Setting the photoelectron clock through molecular alignment

The interaction of strong laser fields with matter intrinsically provides a powerful tool for imaging transient dynamics with an extremely high spatiotemporal resolution. Here, we study strong-field ionisation of laser-aligned molecules, and show a full real-time picture of the photoelectron dynamics in the combined action of the laser field and the molecular interaction. We demonstrate that the molecule has a dramatic impact on the overall strong-field dynamics: it sets the clock for the emission of electrons with a given rescattering kinetic energy. This result represents a benchmark for the seminal statements of molecular-frame strong-field physics and has strong impact on the interpretation of self-diffraction experiments. Furthermore, the resulting encoding of the time-energy relation in molecular-frame photoelectron momentum distributions shows the way of probing the molecular potential in real-time, and accessing a deeper understanding of electron transport during strong-field interactions.

Numerical simulations of the full LIED dynamics have been performed from first-principles within the timedependent density functional theory (TDDFT) [1] framework as implemented in the real-space real-time Octopus code [2]. In TDDFT, the dynamics of an interacting manyelectron system is cast into the manageable problem of a fictitious non-interacting system under the effect of a time-dependent potential such that the non-interacting and the interacting systems have the same time-dependent density.
Since core electrons are expected to play a marginal role in the experiment we consider only valence electrons and account for inner-shell electrons by the effect of normconserving Troullier-Martins pseudopotentials. To obtain a good description of ionisation, we employed a local density approximation (LDA) functional with the average density self-interaction correction (ADSIC) [3], which corrects the asymptotic decay and provides a first and second ionisation energy of 11.65 eV and 15.69 eV, in good agreement with experimental values [4]. During the simulations the nuclei are held fixed in the equilibrium positions, r C-S = 156.1 pm and r C-O = 115.6 pm.
The TDDFT equations are discretised in real-space with a cartesian grid of spacing 0.4 a. u. with a cylindrical shape of radius 50 a. u. and length 260 a. u. aligned along the laser-polarisation direction. The solution of the electron dynamics is obtained by using a discretised real-time evolution with a time step of 0.08 a. u.. The calculations are performed with a 30 fs laser pulse. Complex absorbing boundaries of varying thicknesses, 40 a. u. from the caps of the cylinder and 10 a. u. on the radial borders, are placed at the edges of the simulation box to prevent spurious reflections [5].
The photoelectron spectrum is calculated by collecting the flux of the photoionisation current through a spherical surface of radius 40 a. u. with the tSURFF method [6,7]. This approach gives access to the momentum resolved photoelectron probability I(p) from which, by integrating along the direction perpendicular to the detector, it is possible to obtain the angular distribution of the experiment: I(p X , p Y ) = dp Z I(p).
The non-perfect molecular alignment in the laboratory frame is accounted for by sampling the relative angle θ between the laser polarisation and the molecular axis from 0 • to 90 • in steps of 10 • , as shown in Supplementary Figure 1. This procedure requires a separate simulation for each θ. The photoelectron spectra for a given configuration, parallel or perpendicular, are obtained by averaging the photoelectron distributions I θ (p) with weights n θ (θ − θ /⊥ ) = exp(− sin(θ − θ /⊥ ) 2 /(2σ 2 )), σ 2 = 1 − cos 2 θ 2D , and cos 2 θ 2D = 0.9.
Furthermore, to account for the rotation of the molecule about the polarisation axis for parallel alignment we impose cylindrical symmetry of the photoelectron distribution about Y by averaging over φ:Ī φ (p) = (2π) −1 2π 0 dφ R φ I(p) with the operator R φ of rotation in the X, Z plane. The final spectrum is obtained as follows: To account for experimental-background in the simulations, a constant offset of 2 × 10 −8 was added to the energy distributions, see Supplementary Figure 2. For comparison, the spectra for perfectly aligned configurations are reported in Supplementary Figure 3. We point out that the background correction shifts the numerically obtained cutoffs to lower energy, but does not affect the general behaviour nor the difference of the cutoffs between the parallel and perpendicular configurations.
From the numerical simulations the crucial role of the usually neglected electron-electron interaction for correctly describing the cutoff region in the parallel configuration became evident. Supplementary Figure 4 a shows the decomposed contributions of the Kohn-Sham HOMO and HOMO-1 orbitals, which highlight their distinct contributions to two distinct cutoffs, which are strongly separated in intensity. In particular, the faint 10 U p cutoff for the parallel case actually appears to be uniquely determined by the HOMO-1, which does not have a node along the molecular axis, whereas contributions from the HOMO were strongly suppressed by the presence of a node along the molecular axis, i. e., parallel to the laserpolarisation axis. Second, the independent particle simulation obtained by propagating the system with the Hartree, exchange, and correlation potentials frozen, mimicking the widely used single-active electron model, presents a qualitatively different picture, see Supplementary Figure 4 b. In particular the contribution of the HOMO-1 is highly overestimated and for the parallel alignment the 10 U p cutoff is restored, in clear contradiction with the experiment. These results also confirm the importance of the coherent interaction between different orbitals in strong-field ionisation [8].

SUPPLEMENTARY NOTE 2: SEMICLASSICAL TRAJECTORY SIMULATIONS
Semiclassical trajectory simulations were carried out employing a simplified MO-ADK (molecular-orbital Ammosov-Delone-Kraǐnov) approach, to create the initial wavepacket, in conjunction with a classical continuum propagation in the combined laser-electric and Coulomb field of the cation.

Initial electron wavepacket
In general, the phase-space distribution of the initial electron wavepacket was created in a similar fashion as described before [9]. The ionisation probability in dependence of the instantaneous electric field was obtained through the quasistatic ADK tunnelling theory supplemented by an empirical extension to the barriersuppression regime [10]. Electric-field dependent ionisation potentials, I p ( ), were computed through secondorder perturbation theory assuming that ionisation occurs exclusively from the HOMO, as supported by the TDDFT results. I (0) p is the field-free ionisation potential, ∆ µ and ∆α are the differences of dipole moment and polarizability tensor between cationic and neutral species, respectively. Here, the measured field-free ionisation potential of I (0) p = 11.19 eV [4] was used in combination with calculated neutral and cationic dipole moments and polarizabilities [9]. In Supplementary Figure 5 the resulting time-of-birth distribution of a typical electron wavepacket is shown.
The classical tunnel exit was composed as r 0 = − I p / 2 [11]. The initial momentum distribution in the plane transverse to the electric field vector at the instance of ionisation was modelled according to the atomic ADK tunnelling theory with additional imprint of the initial electronic state's nodal structure. The initial momentum component along the electric field vector was obtained through nonadiabatic tunnelling theory [12]. For the parallel-alignment case the nodal line of the HOMOs along the molecular axis imprints onto the momentum distribution at birth. Accordingly, the initial transverse momentum distribution for the parallel alignment case was described as For the perpendicular alignment case the nodal plane orthogonal to the molecular axis imprints onto the initial momentum distribution. Since it splits the HOMO's electron density unequally with a ratio of 85:15 [9], its imprint was described as a nodal plane with a damped peculiarity with ± representing the integral electron densities on the two sides of the nodal plane. Hence, the distribution of initial transverse momenta for the perpendicular alignment case was set up as Supplementary Figure 6 illustrates exemplary transverse momentum distributions for both alignment cases.

Classical propagation
The electron wavepacket at birth into the continuum was sampled from the initial phase-space distribution through rejection sampling, expanded as a coherent superposition of partial plane waves and, subsequently, propagated in the combined electric laser and the cation's Coulomb field. The singly charged cation was represented by a point charge +e.
The asymptotic electron phase after exposure to the combined electric field of laser and point charge was obtained through [13] At short distances between the electron and the point charge, r < r 2b , the laser-electric field becomes negligible and the description of the electron motion reduces to a two-body problem. Here, the threshold distance, r 2b , was chosen such that the corresponding Coulomb field is 1000 times larger than the peak electric field of the laser. Accordingly, for the experimental peak electric field of 0 ≈ 0.048 a. u. this threshold distance becomes r 2b = 1/ √ 1000 0 ≈ 0.14 a. u.. The motion of the electron within the spherical volume of radius r 2b around the point charge could then be described conveniently as a Kepler orbit. This approximation allows for direct computation of the electron's properties at its symmetric exit point from the sphere: Its position and momentum vector at exit as well as its time of flight between entry and exit of this sphere can be computed fully analytically. The phase accumulated during its passage through the sphere is accessible through low-effort numerical integration. In Supplementary Figure 7 a typical trajectory of an electron is shown as it performs a swing-by around the point charge.     Figure 7. Example trajectory in the combined laser-electric and Coulomb field of a point charge +e (green dot), numerically propagated for r > r 2b (solid red line) and analytically approximated for r ≤ r 2b by means of a pure two-body interaction (dashed red line). By deduction of the orbital centre c and the distance at closest approach, the periapsis rp, the symmetric exit position r1, the corresponding momentum vector p1, and the time spent within the sphere can be obtained analytically.