Abstract
The electrooptical properties of most semiconductors and insulators of technological interest are dominated by the presence of electronhole quasiparticles, called excitons. The manipulation of excitons in dielectrics has recently received great attention, with possible applications in different fields including optoelectronics and photonics. Here, we apply attosecond transient reflection spectroscopy in a sequential twofoci geometry and observe subfemtosecond dynamics of a corelevel exciton in bulk MgF_{2} single crystals. Furthermore, we access absolute phase delays, which allow for an unambiguous comparison with theoretical calculations. Our results show that excitons surprisingly exhibit a dual atomic and solidlike character, which manifests itself on different time scales. While the former is responsible for a femtosecond optical Stark effect, the latter dominates the attosecond excitonic response. Further theoretical investigation reveals a link with the exciton subfemtosecond nanometric motion and allows us to envision a new route to control exciton dynamics in the closetopetahertz regime.
Introduction
The quest for new devices capable of surpassing the current technological limits^{1} has pushed the scientific community to explore solutions beyond classical electronics as done in excitonics, spintronics and valleytronics^{2}. Therefore, studying the dynamics of excitons in solids^{3,4,5} becomes a priority task not only to widen our knowledge of fundamental solidstate dynamical phenomena, but also to explore the ultimate limits of these novel technologies^{6,7,8,9,10,11}. While the development of attosecond spectroscopy^{12} has proven the possibility to study subfemtosecond (fs) electron dynamics in solids^{13}, shedding light onto strongfield phenomena and lightcarrier manipulation^{14,15,16}, a clear observation of attosecond exciton dynamics was missing. Besides more conventional femtosecond techniques^{17}, attosecond transient absorption and reflectivity spectroscopy have been employed to study the ultrafast decay processes (fewfs) of coreexcitons^{18,19}, but failed in recording the subcycle dynamics unfolding during light–matter interaction. Here we used attosecond transient reflection spectroscopy (ATRS)^{20} to study attosecond dynamics of a corelevel exciton in bulk MgF_{2} single crystals, a widely used material in optics, characterised by a clear core^{21} and valence^{22} exciton signal which make it an optimal target to investigate exciton dynamics with attosecond techniques. Thanks to the employment of simultaneous and independent calibration experiments, we achieved direct comparison with theoretical simulations, which in turn allowed us to make a clear link between the observed transient features of the system optical response and the nanometric and attosecond motion of the excitons. Holding in general for WannierMott excitons, including valence excitons, our results provide a description of the ultrafast exciton–crystal interaction on subfs time scales and move an important step towards a more complete understanding of attosecond lightexcitonic dynamics.
Results
Figure 1a shows a schematic picture of the experimental setup characterised by a sequential twofoci geometry, used to perform simultaneous attosecond photoelectron and ATRS measurements^{23} (see Methods). The static reflectivity for a MgF_{2} (001) crystal, R_{0}, close to the Mg L_{2,3} edge is probed by an extremeultraviolet (XUV) attosecond (as) pulse (Fig. 1b). In this energy region R_{0} is characterised by a peak (labelled with A), which has been attributed to the formation of excitons after excitation of a Mg^{2+} 2p core electron^{24} (Fig. 1c). The weaker satellite peak at about 54 eV (A′ in Fig. 1b) originates instead from the spin–orbit splitting of the Mg^{2+} 2p core state^{25}. In our experiment we use a 5 fs infrared (IR) pulse (centre wavelength 750 nm, peak intensity 10^{12}–10^{13} W/cm^{2}) to drive the crystal out of equilibrium (Fig. 1c). The induced ultrafast exciton dynamics are probed with 250as pulses by monitoring the sample differential reflectivity ΔR/R in the XUV range, defined as the difference between pumped and unpumped reflectivity, divided by the latter: [R_{IR} − R_{0}] /R_{0}. The results as a function of photon energy E and pumpprobe delay t, are shown in Fig. 1d. At small values of t, we observe rich transient features which unfold either on fewfs or attosecond time scales (hereafter referred to as slow and a fast component, respectively). While the former is mainly located around the excitonic features A and A′, the latter oscillates at twice the IR frequency and extends over the full energy range under consideration, becoming more evident in the conduction band (CB) region. The weak differential signal already present at negative delays around 54.4 eV originates from a nonvanishing IR electric field. It goes below the experimental noise level for delays smaller than −15 fs and does not influence our analysis (see Supplementary Note 6). The upper panel in Fig. 1d shows the square of the IR vector potential, A_{IR}^{2}, as retrieved from the simultaneous streaking trace (see Supplementary Note 7). Knowing the associated pump field time evolution, we calibrated the pumpprobe delay axis in order to have the zero delay coinciding with the maximum of the IR electric field squared (equivalently, a zero of A_{IR}^{2}). This allowed us to set an absolute reference for our measurements and study the precise timing of the system dynamics.
To reach a complete understanding, we calculated the quantum dynamics with a WannierMott (WM) exciton model^{26,27} (see Methods) and chose the same reference for the delay zero, such that we could directly compare experimental and theoretical results. The calculated ΔR/R reported in Fig. 1e accurately reproduces the experimental data of Fig. 1d, suggesting dynamics which go beyond the optical Stark effect^{28}. While the calculated and experimental ΔR/R fall in the same order of magnitude, there are appreciable differences in their absolute values which may originate mainly from an underestimation of the pump intensity. In addition, we observe that the calculations overestimate the oscillatory part in the CB with respect to the slow component. This may originate from a combination of unavoidable experimental imperfections (any instability will reduce the contrast of the fast oscillation) and the approximations made in our theoretical model. The observed discrepancies have anyway no effect over the Results and Discussion reported in this paper.
Discussion
The slow component of the experimental ΔR/R (Fig. 2a) is characterised by a series of positive (red) and negative (blue) features, which develop around delay zero and fully disappear within 10–15 fs. These features originate mainly from IRinduced optical Stark effect of the excitonic transitions, and the presence of dark excitonic states which become optically active around zero pumpprobe delay^{19}. The exciton optical Stark effect is the analogous of the wellknown optical Stark effect in atoms where absorption or emission of nonresonant photons can lead to a shift of the atomic energy levels^{29}. In analogy with a photondressed twolevel system, the optical Stark effect originates from the coupling between the bright and dark excitonic states. Since the photon energy (~1.6 eV) is bigger than the spacing between the two states (~1.3 eV), the associated transition is redshifted, which translates in a blue shift of the Mg 2p—bright exciton transition.
Figure 2b, c shows the calculated slow component of ΔR/R considering only the excitonic states reported in Fig. 1c (“pure excitonic” contribution) or only the crystal CB (“pure crystal” contribution) (see Methods). We find that “pure exciton” calculations qualitatively reproduce the main experimental features while the crystal response does not (Fig. 2d).
Since in our pureexciton model the exciton is described with atomiclike states, solidlike interactions with the IR field (e.g. intraband motion) are not possible. Therefore, the good agreement between the calculations of Fig. 2b and the experimental data of Fig. 2a supports the attribution of the origin of the slow component of ΔR/R to the optical Stark effect and shows that the fewfs dynamics of the optical response can be understood, on first approximation, by considering only the atomiclike character of the exciton quasiparticle.
From the delaydependent optical Stark shift, it is possible to study the exciton decay process^{19}. We found the bright exciton to decay rather quickly with an effective time constant of 2.35 ± 0.3 fs (see Methods), comparable to what observed for other insulators like SiO_{2}^{18} and MgO^{19}. A deeper analysis, which could reveal the interplay between Auger decay and phonon coupling, needs either measurements with a shorter IR pulse, or more complex reconstruction procedures, both of which go beyond the scope of this work.
While the fact that some optical properties can be described by the exciton atomic character is typical for a quasiparticle characterised by a binding energy of 1.4 eV^{21}, the analysis of the subfs dynamics reveals an unexpected result. The fast component of the transient reflectivity spectrogram is reported in Fig. 3a, showing clear oscillations at twice the IR frequency, which are fully reproduced by our simulations (Fig. 3b). Both the amplitude and the tilt of the oscillations cannot be reproduced considering solely the exciton contribution (Fig. 3c). These oscillations have a Vshaped dispersion which resembles what is found for the dynamical FranzKeldysh effect (DFKE) high into the CB of diamond^{30,31}, suggesting a clear link with intraband motion of virtual charges. We note that the interaction regime significantly differs from DFKE in valence excitons exposed to THz pulses^{32,33} where the pump field evolves on a much slower time scale. Furthermore, even if the lower 2p state is nondispersive, the energy of the electron–hole particle follows the upper CB state and exhibits a parabolic profile. This explains why a Vshaped structure is observed even when the initial state is not a dispersive valence state, but an atomiclike core level.
Thanks to our independent pumpprobe delay calibration, we can make a further step towards a complete comprehension of such a rich exciton dynamics and study the phase delay τ between the oscillations in the transient signal and the square of the IR electric field E_{IR}^{2} (see Methods). Figure 3d presents the experimental τ obtained as a weighted average over four independent measurements (black solid curve) compared with the calculated phase delay in case of full model (red solid curve), pure exciton (orange dotted curve) or purecrystal response (dashed grey curve). In all cases, the shaded area represents twice the standard deviation originating from the measurement error or from the uncertainty of the phase extraction method. The full model accurately reproduces the experiment both on a qualitative and quantitative level. However, in contrast to what we observed for the slow component of ΔR/R, now the pure excitonic contribution alone fails in capturing the measured τ, even qualitatively. This strongly indicates that the fast component of the differential reflectivity is dominated by the solid nature of the exciton, despite its localised character and in stark contrast to its slower atomiclike response. Indeed, the oscillations found in the pure exciton model (Fig. 3c) originate from an atomic effect: the instantaneous optical Stark shift. On the contrary, the oscillations observed in the purecrystal model arise from the intraband motion of virtual carriers (DFKE). In principle, real carrier motion at the CB bottom cannot be neglected^{31}. Nevertheless, as observed in GaAs crystals^{21}, while the total carrier injection rate into the CB is considerably affected by the interplay of inter and intraband transitions, the phase of the ultrafast optical response is mainly dictated by the motion of the virtual carriers. Therefore, we conclude that the latter plays a major role in sculpting the energy dispersion of the oscillation phase reported in Fig. 3a. The fact that the full system response (red curve in Fig. 3d) resembles the bare crystal case (dashed grey curve in Fig. 3d) but shifted in energy, suggests the following physical interpretation. The fewcycle IR pulse dresses the crystal CB inducing virtual charges intraband motion (DFKE). In turn, this alters the exciton dynamical properties causing the quasiparticle to oscillate in the IR field with a phase relation similar to that of the bare crystal.
If the proposed picture is correct, we expect the observed transient features in the optical response to correspond to an actual movement of the exciton on attosecond (as) and nanometric scales. To tackle this, we calculated the excitonic dipole in real time and found that it oscillates almost at the same frequency of the IR field during interaction. The results are shown in Fig. 4a for the case of the full system (red curve) or considering only the quasiparticle (orange curve). By evaluating the phase delay for the dipole with respect to the IR field (black dotted curve), we found the oscillations of the pure exciton dipole to be delayed by 252 ± 69 as with respect to the full system response (square marks in Fig. 4b). Remarkably, a similar shift of 272 ± 57 as is observed between the oscillations of ΔR/R evaluated at the energy of the excitonic transition and calculated with the full model or considering only the excitonic contribution (full circles in Fig. 4b). Therefore, our findings suggest a strong correlation between the subnm exciton motion in real space and the subfs transient features observed in the differential reflectivity. This not only proves that localised excitons can show properties beyond atomic model, but also opens the possibility to investigate separately the optical Stark effect (atomiclike) and the DFKE (solidlike), which were previously found to compete in timeaveraged measurements^{32}, widening our comprehension of the ultrafast solidstate physics^{34}.
So far we have proven that exciton–crystal interaction can transfer bulk properties to atomiclike core excitons on attosecond time scale. All the more reason, the same is expected to happen for WannierMott valence excitons which, being more delocalised, have a stronger solidlike behaviour (see Supplementary Note 19). The question remains whether the attosecond response will qualitatively change, deviating by the solidlike response, if more localised excitons are taken into considerations. In order to further investigate the role of real space dynamics and exciton localisation we computed the reflectivity phase delay τ for different exciton binding energies E_{b} or, equivalently, Bohr radii a_{0}, which provides a reference for the actual size of excitonic devices^{10,35}. With increasing E_{b} and thus the degree of localisation, τ preserves its V shape whose centre appears to move towards lower photon energies (Fig. 4c), resulting in an overall bigger phase delay difference, Δτ, between the bare crystal and full system responses (Fig. 4d). The energy shift ΔE which minimises Δτ is not exactly equal to E_{b} (Fig. 4e), indicating that τ does not simply experience a rigid shift following the exciton transition. On the one hand this further confirms that the central role of exciton–crystal interaction, on the other hand it opens a way to sculpt and control the excitonic response in the petahertz regime. While the optical response of delocalised electron–hole pairs will react almost in phase with the pump electric field (as underlined by the phase delay τ evaluated at the exciton vertical transition, τ_{ex} in Fig. 4f), more localised excitons will have a slower response, almost out of phase. Due to the exciton–crystal interaction, the CB response will also be significantly affected (see the phase delay τ evaluated at the CB bottom, τ_{CB} in Fig. 4f). Since the exciton binding energy can be continuously tuned around its natural value by an external field, modifying the dielectric screening or inducing a strain^{36,37,38}, one can control the phase delay between the quasiparticle and an optical field on an attosecond level. In particular, for values of E_{b} ≈ 1 eV where a_{0} becomes comparable with the minimum internuclear distance in the MgF_{2} unit cell, it is possible to control the attosecond timing between the exciton and the CB signal, realising a condition when the first is advanced with respect to the IR field while the second is delayed.
As the attosecond response of valence excitons behaves qualitatively in the same fashion (see Supplementary Note 19), our findings point to a possible route for the realisation of a different class of devices where a control over the degree of Coulomb screening^{39} can be used to tune the system response timing with attosecond resolution.
It is worth noticing that we expect intraband motion to be less important for Frenkel excitons like chargetransfer excitons in molecular crystals^{8} or interlayer excitons in layered solids^{7}. Therefore, the subfs optical response of these excitons could qualitatively differ from what is reported here.
To conclude, we investigated ultrafast coreexciton dynamics around the L_{2,3} edge in MgF_{2} single crystal with ATRS. Simultaneous calibration measurements allowed us to perform a direct and unambiguous comparison with theoretical results, addressing the dual nature of the excitonic quasiparticle from a different perspective. In particular, we found that while the exciton dynamics unfolding on the first few femtoseconds originate from the optical Stark effect and can be understood invoking just the atomic character of the quasiparticle, the physical processes happening on an attosecond time scale during light–matter interaction (i.e. intraband motion and DFKE) are typical of the condensed state of matter. Moreover, our theoretical simulations and analysis revealed that the atomsolid duality is general and exists for strongly bound excitons and also for delocalised valence excitons where the absolute timing of the system optical response can be controlled on attosecond time scale by tuning the exciton binding energy. Since these findings are not limited to the chosen target but hold, in general, for excitons originated by dispersive bands in solids, they set a new lever for the coherent control of excitonic properties in the closetopetahertz regime.
Methods
Experimental setup
The setup used for the experiment reported in the main manuscript is described in detail in ref. ^{23}. Single attosecond pulses (SAPs), centred around 42 eV photon energy, and 5 fs IR pulses (central wavelength 750 nm) are first focused onto a Ne gas target. The XUV pulses have a time duration of about 250 as while the IR peak intensity is set between 10^{12} and 10^{13} W/cm^{2}. A timeofflight (TOF) spectrometer records the photoelectron spectra as a function of the delay, t, between the IR and XUV pulses to perform an attosecond streaking experiment^{40}. This allows us to retrieve the temporal characteristics of the two pulses and to obtain a precise calibration of the relative delay t by extracting the exact shape of the IR vector potential. A goldplated toroidal mirror then focuses both beams onto the MgF_{2} crystal, where a thin gold layer deposited on a portion of the sample is used to calibrate the incident XUV photon flux and extract the energydependent sample reflectivity, R_{0}(E). For more details, see the Supplementary Notes 1–5.
Data analysis
To study the different mechanisms underlying the transient features observed in ΔR/R, we decompose the pumpprobe spectrogram in a slow and a fast component. To extract the slow component, we apply a lowpass frequency filter to the reflectivity spectrogram which is constant for frequencies below a cutoff frequency fc and decays with a superGaussian a profile \(e^{\left( {\frac{{f  f_c}}{{2\sigma _f}}} \right)^n}\)with coefficient n = 16 and width σ_{f} = 0.01 PHz. Since the fastest feature observed oscillates at twice the IR frequency 2f_{IR} ≈ 0.75 PHz, we decided to set fc to 1.5f_{IR} = 0.5621 PHz. Once the slow component has been extracted, the fast component of ΔR/R is simply obtained by subtracting the slow component from the total spectrogram. In the case of the experimental data, a highfrequency filter centred at 5f_{IR} = 1.8737 PHz is used to remove the fast noise from the data prior to slow and fast decomposition.
As discussed in the main manuscript, the femtosecond transient features of ΔR/R originate from the optical Stark effect (OSE) induced by the IR electric field. To extract the Stark shift ε from the experimental data, at each delay t, we fitted the sample reflectivity at the presence of the IR pump, R_{IR}(E, t) with six Gaussian bells. Two Gaussians describe the background. Their parameters are derived from the static reflectivity R_{0}(E). The other four Gaussians are used to fit the bright and dark exciton features, doubled because of the Mg^{2+} 2p spin–orbit splitting. As observed for MgO^{19}, the dark excitonic state is responsible for an increase of R_{IR}(E, t) around t = 0 fs, which appears next to the bright excitonic peak, on the lowenergy side, thus overlapping with the bright exciton signal which originates from 2p_{3/2} state. Due to the energy overlap, it is not possible to fit accurately the contribution of the 2p_{1/2}dark state transition as well as all the transitions involving the 2p_{3/2} state. Therefore, we can obtain a reliable estimation of ε(t) only for the brightexciton2p_{1/2} transition which is found to follow the delaydependent energy position of the maximum of ΔR/R around the A feature. The excitonic dipole d(t) is obtained by deconvoluting the delaydependent Stark shift ε(t) with the envelope of the IR electric field^{19}, directly extracted from the simultaneous streaking trace. The Auger decay rate γ and the phonon coupling ϕ can then be evaluated by modelling the excitonic dipole with the function \({\it{d}}({\it{t}})\sim e^{  \gamma t}e^{\phi ({\it{t}})}\)^{18,19}. For more details see Supplementary Note 9.
The absolute phase delay between the fast transient feature of ΔR/R and the IR electric field is evaluated following the approach reported in^{15,31}. First the IR vector potential is extracted from the simultaneous streaking trace by means of a 2D fitting procedure based on the analytical model reported in^{41}. Then the phase difference between the transient features in the differential reflectivity and E_{IR}^{2} is directly evaluated by multiplying the energydependent Fourier transform of the first with the complex conjugate of the Fourier transform of the latter. The product thus constructed automatically peaks at the common frequency between the signals and has a phase equal to their phase difference Δφ. As Δφ could be frequencydependent, we evaluate the average value around 2f_{IR} using the local intensity of the Fourier transform product as weight. The standard deviation is obtained calculating the second momentum of this distribution. Finally, the phase delay τ is given by the ratio between Δφ and the beating frequency 2f_{IR}. The main results reported in Fig. 3c represent an average over four independent transient reflection measurements conducted under similar conditions and weighted by the inverse of their individual experimental uncertainty. The final error accounts both for the mean measurement error and for the statistical deviation between the independent measurements. For further details, see Supplementary Note 7. We note that τ has the opposite sign of the pumpprobe delay modulations (compare the black curves in Fig. 3a, c). This originates from the fact that a positive pumpprobe delay t means that the IR pulse is coming later (IR behaving as a probe), but for the sake of an easier interpretation, we chose a positive τ to mean that the system has a delayed response with respect to the IR field in real time (IR behaving as a pump).
Theoretical model
To investigate the microscopic mechanism of the experimental observation, we describe the dynamical system under the intense IR pulse and the weak XUV pulse based on the following ansatz:
where \(\left \right.\Psi\left(t\right) \left\rangle\right.\) is the wavefunction of the dynamical system, \(\left \right.{{\mathrm{{\Phi}}}_{{\mathrm{GS}}}} \left\rangle\right.\) is the ground state wavefunction of the matter, \(\hat {a}_{c,k}^\dagger \left( \hat {a}_{v,k} \right)\) is a creation (annihilation) operator for the conduction (valence) state at the Bloch wavenumber k, and c_{k}(t) is an expansion coefficient. Here, the vector potential of the IR field is denoted as A_{IR}(t). Note that the ansatz in Eq. (1) is a linear combination of singleparticle singlehole states and is in line with the TammDancoff approximation. Treating the excitation from the ground state to electron–hole states by the XUV field perturbatively, the equation of motion for the coefficient c_{k}(t) is given by
where E_{XUV} (t) is the electric field of the XUV pulse, D_{k} is the transition dipole moment between the ground state and the electron–hole state at the Bloch wavevector k, and H_{kk′} (t) is the electron–hole Hamiltonian. We employ the WannierMott model^{26,27}, and the electron–hole Hamiltonian is given by
where E_{g} is the direct gap of the matter, μ is the effective electron–hole mass, and V_{kk′} is the Coulomb interaction, which we model with the onedimensional soft Coulomb interaction. Note that, by diagonalizing H_{kk′} (t = 0), one obtains exciton states as bound states of electrons and holes as shown in Fig. 1c.
We solve the Schrödinger equation, Eq. (2), with an opensource code^{42} and evaluate physical observables with the timeevolving wavefunction \(\left {{\Psi}\left( {\it{t}} \right)} \right\rangle\), For example, the induced electric current density can be computed as \(J_{{\mathrm{XUV}}}\left( t \right) = \left\langle {{\Psi}\left( {\it{t}} \right)\left {\hat J} \right{\Psi}\left( {\it{t}} \right)} \right\rangle /{\Omega}\), where Ĵ is the current operator and Ω is the crystal volume. Furthermore, the linear susceptibility χ_{exc} (ω) can be evaluated as
where \(\sigma _{{\mathrm{exc}}}\left( \omega \right)\) is the optical conductivity evaluated as the ratio of the current and the electric field in the frequency domain. We model the dielectric function of MgF_{2} by combining the coreexciton susceptibility χ_{exc} (ω) and the valence contribution as
where \({\it{\epsilon }}_{{\mathrm{valence}}}\left( \omega \right)\) is the valence contribution, c is a fitting parameter and Δ_{SO} is the spin–orbit split. By optimising \({\it{\epsilon }}_{{\mathrm{valence}}}\left( \omega \right)\) and c, the dielectric function and the reflectivity of MgF_{2} can be well reproduced by the above model (see Fig. 1b).
With the final goal being to investigate the transient reflectivity of the XUV pulse under the presence of the IR pulse, we compute the electron dynamics by solving Eq. (2) with both the IR and XUV fields. Then, we further compute the current with the timedependent wavefunction \(\left. {{\mathrm{{\Psi}}}\left( t \right)} \right\rangle\). Following the above procedure, Eqs. (4) and (5), the transient dielectric function under the presence of the IR field and the corresponding transient reflectivity can be evaluated. The computed transient reflectivity by the WannierMott model is shown in Fig. 1e.
In order to obtain further insight into the phenomena, we construct two idealised models. One is the pureexciton model, and it is designed to exclude the crystalline nature of the dynamics. The other is the purecrystal model, and it is designed to exclude the atomic nature. The pureexciton model is a threelevel model constructed by the following three states; the ground state \(\left. {{\Phi}_{{\it{{\mathrm{GS}}}}}} \right\rangle\), the bright exciton state and the dark exciton state (see Fig. 1c). Hence the pureexciton model allows us to extract the nature of the discrete energy levels (atomic nature). Note that the pureexciton model is constructed by the subspace of the above WannierMott model because the bright exciton state is the ground state of H_{kk′} (t = 0) and the dark exciton state is the first excited state. The purecrystal model, instead, consists of the conduction band (CB) but excludes the exciton states (see Fig. 1c). Such a model can be realised by setting the electron–hole attraction V_{kk′} to zero, and it is identical to the parabolic twoband model used to discuss the dynamical FranzKeldysh effect^{30}. Therefore, the purecrystal model allows us to extract the contribution purely from the intraband motion (crystal nature) accelerated by the IR field. For a detailed discussion of the theoretical model^{42}, see Supplementary Notes 10–19.
Data availability
All data generated and analysed during this study are available from the corresponding author upon reasonable request.
Code availability
All the custom codes used in this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 848411 title AuDACE). M.L., G.I. and L.P. further acknowledge funding from MIUR PRIN aSTAR, Grant No. 2017RKWTMY. S.S., H.H., U.D.G. and A.R. were supported by the European Research Council (ERC2015AdG694097) and Grupos Consolidados UPV/EHU (IT124919).
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G.D.L, B.M. and G.I. performed the measurements. Together with R.B.V., M.N. and M.L., they also evaluated and analysed the results. G.D.L., B.M., G.I., M.L., F.F. and L.P. designed and built the reflectometer. S.A.S., H.H., U.D.G. and A.R. developed the theoretical models and performed the calculations. M.L. wrote the manuscript. All authors contributed to the analysis and interpretation of the experimental and theoretical results and writing of the manuscript.
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Lucchini, M., Sato, S.A., Lucarelli, G.D. et al. Unravelling the intertwined atomic and bulk nature of localised excitons by attosecond spectroscopy. Nat Commun 12, 1021 (2021). https://doi.org/10.1038/s41467021213457
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DOI: https://doi.org/10.1038/s41467021213457
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