Abstract
Strongfield photoelectron holography and laserinduced electron diffraction (LIED) are two powerful emerging methods for probing the ultrafast dynamics of molecules. However, both of them have remained restricted to static systems and to nuclear dynamics induced by strongfield ionization. Here we extend these promising methods to image purely electronic valenceshell dynamics in molecules using photoelectron holography. In the same experiment, we use LIED and photoelectron holography simultaneously, to observe coupled electronicrotational dynamics taking place on similar timescales. These results offer perspectives for imaging ultrafast dynamics of molecules on femtosecond to attosecond timescales.
Introduction
Timeresolved measurements of molecular dynamics have made substantial progress over the past years. In particular, electronic dynamics in atoms and molecules have become accessible through recent developments in attosecond and strongfield science^{1,2,3,4,5,6,7,8,9,10}. This fundamental progress has so far remained restricted to methods relying on spectroscopy, rather than diffraction. In other words, electronic dynamics have been recorded by measuring either the timedependent populations of electronic eigenstates or their accumulated phase differences, which does not provide any structural information about electronic wave functions. However, the interaction of a molecule with a strong laser field naturally provides access to structure by inducing rescattering between the laserdriven photoelectron wave packet and the parent ion. This socalled laserinduced electron diffraction (LIED)^{11,12} has been used to probe the structure of static atoms and molecules^{13,14,15,16,17,18,19,20}, and to obtain evidence of nuclear dynamics following strongfield ionization^{21}. Photoelectron holography, which results from interference between electrons that have scattered from the parent ion and electrons that have not, has also been used to interrogate the structure of static atoms and molecules^{22,23,24}, and to reveal nuclear dynamics following ionization^{25}. Although the imprint of electronic dynamics on photoelectron distributions has been reported for neutral atoms^{26}, it is not obvious whether the distributions can be interpreted in terms of LIED or holography. As far as molecules are concerned, electronic dynamics have so far escaped observation by any of these powerful methods. As LIED and holography rely both on the wave nature of the electron and on rescattering, they are expected to be sensitive to small variations of the molecular structure. Earlier work on electron interference without rescattering^{27,28,29,30} has already revealed the sensitivity to the symmetry of the bound state.
In this study, we transpose photoelectron holography and LIED from static systems to probing coupled electronic and nuclear dynamics in molecules. We first concentrate on purely electronic dynamics and show that a valenceshell electron wave packet leads to a very strong contrast modulation of the holographic fringes. Our calculations trace the origin of this effect to the time dependence of the momentumspace electron wavefunction, which modulates the amplitude ratio and the relative phases of scattering and nonscattering trajectories. We then investigate the manifestation of coupled electronic and nuclear dynamics taking place on similar time scales. We find signatures of both types of dynamics in photoelectron holography, but LIED is found to be almost exclusively sensitive to the nuclear dynamics. These results suggest avenues for disentangling electronic and nuclear dynamics in molecules, which is particularly interesting in the case of nonadiabatic dynamics, such as those occurring at conical intersections^{31}.
Results
Experiment
We use impulsive stimulated Raman scattering to prepare a coupled electronic and rotational wave packet in the neutral NO molecule. A supersonic molecular beam was formed by expanding a 1% mixture of NO with helium through a pulsed Even–Lavie valve. Several rotational states within the ^{2}Π_{1/2} ground electronic state and the ^{2}Π_{3/2} first excited electronic state (separated by 15 meV) are impulsively prepared by a nonresonant laser pulse (800 nm, 145 fs, 4 × 10^{13} W cm^{−2} peak intensity). The long duration of the laser pulse is chosen to avoid the preparation of vibrational wave packets in NO and NO^{+} (periods of 17 and 14 fs, respectively). The wave packet is subsequently probed by strongfield ionization (800 nm, 35 fs, 2.3 × 10^{14} W cm^{−2}) and the photoions and photoelectrons are detected by a velocitymapimaging spectrometer^{32,33}. We also recorded ion timeofflight measurements (Fig. 1a) by adding <1% Xe to the NO/He mixture and the NO^{+} signal was normalized to the Xe^{+} signal, to eliminate fluctuations of laser parameters and gasjet density.
Figure 1a shows the normalized NO^{+} yield as a function of the pump–probe delay. The rapid regular oscillation originates from the electronic dynamics, illustrated in the inset, whereas the two features around 5 and 10 ps correspond to rotational revivals. The amplitude modulations observed around 8–9.5 ps are a characteristic signature of multilevel quantum beats^{34,35}. Figure 1b shows a cut through the threedimensional photoelectron momentum distribution recorded at a delay of 1.56 ps. Details about the processing of the photoelectron momentum images are given in the Methods section. Figures 1c,d show normalized differences, defined as S(t_{1}, t_{2})= 2(D(t_{2})−D(t_{1}))/(D(t_{2})+D(t_{1})) of photoelectron momentum distributions D recorded at the local minimum (delay t_{1}) and maximum (delay t_{2}) of the photoelectron (or NO^{+}) yield that are dominated by either purely electronic or electronic and rotational dynamics, respectively. The corresponding values of t_{1} and t_{2} are given in the caption of Fig. 1. The purely electronic valenceshell dynamics (Fig. 1c) appears as a pronounced modulation of a holographic pattern, whereas the rotational dynamics manifests itself by additionally modifying the angular distribution of rescattered photoelectrons, clearly visible at the highest momenta of Fig. 1d.
Electronic dynamics
We first concentrate on the purely electronic dynamics. Figure 2a (reproduced from Fig. 1c) shows stripes of alternating orange and yellow colours extending from the centre towards the border of the momentum distribution. We attribute these structures to photoelectron holography, previously observed for stationary states of atoms^{22,23,36}. This assignment is substantiated by comparing the observed fringe pattern with the quantitative prediction of holography. Using the vertical ionization potential of NO and the conditions of our experiments leads to the prediction of regions of constructive (black) and destructive (white) interference calculated following the simple model outlined in ref. 36. These fringes arise from the interference between quantum trajectories of (signal) electrons that have scattered forward from the parent ion and (reference) electrons that have passed by the parent ion without scattering. The lack of quantitative agreement between the experiment and the simple model mainly originates from the absence of the Coulomb potential in the latter, as we further discuss below.
The sudden loss of fringe visibility with increasing momentum constitutes further evidence of the holographic origin of the fringes. The green circle in Fig. 2a marks the maximal momentum that can be acquired by electrons that do not scatter. The blue line represents the 1/e^{2} full width of the tunnelling filter, defined by
where p_{x} and p_{y} are momentum components perpendicular to the laserfield direction and , where I_{p} represents the ionization potential and F is the instantaneous field strength at the moment of ionization. The fringes are only observed within the area defined by the green circle, showing that they involve nonscattered electrons. More precisely, we find that the fringes are even limited to the area within the blue curve, showing that the maximal lateral momentum of the electron wave packet is indeed limited by the tunneling filter, lending support to the concept proposed in refs 15, 37. We emphasize that the holographic fringes observed in Fig. 2a are barely visible in the inverted photoelectron momentum distributions themselves (Fig. 1b). They only become visible in the normalized differences. Figure 2b shows how the contrast of the holographic fringes evolves in time over one period of the electronic dynamics. The holographic fringes in the normalizeddifference images modulate with a contrast close to 100%.
A weak signature of the electronic dynamics can also be observed outside the green circle of Fig. 2a, that is, in the region dominated by the electrons that have scattered backwards from the parent ion. We note that the noise along the central vertical line is an artifact of the inversion procedure. The nearly constant value of the difference signal observed in this region shows that the electronic dynamics does not change the angular distribution function of the rescattered electrons. However, the total signal of the rescattered electrons modulates in time with approximately the same contrast as the total signal. We thus conclude that the electronic dynamics in NO causes a variation of the rescattering probability, but it hardly affects the shape of the photoelectron distribution in the rescattering region. In contrast, the shape of the holographic pattern shows a strong signature of the electronic dynamics.
Model for timedependent photoelectron holography
We now introduce a model for timedependent photoelectron holography and use it to explain our observations. Figure 3a shows the temporal evolution of the timedependent orbital describing the unpaired electron of NO. The displayed orbital wavefunction corresponds to one of two possible values of the spin projection quantum number (Σ=+1/2). The orbital corresponding to the opposite value of Σ rotates in the opposite direction^{8} and is omitted here for clarity. As holography results from the interference of scattered and nonscattered trajectories, we focus on the momentum distribution of electrons perpendicular to the direction of tunnelling, also called lateral momentum distribution. We determine this quantity for the timedependent electronic wave packet by building on the concept of partialFourier transformation for strongfield ionization^{38}. The laser field is linearly polarized along the z axis. A cut through the oneelectron density in the xy plane at the distance z_{0}=−1.83 Å from the origin is shown on the front face of each cube in Fig. 3a. z_{0} corresponds to the position of the local maximum of the combined Coulomb and laserfield potentials using the peak electric field strength. The twodimensional Fourier transform of the wave function at z=z_{0}, multiplied by the tunnelling filter (equation 1) leads to the lateralmomentum distribution of the continuum photoelectron wave packet after tunnelling, shown in Fig. 3b:
The corresponding angleaveraged distribution function M(p_{r}, t), shown in Fig. 3c, is defined by
where
and .
The timeevolving lateralmomentum distribution shown in Fig. 3b has a transparent physical interpretation. As time evolves, the orbital wave function of the unpaired electron shown in Fig. 3a rotates around the molecular axis, leading to the periodic appearance and disappearance of a nodal plane containing the direction of tunnelling (z).
We further support the interpretation of the observed holographic fringes by turning to classicaltrajectory calculations^{39} that are described in more detail in the Methods section. We consider the two cases where the oneelectron density is aligned either parallel to the polarization direction of the ionizing laser pulse (t=t_{0}) or perpendicular to it (t=t_{0}+T/2). We propagate classical trajectories in the combined Coulomb and laser field, using the experimental parameters for the latter. The initial momenta sample the distribution function derived above for the highestoccupied molecular orbital of NO obtained from quantumchemical calculations using the GAMESS package. Figure 4 shows the calculated final momentum distributions at the detector for (a) the electron density being aligned along the direction of the ionizing laser polarization, (b) the electron density being aligned perpendicular to this direction and (c) the normalized difference of (a) and (b). The momentum distribution (a) shows the characteristic features of strongfield ionization by a nearinfrared laser pulse, that is, the abovethreshold ionization peaks, which extend up to 2U_{p} and the holographic fringes. The momentum distribution (b) is significantly weaker in amplitude (note the different colour scales), broader in the lateralmomentum direction and additionally displays a node at p_{x}=0. This feature reflects the nodal plane of the molecular orbital, which contains the laser polarization in this specific configuration. This feature is not visible in the experimental data, because the fraction of excited molecules is on the order of 1% (refs 34, 35) and the molecules are not aligned at the considered delays, whereas 100% excitation and perfect alignment was assumed in the calculations.
Figure 4c shows the normalized difference of Fig. 4a,b and is in excellent agreement with the experimental result (Fig. 2a). In particular, the calculation nicely displays all characteristic properties of the experimentally observed holographic structures. Interestingly, the comparison of these calculations with the simple model from ref. 36 shows the same discrepancies as Fig. 2a, that is, the model overestimates the spacing of the holographic fringes along the lateralmomentum direction. The improved agreement between the experiment and our trajectory calculations must therefore originate from the inclusion of the Coulomb potential in the latter.
These complete threedimensional trajectory calculations further show that the electronic dynamics indeed modulate the contrast of the holographic fringes. As the calculations were done using a purely Coulombic potential, the observed timedependent holography can be uniquely attributed to the timedependent momentumspace structure of the outermost orbital of NO as illustrated in Fig. 3. The time dependence of the orbital is translated into a timedependent lateralmomentum distribution of the photoelectron wave packet after tunneling, which results in a high contrast modulation of the holographic fringes. The good agreement between theory (Fig. 4c) and experiment (Fig. 2a) also shows that the molecular properties of the potential seen by the continuum electron are not crucial in defining the observed holographic pattern. This insensitivity may turn out to be an advantageous filter that makes timedependent photoelectron holography specifically sensitive to a particular aspect of the electronic dynamics that it records, that is, the lateral momentum distribution.
Coupled electronic and nuclear dynamics
We now turn to the observation of coupled nuclear and electronic dynamics occurring around a rotational revival. Figure 1a compares the measured NO^{+} signal with calculations of the rotational dynamics of NO. At the early delays discussed so far (1–4 ps), rotational dynamics can be neglected, because the axis distribution does not change over the period of the electronic dynamics. In contrast, electronic and nuclear dynamics occur on similar time scales around a rotational revival, which offers the opportunity to study the manifestation of coupled electronic and nuclear dynamics. Figure 5a compares the photoelectron yield integrated over all momenta around the revival of the rotational dynamics (top) with the extrapolated electronic quantum beat (centre) and the calculated rotational dynamics (bottom) taken from Fig. 1a. These calculations have been done with the methods described in refs 34, 35. They include the rotational levels of both populated electronic states of NO and fully account for the coupled electronic and rotational dynamics on the relevant time scales.
Figures 5b–d show normalized differences of photoelectron momentum distributions S(t_{2}, t_{1}), S(t_{2}, t_{3}) and S(t_{2}, t_{4}), respectively. These delays were chosen to highlight the different manifestations of coupled electronic and nuclear dynamics. The delays t_{1} and t_{2} both lie close to the minimum of , which means that the axis distribution undergoes little change within this time interval. However, the interval [t_{1}, t_{2}] samples a significant fraction of the electronic quantum beat. Between t_{2} and t_{3}, both electronic and nuclear dynamics evolve. Between t_{2} and t_{4}, the electronic dynamics have evolved for half of a period and the rotational dynamics have undergone their maximal evolution, from antialignment to alignment.
We first note that the minimum of the photoelectron and NO^{+} yield (t_{2}) lies between the minima of (close to t_{1}) and that of the extrapolated purely electronic quantum beat (close to t_{3}). This is a first indication that both electronic and nuclear dynamics influence the ionization probability. Much more details are visible in the photoelectron images. We focus our discussion on three aspects of these images. First, Fig. 5b–d all display holographic signatures similar to those of Fig. 2a. These features coincide with the location of the measured main holographic maxima in Fig. 2a, which are reproduced here as black dashed lines. The excellent agreement shows that the structures observed in Fig. 5b–d also correspond to photoelectron holography. These holographic fringes are not unexpected because all three intervals sample substantial fractions of an electronic quantum beat.
Second, each of these six main holographic fringes displays an additional extremum in the radial dimension, which was not observed in Fig. 2a. Comparing Fig. 5b with Fig. 5c,d, we observe a maximum along the radial dimension in the case of Fig. 5b and a minimum in Fig. 5c,d. These radial extrema cannot be explained by purely electronic dynamics because of the differences observed in comparison with Fig. 2a. We note that normalizeddifference images evaluated between delays sampling fractions of the purely electronic quantum beat in the range of 1–2 ps all look qualitatively identical to Fig. 2a (not shown). The additional structures on top of the holographic fringes can therefore only originate from the simultaneous effect of electronic and nuclear dynamics, which is thus found to leave a characteristic imprint on timedependent photoelectron holography.
Third, Fig. 5d additionally shows pronounced structures in the region of the rescattered photoelectrons (outside the green circle). This shows that the distribution shape of the rescattered electrons with energies above 2U_{p} is also modulated, which was clearly not observed during the purely electronic dynamics (compare Figs 2a and 5d). We further investigate this aspect by analysing the rescattered electrons. Outside the green circle in Fig. 5d, alternating regions of negative (blue) and positive (red) colour are observed. We analyse this region using the LIED approach illustrated with a black circle and arrow^{21,40,41}. Electrons that have elastically scattered with the momentum p_{s} in the direction α acquire the additional momentum p_{L}=−A(t_{r}) from the laser field, equal to the opposite value of the vector potential at the moment of rescattering (t_{r}). The value of the normalized difference along such circles is displayed in Fig. 5f and compared with calculations (Fig. 5e). We use the variational Schwinger method implemented in ePolyScat^{42,43} to accurately solve the scattering problem of electrons from aligned NO^{+} molecules. Our calculations also take into account the angledependent strongfield ionization rate of NO^{44} and the molecularaxis distribution corresponding to our experimental conditions. The prediction shown in Fig. 5e qualitatively agrees with the experimental result in Fig. 5f. Both the general shape of the curves and the shift of the minimum to larger angles for increasing scattering momenta are well reproduced. The calculation however overestimates the contrast of the normalized difference, which can originate from the limited accuracy of the electronmolecule scattering calculations and/or the strongfield ionization rates. We have verified that uncertainties in the molecular axis distributions cannot explain the observed discrepancy.
Discussion
We have reported on the observation of valenceshell electron and coupled electronicnuclear dynamics using strongfield photoelectron holography and rescattering. We will separate our conclusions into two parts, that is, conclusions that we expect to be general and conclusions that we view as specific to the broad class of radicals (such as NO, NO_{2}, OH and so on). We have shown that photoelectron holography can be used as a probe of valenceshell dynamics in molecules, and that it is particularly sensitive to the components of the momentum wave function perpendicular to the direction of tunnelling. These components indeed control the relative probability and phase of rescattering relative to nonrescattering trajectories, which directly control the contrast and structure of the holographic pattern, respectively. We have further shown that coupled electronic and nuclear dynamics leave a characteristic imprint on the holographic pattern. This suggests the application of photoelectron holography to probe coupled electronic and nuclear dynamics in polyatomic molecules. We expect these two conclusions to be general. As NO has a single unpaired electron, we expect some of our observations to apply to radicals. Specifically, we have observed that the electronic dynamics leave no signature on the angulardistribution shape of the rescattered electrons (see Fig. 2a), such that LIED provides a virtually backgroundfree measurement of nuclear dynamics. This observation constitutes an experimental demonstration of the fundamental quantummechanical principle that the laserdriven electron wave packet must be described as scattering from its parent ion, rather than from the neutral molecule. In systems featuring a single unpaired electron, such as NO, the parent ion does not display electronic dynamics, resulting in a delayindependent LIED. In the more frequent case of closedshell molecules, the returning electron will scatter off an ion supporting electronic dynamics, such as charge migration^{10}. Our results suggest that in this case, the electronic dynamics of the neutral molecule will be observable in holography, which will however additionally contain signatures of the ionic dynamics (both electronic and nuclear), contributed by the rescattering quantum trajectories. In contrast to holography, LIED will mainly reflect the dynamics of the ion, because the angle dependence of the rescattered electrons is dominated by the elasticscattering crosssection^{41}. More broadly, these considerations show how the fundamental principles uncovered in our pump–probe experiment can be transferred to the regime of subcycle temporal resolution to observe, for example, coupled structural and electronic rearrangements on femtosecond to attosecond timescales.
Methods
Data processing
The twodimensional photoelectron momentum images were processed as follows. A constant background was subtracted from the entire image to account for dark counts and residual stray light. Then, the images were symmetrized. Finally, twodimensional momentum distributions were obtained by iterative Abel inversion using a computer program developed in our group^{33}. A typical photoelectron momentum distribution is shown in Fig. 1b. The momentum distributions shown in Figs 1c,d, 2a and 5b–d are normalized differences between distributions measured at two different pump–probe delays, as defined in the text. The analysis of the LIED signals proceeded as follows. For a given photoelectron momentum at rescattering p_{s}, the corresponding time of rescattering t_{s} was determined using the long trajectories in the classical recollision model (see, for example, ref. 41). The short trajectories were discarded, because they are suppressed by the lower strongfield ionization rate at the corresponding times of ionization. The additional momentum shift p_{L} that the electrons acquire from the laser field after rescattering was calculated according to p_{L}=−A(t_{r}), using the vector potential corresponding to the laser field used in our experiment.
Trajectory calculations
We calculate photoelectron momentum distributions for different times during the electronic wavepacket dynamics using the semiclassical twostep model for strongfield ionization^{39}, which is a MonteCarlo method involving interfering trajectories. Electron trajectories are launched throughout the laser pulse with probabilities corresponding to the instantaneous tunnel ionization rate. As initial conditions we use the lateral momentum distributions derived above (see equation 3) in combination with zero initial momentum along the field and initial electron position at the tunnel exit of a triangular barrier determined by the instantaneous electric field. As the method^{39} is based on Newtonian trajectories moving in a cylindrically symmetric potential formed by the laser field and a Coulomb potential, the simulation can be restricted to trajectories in a plane containing the field direction. However, electrons departing towards opposite sides at the time of ionization may end up with the same final momentum and interfere. Thus, initial conditions for both negative and positive lateral momenta must be defined. The lateral distribution is chosen to be symmetric in both cases that we consider, t=t_{0} and t=t_{0}+T/2. In the former case, the initial phases are taken equal on both sides, but in the latter case a π phase difference is used, in view of the orbital symmetry (see Fig. 3a,b). Similarly, in the former case, the initial phases have a π difference for opposite directions of the ionizing field and they are taken equal for both field directions in the latter case. The laser parameters for the simulation are the experimental ones except that a short sin^{2}shaped envelope with total length of eight optical cycles is used instead of the longer experimental pulses.
Data availabilty
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Additional information
How to cite this article: Walt, S. G. et al. Dynamics of valenceshell electrons and nuclei probed by strongfield holography and rescattering. Nat. Commun. 8, 15651 doi: 10.1038/ncomms15651 (2017).
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Acknowledgements
We gratefully acknowledge funding from the Swiss National Science Foundation (Project Number 200021_138158 and the NCCRMUST), ETH Zürich and an ERC Starting Grant (Project Number 307270ATTOSCOPE). N.B.R. acknowledges support from a Swiss Government Excellence Scholarship. N.S.S. and M.L. acknowledge the support within the QUESTLeibniz Research School Hannover.
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Author notes
 Marcos Atala
Present address: Max Planck Institute for the Structure and Dynamics of Matter, Center for Free Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
Affiliations
Laboratorium für Physikalische Chemie, ETH Zürich, VladimirPrelogWeg 2, HCI E 237, 8093 Zürich, Switzerland
 Samuel G. Walt
 , Niraghatam Bhargava Ram
 , Marcos Atala
 , Aaron von Conta
 , Denitsa Baykusheva
 & Hans Jakob Wörner
Institut für Theoretische Physik, Leibniz Universität Hannover, 30167 Hannover, Germany
 Nikolay I ShvetsovShilovski
 & Manfred Lein
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Contributions
S.G.W. and H.J.W. conceived and designed the experiments. S.G.W., N.B.R. and A.v.C. performed the experiments and analysed the data. S.G.W., D.B. and M.A. developed the model calculations and the rescattering calculations. N.I.S.S. and M.L. implemented the semiclassicaltrajectory calculations. All authors contributed to writing the manuscript.
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The authors declare no competing financial interests.
Corresponding author
Correspondence to Hans Jakob Wörner.
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Optical and Quantum Electronics (2018)
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