{H}}_2}^{+}$ and ${\mathrm{H}}_2$ spectra with moving nuclei. We show that the signatures of light-induced structures are closely related to the nuclear dynamics of the system through the shapes and relative arrangement of the Born-Oppenheimer potential-energy curves.

Figure 1

ATAS spectrum $\tilde{S}(\omega ,\tau )$ [Eq. (9)] calculated for the fixed nuclei model of ${{\text{H}}_2}^{+}$, as a function of the photon energy, for a range of delays between the XUV and NIR pulses. The ATAS spectrum is normalized such that maximum absorption is at $\tilde{S}(\omega ,\tau )=-1.0$. For photon energies between 14.0 and 20.3 eV the ATAS spectrum has been multiplied by a constant factor of 50 to highlight the LIS doublet in that region (see text). The white line in the upper right part of the figure indicates the intensity profile of the XUV pulse. Pulse parameters: $T_{\text{XUV}}=330$ as, $T_{\text{NIR}}=14.0$ fs, ${\lambda }_{\text{XUV}}=50$ nm, ${\lambda }_{\text{NIR}}=700$ nm, $I_{\text{XUV}}=5\times {10}^7 {\mathrm{W}/\mathrm{cm}}^2, I_{\text{NIR}}=2\times {10}^{12} {\mathrm{W}/\mathrm{cm}}^2$.

Figure 2

Normalized ATAS spectrum $\tilde{S}(\omega ,\tau )$ [Eq. (9)], calculated using the moving nuclei model [Eqs. (24) and (25)], for 1D ${{\text{H}}_2}^{+}$, as a function of the photon energy, for a range of delays between the XUV and NIR pulses. The white line to the right shows the intensity profile of the XUV pulse. The pulse parameters are as in Fig. 1.

Figure 3

ATAS spectrum $\tilde{S}(\omega ,\tau )$ [Eq. (9)] (solid, black lines) and field intensity (dashed, red lines) as functions of the photon energy. In the left column pulses with $N_c=1$ in Eq. (13) have been used in the calculations, while pulses with $N_c=3$ have been used in the right column. The central carrier frequencies of the pulses used in the calculations are 6 eV for (a) and (b), 10 eV for (c) and (d), 20 eV for (e) and (f), 30 eV for (g) and (h), and 50 eV for (i) and (j).

Figure 4

Time-dependent dipole moment from the full TDSE calculation (solid, black line) and the approximation $d_{-}^{\left(1\right)}$ of Eq. (36) (dashed, red line) calculated for the 1D model of ${{\text{H}}_2}^{+}$ and a single-cycle 30-nm pulse. The intensity of the pulse is ${10}^7 {\mathrm{W}/\mathrm{cm}}^2$. The light blue dotted line shows a Gaussian fit to the norm of the extrema in the damped dipole moment. The first data point included in the fit is the extremum at $t\simeq 0.7$ fs. The XUV pulse is centered at $t=0.5$ fs.

Figure 5

Normalized ATAS spectrum $\tilde{S}(\omega ,\tau )$ [Eq. (9)] calculated for 3D ${\mathrm{H}}_2$: (a) for fixed nuclei, (b) for moving nuclei. The shaded areas indicate photon energy regions where the spectra are expected to be influenced by the finite electronic basis (see text). The white line to the right indicates the intensity profile of the XUV pulse. The dots and triangles to the very right in (b) are located at the field-free energies of the vibrational states corresponding to the BO curves $B^u$ (black dots) and $C^u$ (black triangles). Pulse parameters: $T_{\text{XUV}}=530$ as, $T_{\text{IR}}=10.7$ fs, ${\lambda }_{\text{XUV}}=80$ nm, ${\lambda }_{\text{IR}}=1600$ nm, $I_{\text{XUV}}=5\times {10}^7 {\mathrm{W}/\mathrm{cm}}^2, I_{\text{IR}}={10}^{12} {\mathrm{W}/\mathrm{cm}}^2$.

Figure 6

Absolute value of the Fourier transform (40): (a) for relatively low frequencies and (b) for higher frequencies. The dashed line in (a) indicates the local energy separation between vibrational states in the $B^u$ curve corresponding to the local vibrational period of Eq. (41). The lines in (b) are $E_i-{\omega }_{\tau }$ [see Eqs. (40) and (42)] when $E_i$ is either the energy of one of the seven lowest vibrational states of the $C^u$ surface (dash-dotted lines), or one of the five lowest vibrational states of the $B^{\prime u}$ surface (dash-dotted lines), or one of the three lowest vibrational states of the $D^u$ surface (dotted lines). See Ref. [30] for information on the energy levels.

Figure 7

Normalized ATAS spectrum $\tilde{S}(\omega ,\tau )$ [Eq. (9)] calculated at $\tau =0$ for (a) ${{\text{H}}_2}^{+}$ and the artificial ${{\text{H}}_2}^{+}$ molecule (see text) with (b) $R_T=0$, (c) $R_T=0.234$ and (d) $R_T=0.625$. The four insets show the BO curves $E_g\left(R\right)$ and $E_d\left(R\right)$. In the insets $R_0$ is indicated by a dashed gray line. The pulse parameters are as in Fig. 1, but with a fixed delay $\tau =0$ between NIR and XUV pulses.

Figure 8

Normalized ATAS spectrum $\tilde{S}(\omega ,\tau )$ [Eq. (9)] calculated for the fixed nuclei model of ${{\text{H}}_2}^{+}$. In the upper panel the Fourier transform of the time-dependent dipole moment is found from a full TDSE calculation [Eq. (14)], while in the lower panel Eq. (A3) was used. The pulse parameters are as in Fig. 1.