Symmetry of molecular Rydberg states revealed by XUV transient absorption spectroscopy

Transient absorption spectroscopy is utilized extensively for measurements of bound- and quasibound-state dynamics of atoms and molecules. The extension of this technique into the extreme ultraviolet (XUV) region with attosecond pulses has the potential to attain unprecedented time resolution. Here we apply this technique to aligned-in-space molecules. The XUV pulses are much shorter than the time during which the molecules remain aligned, typically \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<$$\end{document}<100 fs. However, transient absorption is not an instantaneous probe, because long-lived coherences re-emit for picoseconds to nanoseconds. Due to dephasing of the rotational wavepacket, it is not clear if these coherences will be evident in the absorption spectrum, and whether the properties of the initial excitations will be preserved. We studied Rydberg states of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{2}$$\end{document}2 and O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{2}$$\end{document}2 from 12 to 23 eV. We were able to determine the polarization direction of the electronic transitions, and hence identify the symmetry of the final states.

of the main text shows the induced polarization for the case when the rotational constant of the upper and lower states are different, which is typical of almost all systems. In Supplementary Fig. 2 we show a calculation when B 0 = B 1 = 2 cm −1 . Because both the ground and excited state molecules are "rotating" at the same speed, the signal appears less chaotic. The periodicity of the ground state alignment ( Supplementary Fig. 1) is more apparent. Both Supplementary Fig. 2 and Fig. 4 of the main text show the polarization signal is greatest immediately after excitation for aligned molecules, whereas for anti-aligned molecules, the signal is at first small, then increases as the molecules move into alignment 300 fs later. The overall signal is greater when the probe pulse arrives during the molecular alignment.
This agrees with our experimental result: parallel transition shows stronger absorption when the XUV pulse arrives during a time of maximum alignment to the polarization direction.

Molecular Alignment
When linear molecules are irradiated by a pump pulse whose duration τ = 70 fs is much shorter than the rotational period, nonadiabatic field-free alignment is achieved [1,2]. Each initial rotational eigenstate |J 0 , M 0 will expand to a rotational wave packet Ψ J0,M0 (t) = where T is the rotational temperature and ρ J0,M0 (T ) is the Boltzmann weight function for the |J 0 , M 0 initial state. To calculate the rotational wavepacket created by the pump pulse, we use I = 3×10 13 W/cm 2 , τ = 70f s and a rotational temperature of 100 K. The calculated cos 2 θ (t) − 1/3 is shown in Fig. 2(e) and Fig. 3(e) of the main manuscript.

Absorption Cross Section
The absorption cross section contains contributions from both parallel and perpendicular transitions in the molecular frame, depending on the degree of alignment: where σ (E) and σ ⊥ (E) are absorption cross sections for parallel and perpendicular contributions, cos 2 θ (t) is the alignment degree . For an isotropic medium, cos 2 θ (t) = 1 3 , and so σ(

Bound-Bound and Bound-Free Transitions
We consider two types of dipole-allowed transitions initiated by the XUV: bound-bound and boundfree. As N 2 and O 2 are linear molecules of D ∞h symmetry, with the help of the direct product table of Supplementary reference [4], all the bound-bound transitions are pure parallel or pure perpendicular.
As our XUV spectrum covers several ionization thresholds of N 2 and O 2 , bound-free transitions will induce a continuum absorption background. These bound-free transitions can be of dominant parallel or perpendicular character. The total absorption cross section at different alignments is defined as: The three terms in the summation correspond to bound-bound parallel, bound-bound perpendicular and bound-free transitions, respectively.

N 2 Example
For N 2 , the black line in Supplementary Fig. 3(a) shows the high resolution absorption spectrum adapted from [5]. We get the values of σ m (E) and σ n⊥ (E) from the discrete part of the absorption cross section. Modelling the bound-free part is a bit more complicated. σ c (E) and σ c⊥ (E) are boundfree absorption cross sections for parallel and perpendicular contribution, c can be HOMO, HOMO-1 and HOMO-2 orbitals in our situation. The continuum baseline shown by the magenta line Supplementary   Fig. 3(a) gives the total bound-free absorption cross section. Combining with the ionization ratios for different orbitals [6,7], we can get the partial bound-free absorption cross section: σ HOMO (E), σ HOMO−1 (E) and σ HOMO−2 (E), shown in Supplementary Fig. 3(b). But σ c (E) and σ c⊥ (E) for each orbital are still unknown. In principle one can calculate σ c (E) and σ c⊥ (E), but there are differences according to the computational method used [8][9][10][11]. As these bound-free absorption cross sections usually vary slowly with photon energy, we define a constant ratio

Beer-Lambert Law and Resolution Considerations
The XUV spectrum after it interacts with the macroscopic gas sample was calculated by using Beer-Lambert law: where I 0 (E) is the measured XUV spectrum without absorption gas shown by the black line in Fig. 1(b) of the main manuscript, ρ is the gas density, z is the medium length, σ total (E, t) is the total absorption cross section shown in Supplementary Eq. (6). We determine the gas density by using the smooth continuum part of the measured static absorption of isotropic molecules, and infer ρ(N 2 ) = 1.177 × 10 18 cm −3 and ρ(O 2 ) = 9.77 × 10 17 cm −3 for a medium length of 0.5 mm. The calculated transmitted spectrum is convolved with the experimental spectral resolution to get: where G(E) = e −4 ln(2)( E δE ) 2 and δE = 70 meV.

Differential Absorption Spectrum
For isotropic sample, the total absorption cross section was given by the high resolution synchrotron data, shown by the black line in Supplementary Fig. 3(a) for N 2 . Combining with Supplementary Eq. (7) and Supplementary Eq. (8), we can get the XUV spectrum after transmission through the isotropic sample For aligned sample, the total absorption cross section was calculated by Supplementary Eq. (6), and we can calculate the XUV spectrum after transmission through the aligned sample I on (E, t).
The predicted differential absorption spectrum is calculated in the same way as for the experimental data: Comparing with the measured ∆OD, α HOMO , α HOMO−1 and α HOMO−2 were determined by a fit-

Supplementary Note 3: Residual NIR Driving Pulse Intensity
We estimated the NIR intensity by comparing the static absorption of N 2 measured with an indium filter and with a pinhole (the NIR alignment pulse was blocked in both measurements). As shown in Supplementary Fig. 4, the absorption spectrum were normalized by the peak nearest to the ionization potential (15.58 eV). The residual NIR driving field will shift the absorption peak to higher energy due to Stark effect (the relative peak heights are also different in these two measurements, because the XUV spectrum shape will change after passing through the In filter). The shift amount is ∼ 12 meV, from : U p = e 2 E 2 /4mw 2 and I = cε 0 E 2 /2, where U p is the poderomotive energy, e is the electronic charge, E is the electric field, m is the mass of the electron, w is the pulse frequency, I is the pulse intensity, ε 0 is the permittivity of vacuum. The estimated residual NIR intensity is 2 × 10 11 W/cm 2 . The absorption linewidths in two measurements are similar which also indicate the residual NIR pulse is weak.