Unravelling the intertwined atomic and bulk nature of localised excitons by attosecond spectroscopy

The electro-optical properties of most semiconductors and insulators of technological interest are dominated by the presence of electron-hole quasiparticles called excitons. The manipulation of these hydrogen-like quasi-particles in dielectrics, has received great interest under the name excitonics that is expected to be of great potential for a variety of applications, including optoelectronics and photonics. A crucial step for such exploitation of excitons in advanced technological applications is a detailed understanding of their dynamical nature. However, the ultrafast processes unfolding on few-femtosecond and attosecond time scales, of primary relevance in view of the desired extension of electronic devices towards the petahertz regime, remain largely unexplored. Here we apply attosecond transient reflection spectroscopy in a sequential two-foci geometry and observe sub-femtosecond dynamics of a core-level exciton in bulk MgF$_2$ single crystals. With our unique setup, we can access absolute phase delays which allow for an unambiguous comparison with theoretical calculations based on the Wannier-Mott model. Our results show that excitons surprisingly exhibit a dual atomic- and solid-like 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 and originates by the interaction with the crystal. Further investigation of the role of exciton localization proves that the bulk character persists also for strongly localised quasi-particles and allows us to envision a new route to control exciton dynamics in the close-to-petahertz regime.

The quest for new devices capable of surpassing the current technological limits 14 has pushed the scientific community to explore solutions beyond classical electronics as done in excitonics 4 , spintronics and valleytronics 15 . Therefore, studying the dynamics of excitons in solids becomes a priority task not only to widen our knowledge of fundamental solid-state dynamical phenomena, but also to explore the ultimate limits of these novel technologies 16,17 . While the development of attosecond spectroscopy 18 , has proven the possibility to study sub-femtosecond electron dynamics in solids 19 , shedding new light onto strong field phenomena and light-carrier manipulation [20][21][22] , a clear observation of attosecond exciton dynamics was missing. Besides more conventional femtosecond techniques 23 , attosecond transient absorption and reflectivity spectroscopy have been employed to study the ultrafast decay processes (few-femtoseconds) of core-excitons 24,25 , but failed in recording the sub-cycle dynamics unfolding during light-matter interaction. Here we used attosecond transient reflection spectroscopy (ATRS) 8 to study attosecond dynamics of a core-level exciton in bulk MgF2 single crystals, characterized by a binding energy similar to the interlayer excitons of twodimensional materials 17 . 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. Our results thus provide a novel description of the ultrafast exciton-crystal interaction on sub-femtosecond time scales and move an important step forward attosecond excitonics. Figure 1a shows a schematic picture of the experimental setup characterized by a sequential two-foci geometry, used to perform simultaneous attosecond photoelectron and ATRS measurements 9 (see Methods). The static reflectivity for a MgF2 (001) crystal, R0, close to the Mg L2,3 edge is probed by an extreme-ultraviolet (XUV) attosecond pulse (Fig. 1b). In this energy region R0 is characterized by a peak (labelled with A), which has been attributed to the formation of excitons after excitation of a localized Mg 2+ 2p core electron 26 (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 27 . In our experiment we use a few-femtosecond and intense IR pulse to drive the crystal out of equilibrium (Fig. 1c), initiating ultrafast exciton dynamics later probed 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: [RIR -R0] /R0. The results as a function of photon energy E and pump-probe delay t, are shown in Fig. 1d. At small values of t, we observe rich transient features which can be decomposed in a slower and a faster component. While the former unfolds on a few-femtosecond time scale and 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 upper panel in Fig. 1d shows the square of the IR vector potential, A 2 IR, as retrieved from the simultaneous streaking trace (see supplementary material). Knowing the associated pump field time evolution, we calibrated the pump-probe delay axis in order to have the zero delay coinciding with the maximum of the IR electric field squared (equivalently, a zero of A 2 IR). 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 Wannier-Mott (WM) exciton model 10,11 (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.
The slow component of the experimental ΔR/R (Fig. 2a) is characterized 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 IR-induced optical Stark effect of the excitonic transitions, from which it is possible to evaluate the exciton decay time 24 and its coupling with phonons 25 . For the main excitonic transition A, we found the Auger decay term 28   ("pure excitonic" contribution) or only the crystal CB ("pure crystal" contribution) (see Methods).
We find that "pure exciton" calculations exhaustively reproduce the experimental dispersive profiles which are fully reproduced by our simulations (Fig. 3b). These oscillations have a V-shaped dispersion which resembles what is found for the dynamical Franz-Keldysh effect (DFKE) high into the CB of diamond 30,31 . While this suggests a clear link with intra-band motion of virtual charges, we note that the present case differs significantly as the V-shaped structure in MgF2 is located across the CB bottom and centred on the excitonic transition A, where the presence of real carriers can modify the system response 21,32 .

Fig. 3 | Core-exciton dynamics on an attosecond time scale. a, Fast component of the experimental ΔR/R. The phase delay, τ, between the oscillations and EIR 2 is displayed in black. The solid curve represents the mean over four independent measurements and the shaded area twice its standard deviation. b, Same as a but for the full calculation results. c, Comparison between the experimental (black solid) and calculated τ, considering the full system response (red solid), or addressing separately the crystal (grey dashed) and excitonic (orange dotted) contributions.
Thanks to our unique experiment 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 2 IR (see Methods). Figure 3c 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 pure crystal 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. The system response is instead closer to the bare crystal bulk-case, showing the same shape, but shifted towards lower photon energies. 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 atomic like response. Thus, by addressing separately the fast (attosecond) and slow (femtosecond) components of the optical response with our time-resolved attosecond measurements, we are able to disentangle the optical Stark effect (atomic-like) and the DFKE (solid-like) in excitons, which were previously thought to compete in time-averaged measurements 33 .  Fig. 4a for the case of the full system (red curve) or considering only the quasi-particle (orange curve). By evaluating the phase delay for the dipole with respect to the IR field (black dotted curve), in the same fashion as done for ΔR/R, 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). This value is in agreement with the difference of 272 ± 57 as observed between the phase delay τ of Fig. 3c, 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 sub-nm exciton motion in real space and the sub-fs transient features observed in the differential reflectivity. The few-cycle IR pulse dresses the crystal CB inducing intra-band motion (DFKE). In turn, this alters the exciton dynamical properties causing the quasi-particle to oscillate almost in phase with AIR. This not only proves that localized excitons can show properties beyond atomic model, widening our comprehension of the ultrafast solid-state physics, but also opens the possibility to investigate separately different competing physical mechanisms, ultimately allowing for a full optimization of light-matter interaction in future devices.
Even if we proved that exciton-crystal interaction can transfer bulk properties to atomic-like excitons on attosecond time scale, the question remains whether this dual nature is affected by the localization degree of the quasi-particle. In order to further investigate the role of real space dynamics and exciton localization we computed the reflectivity phase delay τ for different exciton binding energies Eb or, equivalently, Bohr radii a0, which provides a reference for the actual size of excitonic devices 6,34 .
With increasing Eb and thus the degree of localization, τ preserves its V shape which 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 minimizes Δτ is not exactly equal to Eb (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 new way to sculpt and control the excitonic response in the petahertz regime. While the optical response of delocalized electron-hole pairs will react almost in phase with the pump electric field (τex in Fig. 4f), more localized 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 τCB 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 [35][36][37] , one can control the phase delay between the quasi-particle and an optical field on an attosecond level. In particular, for values of Eb ≈ 1 eV where a0 becomes comparable with the minimum inter-nuclear distance in the MgF2 unit cell, it is possible to control the attosecond timing between the exciton and the CB signal, realizing a condition when the first is advanced with respect to the IR field while the second is delayed. As those are the typical binding energies of interlayer excitons in Van der Waals heterostructures 17 , our findings point to a possible route for the realization of new devices where a control over the degree of Coulomb screening 38 can be used to tune the system response timing with attosecond resolution.
We investigated ultrafast core-exciton dynamics around the L2,3 edge in MgF2 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 quasi-particle from a new 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 quasi-particle, the physical processes happening on an attosecond time scale during light-matter interaction (i.e. intra-band motion and DFKE) are typical of the condensed state of matter. Moreover, our theoretical simulations and analysis revealed that the atom-solid duality is general and exists also for strongly bounded excitons where the absolute timing of the system optical response can be controlled on attosecond time scale by tuning the exciton binding energy. These findings set a new lever to the development of innovative optical devices for petahertz optoelectronics mediated by excitonics.

Experimental setup:
The setup used for the experiment reported in the main manuscript is described in detail in Ref. 9 . Single attosecond pulses (SAPs), centred around 42-eV photon energy, and 5-fs infrared (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 time-offlight (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 39 . 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 gold-plated toroidal mirror then focuses both beams onto the MgF2 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 energy-dependent sample reflectivity, R0(E). For more details, see the related sections in the Supplementary Information.

Data analysis:
To study the different mechanisms underlying the transient features observed in ΔR/R, we decompose the pump-probe spectrogram in a slow and a fast component. To extract the slow component, we apply a low-pass frequency filter to the reflectivity spectrogram which is constant for frequencies below a cut-off frequency fc and decays with a supergaussian a profile � − 2 � with coefficient n = 16 and width = 0.01 PHz. Since the fastest feature observed oscillates at twice the IR frequency 2fIR ≈ 0.75 PHz, we decided to set fc to 1.5fIR = 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 high-frequency filter centred at 5fIR = 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, RIR(E, t) with six Gaussian bells. Two Gaussians describe the background. Their parameters are derived from the static reflectivity R0(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 25 , the dark excitonic state is responsible for an increase of RIR(E, t) around t = 0 fs, which appears next to the bright excitonic peak, on the low energy side, thus overlapping with the bright exciton signal which originates from 2p3/2 state. Due to the energy overlap, it is not possible to fit accurately the contribution of the 2p1/2-dark state transition as well as all the transitions involving the 2p3/2 state. Therefore, we can obtain a reliable estimation of ε(t) only for the bright-exciton -2p1/2 transition which is found to follow the delay-dependent energy position of the maximum of ΔR/R around the A feature. The excitonic dipole d(t) is obtained by deconvoluting the delay-dependent Stark shift ε(t) with the envelope of the IR electric field 25 , 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 ( )~− ( ) 24,25 . The absolute phase delay between the fast transient feature of ΔR/R and the IR electric field is evaluated following the approach reported in 21,31 . 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 40 . Then the phase difference between the transient features in the differential reflectivity and EIR 2 is directly evaluated by multiplying the energy dependent 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 frequency-dependent, we evaluate the average value around 2fIR 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 ration between Δφ and the beating frequency 2fIR. The main results reported in Fig. 3c represent average over 4 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 the Supplementary Information. We note that τ has the opposite sign of the pump-probe delay modulations (compare the black curves in Fig. 3a and Fig. 3c). This originates from the fact that a positive pump-probe 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 as 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 |Ψ( )⟩ is the wavefunction of the dynamical system, |Φ GS ⟩ is the ground state wavefunction of the matter, � , † � � , � is a creation (annihilation) operator for the conduction (valence) state at the Bloch wavenumber, , and ( ) is an expansion coefficient. Here, the vector potential of the IR field is denoted as ( ). Note that the ansatz in Eq. (1) is a linear combination of single-particle singlehole states and is in line with the Tamm-Dancoff approximation. Treating the excitation from the ground state to electron-hole states by the XUV field perturbatively, the equation of motion for the coefficient ( ) is given by where ( ) is the electric field of the XUV pulse, is the transition dipole moment between the ground state and the electron-hole state at the Bloch wavevector, , and ′ ( ) is the electron-hole Hamiltonian. We employ the Wannier-Mott model 10,11 , and the electron-hole Hamiltonian is given by where is the direct gap of the matter, is the effective electron-hole mass, and ′ is the Coulomb interaction, which we model with the one-dimensional soft Coulomb interaction. Note that, by diagonalizing ′ ( = 0), one obtains exciton states as bound states of electrons and holes as shown in Fig. 1c.
By solving the Schrödinger equation, Eq. (2), one can compute the time-evolving wavefunction |Ψ( )⟩ with Eq. (1) and physical observables can be computed by making use of it. For example, the induced electric current density can be computed as ( ) = ⟨Ψ( )� ̂� Ψ( )⟩/Ω, where ̂ is the current operator and Ω is the crystal volume. Furthermore, the linear susceptibility ( ) can be evaluated as where ( ) 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 MgF2 by combining the core-exciton susceptibility ( ) and the valence contribution as where ( ) is the valence contribution, is a fitting parameter, and Δ is the spin-orbit split. By optimizing ( ) and , the dielectric function and the reflectivity of MgF2 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 time-dependent wavefunction |Ψ( )⟩ . Following the above procedure, Eq. (4) and Eq. (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 Wannier-Mott model is shown in Fig. 1e. In order to obtain further insight into the phenomena, we construct two idealized models. One is the pure-exciton model, and it is designed to exclude the atomic nature of the dynamics. The other is the pure-crystal model, and it is designed to exclude the crystalline nature. The pure exciton model is a three-level model constructed by the following three states; the ground state |Φ ⟩, the bright exciton state, and the dark exciton state (see Fig. 1c). Hence the pure-exciton model allows us to extract the nature of the discrete energy levels (atomic nature). Note that the pure exciton model is constructed by the subspace of the above Wannier-Mott model because the bright exciton state is the ground state of ′ ( = 0) and the dark exciton state is the first excited state. The pure crystal model, instead, consists of the conduction band (CB) but excludes the exciton states (see Fig. 1c). Such model can be realized by setting the electron-hole attraction ′ to zero, and it is identical to the parabolic twoband model used to discuss the dynamical Franz-Keldysh effect 30 . Therefore, the pure crystal model allows us to extract the contribution purely from the intra-band motion (crystal nature) accelerated by the IR field. For a detailed discussion of the theoretical model, see Supplementary