Strong-field ionization and AC-Stark shifted Rydberg states in OCS

In figure 1 (a) we show intensity-dependent ATI spectra for randomly-oriented OCS from recent experiments [19]. To guide the eye, the spectra are complemented with black, full lines that show how the energy of the ATI peaks would change with laser intensity, assuming the electrons were ionized directly from the HOMO without participation of resonances. The kinetic energy of the photoelectron for each line is calculated using the equation

where $n$ is the number of absorbed photons, $I_{p}$ the ionization potential, and the pondermotive shift $U_{p}=E_{0}^{2}/(4\omega^{2})$. The first ATI peak, observed at laser intensities in the range 30$-$45 TW/cm² corresponds to the OCS molecule absorbing $n=9$ photons (as indicated by the label $n=9$). For the OCS molecule in the intensity range 30$-$60 TW/cm², the $n=9,10,11$, and 12 ATI peaks shift to lower energy upon increasing the laser intensity, suggesting that at these intensities the HOMO electron is directly ionized into the continuum, without contributions from excited states. At laser intensities exceeding 60 TW/cm², however, new features appear in the spectra which have been associated with ionization via excited states. These are labeled peak1, peak2, and peak3 in figure 1 (a) and centered at fixed kinetic energies of 0.7, 1.2 and 2.2 eV. The feature peak1 is clearly visible in the intensity range 60$-$80 TW/cm². The feature peak3 is a progression of peak1 caused by absorption of an additional photon. These features (peak1 and peak3) have been attributed in reference [19] to (6+4)- and (6+5)-photon resonant ionization through the ${}^{1}\Delta$ excited state of OCS: at the considered intensities the HOMO will be excited into the ${}^{1}\Delta$ excited state upon absorption of 6 photons and ionized into the continuum upon absorption of additional 4 photons. Regarding the feature labeled peak2, it is fixed at a kinetic energy of 1.2 eV and, compared with peak1 and peak3, appears in the ATI spectra at higher intensities in the range 80$-$95 TW/cm². This feature (peak2) has been associated with (8+3)-photon ionization through the ${}^{1}\Pi$ excited state of OCS [19]. The energy positions of these resonance-mediated features (peak1, peak2, and peak3) are intensity-independent. In reference [19] this characteristic is explained by assuming that the excited resonance states are AC-Stark shifted by the ponderomotive potential, and therefore their energy follow the ponderomotive-potential-shifted ionization threshold, $I_{p}+U_{p}$. We note that while the ionization threshold and highly-excited states converging to this threshold are AC-Stark shifted by $U_{p}$, it is not to be expected that excited states located as far as $\sim 3,4$ or 5 photons below threshold follow the simple $U_{p}$ Stark shift. Such reasoning could therefore lead to a questioning of the accuracy of the assignment of the relatively low-lying excited states to the REMPI pathways discussed above.

The intensity-dependent ATI spectra of OCS, obtained from our TDSE calculations, are shown in figure 1 (b). The numerical cost of the extensive scan over laser peak intensities prevents a consideration of laser focal volume and orientation effects. Accordingly, we only show results for OCS with the molecular axis aligned with the linear polarization direction, i.e., at fixed orientation angle $\beta=0^{\circ}$, and fixed peak intensities. In spite of these restrictions, we find qualitative similarities between our theoretical results (cf. figure 1 (b)) and the experimental data: at laser intensities in the range 30-60 TW/cm², the ATI peak positions and their shift with laser intensity suggest that the HOMO electron is directly ionized without participation of high-lying resonances, in agreement with the experimental findings of reference [19]. Moreover, features associated with resonant ionization via $U_{p}$ Stark shifted high-lying states, in particular peak1 and peak2, are clearly seen in our theoretical calculations in the same intensity range, 60$-$95 TW/cm², as reported experimentally. The feature peak3 is not well resolved in our TDSE calculations, therefore it will not be discussed here.

Now, a comparison between our theoretical results and the experiment reveal some differences. First of all our SAE model for OCS underestimates the energy of the HOMO: our prediction of the HOMO energy is -0.35 a.u. compared with a literature value of -0.42 a.u., see section 2. This means that we need less photons to ionize than experimentally, and the observed shift in energy explains the difference in the photon-absorption numbers assigned to the ATI peaks in figure 1 following from the direct ionization model of equation (2). Our results show that it takes a minimum of 8 photons to direct ionization from the HOMO orbital. So, the theoretical direct ATI channels in figure 1(b) are labeled by $n=8,9,10,11,12$. Moreover, turning to the resonance features (peak1 and peak2) in figure 1, the energies of these peaks from TDSE calculations seems to be shifted to lower values in comparison with the experiment: peak1 (peak2) is observed experimentally at electron energy of $\sim 0.7$ eV ($\sim 1.2$ eV), whereas TDSE calculations predict this peak at $\sim 0.5$ eV ($\sim 0.9$ eV). A possible explanation for this discrepancy is errors in the energies of excited states of OCS predicted by the SAE model. This is both plausible and consistent with our preceding discussion of the SAE performance for OCS, namely that our estimate of the ionization potential for the HOMO orbital being somewhat lower than the literature value. We will come back to the experimental and theoretical excited-state energy positions in section 3.3 below.

In figure 2(a), we show the experimental 2D ATI spectra of OCS at 400-nm wavelength as obtained from reference [19]. From the figure, one can see 4 ATI peaks resulting from absorption of, respectively, $n=4,5,6$, and 7 photons. In reference [19], analysis of contributions from excited states was only considered for the lowest ($n=4$) ATI peak, it was not considered for the higher order ATI peaks ($n=5,6$, and 7) simply because the ionization yield of these ATI peaks is diminished by going to lower laser intensities. Here, we will follow the same approach and will only focus on the imprint of resonance ionization in the first ATI peak. In figure 2 (a), the electron kinetic energies calculated based on the direct ionization model of eqaution (2) are shown as solid, black lines. According to the direct ionization model, the $n=4$ ATI peak is expected to shift slightly to lower kinetic energy with increasing intensity. However, at the considered intensity range (3.5$-$34 TW/cm²), the ATI peak seems to be fixed at an intensity-independent energy of about 0.9 eV. This ATI feature was labeled peak4 in figure 2 (a) and its energy position is represented by the dashed, black line in the figure. This feature was been attributed in reference [19] to (3+1)-photon resonant ionization via the 4p$\sigma$ $\Pi$ Rydberg state.

In figure 2(b), we show the 2D ATI spectra for OCS as obtained from the present TDSE calculations at a fixed orientation angle of $\beta=0^{\circ}$. The OCS molecule was probed by 400-nm laser light at intensities in the range 3.5 – 35 TW/cm². With increasing intensity, our calculations do not show the higher-order ATI peaks ($n=5$, $n=6$, $n=7$) as clearly as observed experimentally. This is most likely because our calculations were conducted at fixed molecular orientation. The black solid lines indicate the intensity-dependent positions of the ATI peaks in the case of direct ionization, without contribution from excited states, i.e., the positions predicted from equation (2). Our TDSE calculations of the first ATI peak ($n=4$ peak) show that this peak corresponding to peak4 in figure 2(a) is fixed at an energy of 2.1 eV in particular at laser intensities up to 20 TW/cm². We note that in the experimental measurements, the resonance peak (peak4) was observed at a kinetic energy of 0.9 eV. As discussed in the previous section, a possible explanation for this discrepancy is errors in the energies of excited states of OCS as predicted by the SAE model. This aspect will be discussed further in the following.

As can be seen from the experimental results at 800-nm wavelength in figure 1 (a), the resonance features peak1 and peak2, are observed at different intensity ranges. In particular, peak1 is observed in the intensity range 60$-$80 TW/cm² and peak2 in the range 80$-$95 TW/cm². Regarding the experimental results at 400-nm wavelength in figure 2 (a), the resonance feature (peak4) is observed at intensities in the range 5$-$15 TW/cm², see dashed line in figure 2 (a).

In the previous section, we established a qualitative agreement between the theoretical intensity-dependent ATI spectra obtained by our TDSE calculations and experimental results. In particular, the ATI features associated with resonance ionization were clearly identified in our calculations. In the following, we conduct analysis of excited state population based on our TDSE calculations at laser peak intensities where peak2 (at 800 nm) and peak4 (at 400 nm) are observed. To help identify which excited states contribute to ionization at the $\beta=0^{\circ}$ orientation, we show in figure 3 population of the excited states of OCS after probing the molecule by five-cycle (800 nm) and eight-cycle (400 nm) pulses with a peak laser intensity of 9 TW/cm² at 400-nm and 88 TW/cm² at 800-nm wavelength. At $\beta=0^{\circ}$, the laser only couples excited states with the same $\pi$ symmetry as the HOMO of OCS. Notice that we use the lower-case letter $\pi$ to describe the symmetry of the active orbital and we will use the wording ’orbital’ and not ’state’, since the latter is reserved for the many-electron complex. As can be seen from figure 3, when our OCS model is probed by the 800-nm pulse, three excited orbitals of $\pi$ symmetry are populated, with the most prominent at an energy of -0.022 a.u. Notice that ionization from this excited orbital by absorption of a single photon results in the first ATI peak at energy of $E_{kin}=-0.022+\omega~{}\approx 1.0$ eV. The expected $E_{kin}$ value is in agreement with our theoretical calculations of the peak2 position in figure 1(b). Notice that the excited orbitals we identify at 800-nm will ionize by a single-photon absorption. However, in reference [19], the resonance features was assigned to an excited state that would require absorption of additional 3 photons to ionize. This means that this excited-state resonance is located relative far away from the ionization threshold, and it is not obvious that it will satisfy the $U_{p}$-energy shifting assumed in the data analysis.

In the case when our OCS model is probed by 400-nm pulses with a peak intensity of 9 TW/cm², the excited orbital at energy of -0.029 a.u. shifts into 3-photon resonance with the HOMO. Ionization by single-photon absorption from this excited orbital results in the first ATI peak at energy of $E_{kin}\approx 2.2$ eV. The expected $E_{kin}$ value from this analysis is in agreement with our theoretical predictions of the $n=4$ position in figure 2(b). Notice that the results of these analyses are also consistent with the assumption that the AC-Stark shift of the resonantly highly excited orbitals can be well-described by $U_{p}$. Accordingly, the kinetic energies of electrons ionized from these orbitals remain constant upon increasing the laser intensity as was observed for OCS experimentally [19] and as is now confirmed by the present TDSE results in figures 1 and 2.