Attosecond coherent electron motion in Auger-Meitner decay

Our experimental setup is shown in Fig. 1a. Isolated soft x-ray attosecond pulses from a free-electron laser (12), tuned near the oxygen 1$s\to \pi$ resonance in nitric oxide (NO) ($\sim$530–540 eV), irradiated a gas target in the presence of a circularly polarized, 2.3 $\mu$m, $5\times {10}^{12}$ W/cm${}^2$ laser field. The momentum distribution of the resultant photoelectrons was recorded by a co-axial velocity map imaging spectrometer (c-VMI) (13). Interaction with the x-ray pulse produced electrons from several different photoionization channels: direct ionization of nitrogen $K$-shell electrons, $KLL$ AM emission resulting from the nitrogen $K$-shell vacancy, and resonant oxygen AM emission following O 1$s\to \pi$ excitation. These channels are labeled in Fig. 1b, which shows the electron momentum distribution recorded without the 2.3 $\mu$m laser field. The 1$s\to \pi$ excitation in nitric oxide corresponds to the promotion of an oxygen $1s$ electron to the degenerate $2\pi$ molecular orbital, which is already partially occupied by an unpaired valence electron. The resonant AM emission following this excitation has a dominant feature corresponding to channels where one of the degenerate $2\pi$ electrons participates in the decay, leading to excited cationic states. There is a small contribution from the channel where both $2\pi$ electrons participate, resulting in a $2{\pi }^0$ ground configuration of the cation (14).

The circularly polarized laser field maped the temporal profile of the electron emission on to the momentum measured at the detector. When electrons were released from the molecule following interaction with the x-ray pulse, their trajectory was altered by the presence of the infrared laser field, similar to the principle of a time-resolving streak camera (15, 16). This interaction altered (or ‘streaked’) the final electron momentum, which was measured at the detector. In a semi-classical approximation, the final momentum of an ionized electron is given bywhere $\overrightarrow{A}\left(t_0\right)=-{\displaystyle {\int }_{-\infty }^{t_0}{\overrightarrow{\mathcal{E}}}_L\left(t^{\prime }\right)}d{t}^{\prime }$ is the vector potential of the circularly polarized laser field, ${\mathcal{E}}_L\left(t\right)$, at the time of ionization $t_0$, $e$ is the charge of an electron (1 atomic unit) and ${\overrightarrow{p}}_0$ is the momentum of the electron in the absence of the infrared laser field. All the quantities are expressed in atomic units.

In our measurement the temporal duration of the circularly polarized ‘streaking’ laser field ($\sim 100$ fs) was much longer than the laser period ($T_L=7.7$ fs). This fact implies that over the timescale of a single laser period the vector potential had nearly constant amplitude ($\left|\overrightarrow{A}\right|$) but a direction that rotated with constant angular velocity $2\pi /T_L$. Thus, Eq. 1 describes how the streaking technique encodes the temporal evolution of the electron emission rate onto the electron momentum spectrum: an electron emitted at $t_i$ will experience a momentum shift in the direction of $\overrightarrow{A}\left(t_i\right)$. Because the period $T_L$ of the circularly polarized laser is well known, if two photoemission features are found to have momentum shifts that differ by an amount $\text{Δ}\theta$, this difference implies that the photoemission events were separated by a time $\text{Δ}\tau$:

This mapping of angle-to-time resembles the face of a clock, which has led to the term ‘attoclock’ being used to describe this type of time-resolved measurement (17–19).

Our method for extracting the temporal profile of the AM electron yield is illustrated in Figs. 1d and 1e. Figure 1d shows the variation in the differential electron yield for measurements with three different x-ray arrival times, or directions of the streaking laser vector potential,${\overrightarrow{A}}_0$. The differential images show the difference between the averaged electron image when the vector potential of the IR laser was chosen to lie along the line labeled ${\overrightarrow{A}}_0$ on the figure (and labeled as ‘$0$ fs’ on the clock face) and the averaged electron image where the IR laser has been intentionally mistimed with the x-rays to ensure there was no effect from the streaking field. To extract the time-dependent emission rate of resonant AM electrons we monitored a small angular region on the detector (black wedge in Fig. 1d) and plot this yield as a function of streak angle, or the angle between the observation bin and the streaking laser vector potential, in Fig. 1e. The observation region was chosen to be slightly higher in momentum than the center of the field-free resonant emission spectrum shown in Fig. 1b. The electron yield in this radial bin therefore mapped to the number of electrons released into the continuum at the time the vector potential ${\overrightarrow{A}}_0\left(t\right)$ is pointed in the angular direction of the observation region. The lower momentum limit of this region is $\sim$6.4 atomic units and the upper limit extends to include all electrons at higher momenta. Equation 2 can be used to convert the streak angle into a time delay, and this value is used to label the clock face in Fig. 1d and the lower horizontal axis in Fig. 1e.

At the LCLS, the synchronization of the streaking laser and x-ray pulse has a jitter of roughly $\sim 500$ fs (20), which is orders of magnitude below the required precision for directly timing the AM process. Thus in order to produce the images shown in Fig. 1d we must employ a single-shot diagnostic of the relative arrival time between the x-rays and laser pulse. As described above, in addition to driving resonant excitation near the oxygen $K$-edge, the attosecond x-ray pulse ionized electrons from the nitrogen $K$-shell of the NO molecule (see Fig. 1b and 1c). This direct photoionization process produced high energy ($\sim 120$ eV) electrons. The photoionization delay between the arrival of the x-ray pulse and the appearance of these fast photoelectrons in the continuum was negligibly small ($\lesssim$ 5 as) compared to the streaking laser period $T_L$ of 7.7 fs (21–23). Therefore the momentum shift observed for the nitrogen $K$-shell photoemission feature provided an accurate, single-shot measurement of the direction of the streaking laser vector potential ${\overrightarrow{A}}_0$ at the time of arrival of the x-ray pulse.

We monitor the AM yield in a small angular region of the detector to avoid introducing artifacts in the extracted time-dependent trace due to angular anisotropy in the AM emission (24). We note that the period of the streaking field was chosen to be longer than the dominant timescale of the AM process. This fact simplifies interpretation of the streaking measurement by limiting the effect of ‘wrapping’, where electrons released into the continuum at time $\tau$ and $\tau +T_L$ experience a similar momentum kick from the streaking field.

The time-dependent electron yield shown in Fig. 1e shows a maximum at $\tau$=0, when ${\overrightarrow{A}}_0$ was directed along the detection direction, and the the core-excited population (and AM emission rate) was at a maximum. In addition to an exponentially decaying electron emission rate, we observed a revival in the time-dependent emission rate at $\tau$=3.5 fs.

We modeled our measurement according to the theory of attosecond streaking of multiple Fano resonances described by Wickenhauser et al. (25, 26). Our model, illustrated in Fig. 2a, included a ground state which is doubly degenerate and was resonantly coupled to four bound states, one of which (${}^2\text{Δ}$) is also doubly degenerate, and thus is labeled as a single state in the figure. These bound states were also coupled to a single, structure-less continuum, which was dressed by the circularly polarized, $2.3\mu$m streaking laser field. The coupling between the bound and continuum states was the result of electron correlation interactions, and drove the AM decay process. The bound states had excitation energies of $531.5$ eV (${}^2{\text{Σ}}^{-}$), $532.6$ eV (${}^2\text{Δ}$) and $533.5$ eV (${}^2{\text{Σ}}^{+}$), which represented the core-excitation spectrum of nitric oxide (27). The continuum coupling constant ($\text{Γ}=170$ meV) was consistent with previous x-ray absorption measurements (27). The relative amplitude between transitions to the bound, core-excited states and direct photoionization of valence electrons to the continuum was represented by the Fano parameter, $q_i$ (see SM) (28). We choose the value for $q_i$ according to the measured absorption spectrum of NO (27).

The coherent bandwidth of the exciting x-ray source was $\sim 5$ eV (12), which was sufficient to span all core-excited bound states in the model. Symmetry constraints do not allow for the coherent population of the ${}^2{\text{Σ}}^{-}$ and ${}^2{\text{Σ}}^{+}$ states since the ${}^2{\text{Σ}}^{-}$ and ${}^2{\text{Σ}}^{+}$ states each coupled to a different component of the doubly degenerate ground state (14). Moreover, each component of the ground state coupled to a different component of the degenerate ${}^2\text{Δ}$ state (14). Thus the model only included coherence between the ${}^2{\text{Σ}}^{-}$ & one of the ${}^2\text{Δ}$ states and the ${}^2{\text{Σ}}^{+}$ & the other ${}^2\text{Δ}$ state, but not the ${}^2{\text{Σ}}^{-}$ & ${}^2{\text{Σ}}^{+}$ states. In both simulation and experiment we tune the central wavelength of the x-ray source across the $1s\to 2p\pi$ resonance (red, green and blue shaded curves, bandwidth drawn to scale with energetic separation of core-excited states).

In simulation we could calculate the energy-resolved continuum wavefunction in the absence of the streaking-field, shown in Fig. 2d, demonstrating the build-up of resonant features. The rate of electron emission (integrated over electron kinetic energy) is shown in Fig. 2e), and we clearly observe an oscillatory emission rate. Finally, in Fig. 2f we show the population of each core-excited state as a function of time, which again shows oscillatory behavior. The periodic modulation of the electron emission rate resulted from the coherent population of the two pairs of excited states ${}^2{\text{Σ}}^{-}$ & ${}^2\text{Δ}$ and ${}^2\text{Δ}$ & ${}^2{\text{Σ}}^{+}$. Electronic coherence between the pairs of excited states resulted in consecutive minima (maxima) in time-dependent ionization rate, due to destructive (constructive) interference between emission from the core-excited states. Because the core-excited wavepacket consisted of states with different angular momentum projections along the molecular axis, the excited state wavepacket produced an excited electron density which ‘rotated’ around the molecular axis, as shown in Fig. 2c.

We directly modeled our experimental observable by computing the asymptotic ($t\to \infty$) momentum distribution of ionized electrons within the Strong-Field Approximation (SFA) (26) (Fig. 2b) and performing the same analysis routine as we applied to the experimental data. The asymmetry parameters describing emission from the oxygen $1s{\to }^2{\text{Σ}}^{-}$, O $1s{\to }^2\text{Δ}$ and O $1s{\to }^2{\text{Σ}}^{+}$ excitations were expected to be different for each of the electronic states (24) and have not previously been measured, meaning the contribution of each channel to emission in the direction of our observation window was not well defined. We fit the simulation to the experimental data using the lower kinetic energy limits of the small detector region defined in Fig. 1e, and the relative contribution from each decay channel at the precise region on the detector, as free parameters. With a separate measurement taken concurrent with the presented data, we determined an error distribution of $\sigma$=30${}^{\circ }$ for single-shot vector potential determination. We accounted for this experimental error by convolution of the time-dependent electron yield with a Gaussian kernel of $\sigma$=30${}^{\circ }$. Further details are provided in the SM. We also accounted for the possibility of a small systematic error in $t_0$ determination between experiment and theory, resulting from the finite temporal profile of the reference nitrogen $K$-shell photoline produced by the attosecond x-ray pulse (12). We identified an offset of $\sim$1.7% of the full detector angle.

Figure 3a shows the vector potential-direction dependent electron yield measured at different x-ray excitation energies (black dots) compared with the simulated yield (solid line). The transient revival at $\tau \sim 3.5$ fs resulting from electronic coherence in the core-excited state was indicated by the gray arrow and is observed in both experiment and simulation. This feature is a quantum beat, occurring at the moment when the quantum phases of the coherently excited ${}^2{\text{Σ}}^{-}$ and ${}^2\text{Δ}$ excitations re-aligned. This alignment caused constructive interference between the two core-excited states, and an increase in AM emission rate. The feature at $\sim$1.3 fs measured at central photon energy 536 eV was possibly due to the temporal build-up of the Fano interference between the resonant and direct excitation channels and has been qualitatively reproduced in further simulation. Analysis of the energetic positions of the Rydberg series converging to the oxygen $K$-edge (27) was not consistent with the interpretation that this modulation was due to further coherent excitation involving Rydberg states.

Figure 3b shows a zoom-in of the revival feature. By tuning the central x-ray photon energy away from the center of the $1s\to 2p\pi$ resonance, we were able to suppress the quantum beat in both experiment (left) and in simulation (right), demonstrating control over the coherent evolution of the core-excited states. The beat was suppressed at higher photon energy due to an increased relative contribution from the direct channel versus the coherently excited resonant decay pathways. In Fig. 3c we compare our measurement, for a central x-ray excitation energy of 533 eV, to our simulation including (deep red) and excluding (pale red) the coherence between the core excited states. As expected, it was only possible to reproduce the revival features by including the coherence between the different core-excited electronic states. The result from incoherent summation of the AM emission from the different core-excited states fails to reproduce the feature at ${150}^{\circ }$ streaking angle.

In conclusion, this work reports the real-time measurement of electronic coherence in the temporal evolution of a core-excited molecule. Electronic coherence imparted a modulation in the time-dependent emission rate of AM electrons, driven by an isolated attosecond soft x-ray pulse from a free electron laser. The AM emission occurred on a few-femtosecond timescale and we time-resolved it using angular streaking. Our measurement provides a testbed for exploring the effect of electronic coherence in the photoexcitation dynamics and subsequent photochemical behavior of molecular systems. The existence of this electronic coherence provides the opportunity to explore inter-atomic site electronic wavepacket coupling, which can reveal interactions between different parts of an extended system (29–31). Measuring this coupling can reveal important information on the system’s fundamental physical properties (32, 33). For example, the spectral makeup of the observed modulations provides rich information on the composition of the excited superposition state. This information opens the possibility to observe the evolution and decay of coherent electronic states in real-time, as they evolve and couple to subsequent nuclear motion in the first stages of a photochemical reaction (34–37).

Funding: S.L., Z. Z., and A.M. acknowledge support from DOE, BES Scientific User Facilities Division Field Work Proposal 100317; J.D. and A. M. were supported by the Laboratory Directed Research and Development Program in support of the Panofsky fellowship. The contributions from T.D., P.H.B., A.K., A.N., J.T.O., T.J.A.W., A.L.W., and J.P.C. were supported by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences (BES), Chemical Sciences, Geosciences, and Biosciences Division (CSGB); E.G.C. was supported by the DOE Laboratory Directed Research and Development program at SLAC National Accelerator Laboratory, under contract DE-AC02-76SF00515. P.R. and M.F.K. acknowledge support by the German Research Foundation via KL-1439/10, and the Fellow program of the Max Planck Society. V.A, J.C.T.B., D.G., J.P.Mar. gratefully acknowledge funding support from UK EPSRC grants No. EP/R019509/1, EP/T006943/1 and No. EP/ I032517/1. N.B., R.O. and A.C.L. acknowledge the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, US Department of Energy, grant no. DE-SC0012376. C.B. acknowledges the Swiss National Science Foundation and the National Center of Competence in Research – Molecular Ultrafast Science and Technology NCCR – MUST. L.F.D. and L.F. acknowledge support from NSF Grant No. 1605042 and DOE DE-FG02-04ER15614. W.H. thanks the German BMBF for funding of the project ‘SpeAR_XFEL’ under the contract number 05K19PE1. Use of the Linac Coherent Light Source (LCLS), SLAC National Accelerator Laboratory, is supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DE-AC02-76SF00515. Author contributions: SL, AM and JPC devised the experimental scheme. S.L., A.M., J.P.C. developed the experimental apparatus. S.L., J.D., J.P.Mac., Z.Z., and A.M. prepared the attosecond X-ray pulses. M-F.L., N.H.S., P.W. prepared the experimental beam-line. All Authors participated in the collection and interpretation of the experimental data. T.D. led the data analysis. S.L., T.D., P.R., and E.G.C. worked on the single-shot ‘streaking’ diagnostic. S.L., T.D., A.M., and J.P.C. prepared an initial version of the manuscript. All authors provided critical feedback in preparing the submitted manuscript. Competing interests: None declared. Data and materials availability: The partially analyzed raw data and the raw data from the calculations is available on the Zenodo repository (38). All (other) data needed to evaluate the conclusions in the paper are present in the paper or the Supplementary Materials.