Observation of the ponderomotive effect in non-valence bound states of polyatomic molecular anions

The ponderomotive force on molecular systems has rarely been observed hitherto, despite potentially being extremely useful for the manipulation of the molecular properties. Here, the ponderomotive effect in the non-valence bound states has been experimentally demonstrated, for the first time to the best of our knowledge, giving great promise for the manipulation of polyatomic molecules by the dynamic Stark effect. Entire quantum levels of the dipole-bound state (DBS) and quadrupole-bound state (QBS) of the phenoxide (or 4-bromophenoxide) and 4-cyanophenoxide anions, respectively, show clear-cut ponderomotive blue-shifts in the presence of the spatiotemporally overlapped non-resonant picosecond control laser pulse. The quasi-free electron in the QBS is found to be more vulnerable to the external oscillating electromagnetic field compared to that in the DBS, suggesting that the non-valence orbital of the former is more diffusive and thus more polarizable compared to that of the latter.

Photodetachment spectra of 4-CP QBS in various pump-control pulse delay at ~10 GW/cm 2 pump laser intensity. Source data are provided as a Source Data file. Fig. 2 Photodetachment spectra of 4-CP QBS in various control pulse intensity at ~10 GW/cm 2 pump pulse intensity. Source data are provided as a Source Data file.

Details of the subtraction of direct detachment in photodetachment spectra
In the photodetachment spectrum, the direct detachment by one-photon transition reveals stepwise feature at electron affinity (EA) as seen in Fig. 1. On the other hands, the stepwise feature from the direct detachment also could be revealed near the most FC-active vibrational state of the DBS or QBS 1 . Since the direct detachment cross-section usually follows the FC-factor, stepwise feature could be revealed when the excitation pulse exceeds the energy of the most FC-active vibrational mode in neutral electronic ground state.
Because of the loosely bound nature of the DB-or QB-electron, the geometry of the DB or QB is barely differed from the neutral ground state geometry 2 . Those makes the stepwise feature located to the blue-side of the most FC-active DBS vibrational peak, shifted with the almost same amount of that of EA. For the DBS or QBS with low binding energies, the vibrational peak of DBS or QBS could be overlapped to the stepwise direct detachment.
This makes the vibrational peak to be more asymmetric to the blue side, consequently makes hard to exact estimation of the extent of ponderomotive shift and broadening. For the 4-CP -QBS and 4-BP -DBS (to be referred below), which have small electron binding energy similar to the picosecond bandwidth (~20 cm -1 ), the most FC-active vibrational peaks have quite asymmetric peak shapes because of aforementioned stepwise feature overlapped to the vibrational peak. We carefully eliminate this feature by fitting in order to evaluate the extent of the ponderomotive shift more precisely. We used error-function to represent the stepwise direct detachment; Eq. S1 To be specific, the direct detachment cross-section is usually expressed by the Wigner's threshold law, which is proportional to the √eKE. However, the Wigner's threshold law could not represent the vibrational aspect of the direct detachment cross-section as mentioned above, rather usually being adopted to the 'electronic' detachment cross-section. In the very narrow energetic range compared to the 'electronic' detachment cross-section, error-function could be reasonable to represent the stepwise feature originated from the vibration.
Supplementary Fig. 3 (a) is the extracting procedure of the 4-CP -QBS 12' 1 vibrational peak from the direct detachment. We subtracted error function in two points: EA and EA+E(v12). This eliminates the stepwise direct detachment features at the EA and the most FCactive vibrational state. Resultant photodetachment spectra are described in Supplementary   Fig. 3 (b). Note that the pump only spectrum has quite symmetric structure after the subtraction of the direct detachment, and the shift and broadening are more clearly observable when the various control pulse intensity was used ( Supplementary Fig. 3 (c) and (d) Notably, as the vibrational Feshbach resonance bands quite stand out compared to the direct-detachment background electron signal, the observed ponderomotive shift in the 0 -30 cm -1 range is little influenced by the subtraction of the direct-detachment within the error limit of ± 2 cm -1 .
Supplementary Fig. 4 Stepwise direct detachment feature-subtracted spectra of 4-CP QBS and their Gaussian fitting at (a) pump only (without control pulse) and (b-h) with control pulse in various intensities.

Measurement of the laser intensity
In order to evaluate the laser intensity on the laser-ion interaction region, we measured laser power before and after the picosecond pulse passes through a couple of the CaF2 windows. From the measured power differences, we could calculate the laser power at the laser-ion interaction region by considering the transmittance of a couple of CaF2 windows.
Eq. S2 Beam waist at the laser-ion interaction region was calculated using following equation; , where z is the position of the laser-ion interaction region from the focus, ω0 is the initial beam waist, and λ is the wavelength of the laser. Because of the low number density of the anion in the ion packet, we loosely focused control laser pulse away from the focal point. The peak power density of each laser shot was calculated by assuming the 1.7 ps pulse width Gaussian beam with 1 kHz repetition rates at 1/e 2 (~13.5 % of peak) beam diameter.
Eq. S4 . Supplementary Fig. 5 Picosecond pump pulse-only photodetachment spectra of the PhO -DBS at low intensity (~10 GW/cm 2 , red) and high intensity (~90 GW/cm 2 , black) near the 11' 1 resonant peak. Source data are provided as a Source Data file.
Supplementary Fig. 6 Ponderomotive shifts of the 11' 1 peak of the PhO -DBS are plotted versus the intensity of the ps control laser pulse at high pump intensity (~90 GW/cm 2 , red) and low pump intensity (~10 GW/cm 2 , orange). The slope of the low pump intensity trend line was caluclated to be (0.274 ± 0.06) η, which is within the error bar obtained from the high pump intensity, (0.260 ± 0.05) η. The error bars represents the experimental errors (± σ) determined from the multiple measurements (n > 10) of the photodetachment spectrum. Source data at low pump power are provided as a Source Data file.