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1 doi: /nature Theory and Computations 1.1 Hamiltonian and Solutions of the Schrödinger Equation The time-dependent Hamiltonian of the system in the optical field is (1) where is the unperturbed Hamiltonian (without the optical field), is electron charge, and is the coordinate operator. We neglect the interaction between electrons because we consider dynamics on very short time scales, on the order of a few optical oscillations, and electron collisions at high electron densities take longer times to come into play. This can be judged, e.g., from the Drude collision rate for metals (silver, as an example) 0.03 ev, which yields collision times on the order of 20 fs. We write the time-dependent Schrödinger equation as, (2) where index numbers the states. The initial condition is that, satisfies the stationary Schrödinger equation (2) with. We consider Schrödinger equation (2) in the direction of the linearly-polarized strong field, i.e., the problem becomes one-dimensional. We assume that the system is periodic with a lattice constant Å. In order to describe the electron dynamics in the system exposed to, we introduce a model Hamiltonian in second quantization formalism, expressed in terms of dipole coupling (also called the length gauge), with, (3),, Δ 4,,, (4),,,,,,, (5) This Hamiltonian consists of the sum of standard tight-binding Hamiltonians (within the nearest-neighbor approximation) for each band, with an additional interband dipole-type independent coupling, which is described by constants, defined below in Eq. (7). In (3),, and, are the creation and annihilation operators of an electron on lattice site in band, is 1

2 RESEARCH SUPPLEMENTARY INFORMATION the offset energy of band, is the bandwidth of band, and is the number of the lattice sites in our one-dimensional model, which is the number of the crystallographic planes of the original system in the direction of. Here, for simplicity, we do not indicate explicitly the spin index. Summation over the spin index was done in all final expressions. The eigenfunctions of the unperturbed Hamiltonian satisfy the Bloch theorem: for the th band, they correspond to the product between a plane wave and a periodic Bloch unit-cell function,. Within the tight-binding approximation above, for a single band in one dimension, the energy dispersion law for the th band has the form 1-3 : E Δ 2 (6) where, or, for conduction and valence bands and, are corresponding subbands; is the wave vector along the direction of. The parameters and Δ in Eq. (4) are chosen to fit the valence and conduction band energies of silica for the unperturbed Hamiltonian, i.e., with. We consider two valence and two conduction subbands with the following band energy offsets:, 2 ev and, 2 ev for the two valence subbands;, ev and, 44 ev for the two conduction subbands. The corresponding bandwidths are Δ, ev and Δ, ev for valence subbands, Δ, 4 ev and Δ, ev for the conduction subbands, respectively. We consider a basis composed of the eigenfunctions, of. Correspondingly, for each band, there are basis functions (for each electron spin). The interband coupling is characterized by the dipole matrix elements, between normalized (in one unit-cell) periodic Bloch functions, z. Such matrix elements between bands and are determined by the expression:,, z, z, (7) where we assume that axis is the direction of electric field and the parameters, are constants, i.e., do not depend on. Note that the dipole matrix elements are,,. The matrix elements, and, control the strength of nonlinear effects: they describe transitions between either fully filled levels or completely empty levels, correspondingly. The constants, are set as:,,,, and,. Constant, is chosen to describe the magnitude of the linear absorption. Parameters, and, are set to reproduce nonlinear effects in recent demonstration of optical-field-induced electric current under similar experimental conditions (Schiffrin et al., submitted for publication). Transitions between the most energetically distant valence and conduction bands are not considered, i.e., we set,. The numerical results are mainly sensitive to parameter, and less sensitive to the parameters, and,. Quantities, Δ and, are the parameters of the model. All theoretical curves presented in this work have been calculated with the same model and the same values of, Δ and,. These model parameters are not used as adjustable parameters. 2

3 RESEARCH With the parameters, Δ and, being set, Eq. (2) is solved via its matrix form within the basis composed of the eigenfunctions of. Dynamics of the system can be described in terms of its causal one-particle Green s function, defined as an operator,,,,, 1,, (8) where, 1 are the Fermi occupation numbers; for the conduction bands, and 1 otherwise, which is justified because temperature. In the experiment, the system is driven by a strong optical field pulse and probed by an attosecond XUV pulse the field of which we denote as ; it is delayed time interval by adjustable with attosecond-scale precision. Optical absorption of the probe XUV pulse, which is considered to be weak, is calculated from the standard time-dependent perturbation theory as a trace, 1,,, (9) where is the Schrödinger dipole operator, which is, in our case, responsible for the XUV transitions that occur from the deep L-band states to the conduction-band states. Note that has the meaning of the absorbed energy of the probe pulse. We can also calculate another quantity of interest, namely, the population of the initial (unperturbed) conduction band, which is expressed in terms of the diagonal elements of the retarded,, and advanced,, Green s functions as,,,,. (10) Note that is the projection operator onto the initial CB states, and is the density matrix of the system. The population is of general physical interest as a measure of the efficiency of the system s interaction with the strong field, though only the residual population is directly measurable. The adiabatic local density of states (LDOS) of the conduction bands is defined as where is Dirac s -function, and δ is the operator of electron density at a point. The absorption spectrum at a given delay is given by, δ, (11) 1 Re,,, (12) where (9).. Note that, where is given by Eq. 3

4 RESEARCH SUPPLEMENTARY INFORMATION 1.2 Finite Difference Time Domain (FDTD) Solution of the Schrödinger Equation and the Maxwell Equations We solve the Maxwell equations numerically by the finite difference time domain (FDTD) method 4,5 for a finite size system with the absorbing boundary conditions. The size of the computational space in the -direction is 6000 nm with the coordinates of the boundaries 000 nm and 000 nm. The dielectric nanofilm is placed at the midpoint of the system, i.e, it is centered at 0. The optical pulse impinges at the left boundary and propagates along the positive direction of the -axis with the polarization of the electric field being aligned along the -axis. In FDTD solutions of the Maxwell equations, we choose the spatial step to be 1 nm, while the time step is 0.7 attoseconds (1 as=10 s). These values provide convergence for both the Maxwell equations and the Schrödinger equation (see the next paragraph). The perfectly matching boundary conditions were used for the FDTD problem. In parallel with the Maxwell equations, we solve the Schrödinger equation (2) numerically (and non-perturbatively) for each basis function via the fourth-order Runge-Kutta method. We impose periodic boundary conditions at the boundary of our nanofilm ( supercell ), i.e., at lattice sites 0 and. The thickness of the supercell plays the role of the thickness of the dielectric nanofilm in our model. Due to the periodic boundary conditions, our quantummechanical problem is equivalent to considering an infinite periodic system (stack) of the nanofilms (supercells) with a constant electric field applied. Note that such a supercell (or, periodic boundary conditions) method is common in microscopic and ab initio theories of thin films, surfaces, interfaces, and adsorbates; see, e.g., Ref. 6 for a review. Most of our calculations were performed for 0, i.e., for 1 nm. The Schrödinger equation and the Maxwell equations are coupled through the light-induced dielectric polarization that is expressed as (13) and that defines the Maxwell boundary conditions, which can be expressed as continuity of the tangential electric and magnetic fields 7. Alternatively and equivalently, the Maxwell boundary conditions can be expressed as continuity of the tangential component of the electric field and the normal component of displacement,, (14) where is the polarization of the system. Here the absolute Gaussian system of units is used (different from the SI system in the main article). 4

5 RESEARCH 2. Supplementary Experimental Data 2.1 Broadband XUV Absorption Time-resolved XUV absorption spectroscopy is performed with broadband, isolated attosecond XUV pulses generated via high harmonic generation (HHG) by focusing carrier-envelopephase-stabilized sub-4-fs visible-infrared laser pulses with 400-µJ pulse energy (FemtoPower Compact Pro, Femtolasers) into a neon gas target. The collinear laser and XUV beam are separated into two arms of a Mach-Zehnder-type interferometer by means of a perforated mirror, which reflects the optical and transmits the XUV pulses, by taking advantage of the much smaller divergence of the XUV radiation. The residual optical radiation in the XUV arm is blocked using a 150-nm molybdenum filter, which also compensates for the chirp of the XUV pulse. Three subsequent Rhodium-coated mirrors at an angle of incidence of 75 degrees reflect some 45% of the broadband radiation. The optical arm includes a telescope to re-collimate the optical beam and a variable aperture to control its intensity on the target. The relative timing between the two arms can be adjusted with a mirror moved by a nm-resolution piezo translator in the optical arm. More details can be found in Ref. 22 of the main text. The XUV and NIR beams are recombined at a second perforated mirror and focused with a grazing-incidence toroidal mirror (Zeiss) on the free-standing chemical-vapor-deposited SiO 2 sample (either 125 or 250 nm thick).the surface roughness of our samples is specified to be less than +/- 20 nm, commensurate with values typical of chemical vapor deposition. The XUV and NIR focal spot sizes on target are 40 µm and 125 µm, respectively, allowing the XUV pulses to sample transmittivity near the optical axis of the laser beam and avoid lateral intensity averaging. The XUV beam transmitted through the sample is spectrally dispersed by a flat-field grating (Horiba Scientific) and projected on a XUV-sensitized camera (see Ref. 22 of the main text). A 0.2-mm-wide slit between the target and the grating leads to a spectral resolution of ~0.35 ev. Our measurements were performed near (to within <10%) optical breakdown intensity. NIR-field-induced breakdown manifested itself as macroscopic mechanical damage of the sample, which became immediately visible in the magnified image of the focal region permanently monitored. Beyond what is shown in Fig. 2 of the main text our studies reveal a strong, broadband bleaching of XUV absorbance over a large fraction of the conduction band that along with spectral shifts responds instantly to the oscillating light field. The data are summarized in Fig. S1(a) and are in excellent agreement with our model predicting transient XUV absorption bleaching to be a broadband effect extending over a bandwidth of 7-8 ev, i.e. almost over the entire CB, and the effect fully recovers over the entire band right after passage of the laser pulse. For these studies we drew on a more energetic driver laser delivering 50% more energetic NIR pulses with similar temporal structure as used for the experiments presented in the main text to increase the XUV flux towards photon energies up to 125 ev. Fig. S1 compares the measured and calculated fieldinduced changes in OD in the range of ev versus XUV delay. The sub-femtosecond sub-structure of the transient absorption spectrogram is not so well resolved as in the measurement shown in the main text, due probably to slightly larger fluctuations of the pulse parameters. However, it is conspicuous in Fig. S1(a) that the XUV absorptivity is affected by the strong NIR field in a similar way over the entire spectral range of ev, i.e. over a band of 7-8 ev, comparable to the bandwidth of the CB. The change in OD builds up on the leading edge and disappears on the trailing edge of the NIR pulse within 5

6 RESEARCH SUPPLEMENTARY INFORMATION near-1-fs windows. For comparison, Fig. S1(b) compares the measured and calculated fieldinduced changes in OD in the range of ev versus XUV delay. To within our measurement accuracy, the XUV absorptivity resumes its original field-free values right after exposure in the entire spectral range investigated. This indicates that only little population survives the NIR field in the conduction band. 6

7 RESEARCH Figure S1. (a) Broad-band strong-nir-field-induced change in XUV optical density (OD) in the conduction band of SiO2. (b) Comparison between measured (blue line) and computed (red dashed line) relative change in OD over the photon energy range of ev as a function of the delay of the attosecond XUV pulse with respect to the strong NIR light pulse inducing the changes. Here we show the change in optical density relative to an absorption spectrum recorded in the absence of the NIR field. 7

8 RESEARCH SUPPLEMENTARY INFORMATION 2.2 Near Infrared Reflectivity and Transmittivity Reflectivity and transmittivity versus intensity Nonlinear resolved reflectance versus laser intensity was recorded with sub-4-fs NIR laser pulses gently focussed (~f/60) onto a thin SiO 2 wedged sample with a thickness of ~ 0.2 mm near its sharp edge. To avoid dispersion in the beam path and plasma generation in the focal region the whole experiment is set up in vacuum. The front surface reflection under Brewster s angle was polarization filtered with a Glan-Thompson polarizer and recorded with a power meter. The intensity variation was achieved by moving the sample along the beam axis with a translation stage (z-scan geometry). With the measured power in the full beam and the near- Gaussian transverse intensity distribution recorded with a CCD-camera the laser intensity hitting the sample was controlled by translating it along the beam axis (Z-scan measurement). Fig. 3(a) in the main text shows the data recorded up to an intensity for which the sample is destroyed instantly after exposure by optical breakdown; we observed that damage usually sets in first inside the sample or at its rear surface, presumably due to self-focusing. The observed increase in reflectivity turns out to be fully reversible as long as continuous illumination is restricted to time intervals shorter than a few seconds. The slow damage is obviously of thermal origin and is interpreted as follows. The intense pulse generates a white-light continuum upon traversing the ~0.1-mm-thick sample. This high-intensity continuum contains short-wavelength components down to the uv, which can be fairly efficiently absorbed by two- and three-photon absorption, creating the seed electrons for subsequent avalanche ionization and lattice heating 8. Given the high repetition rate (3000 laser shots per second, > 0.5 µj pulse energy), resultant accumulating heating of the sample during exposure causes a slow thermal damage of the sample. In order to avoid these undesirable effects in the measurement of the intensity dependence of the transmittivity, we used sub-micron-thin films in this study. Again, we inserted the sample at Brewster s incidence to avoid interference effects. Indeed, we did not observe any slow damage in these nanofilms up to the highest intensity levels approaching and reaching the threshold for optical breakdown. As before, the intensity was varied by the Z-scan technique and the measurement was performed in vacuum Time-resolved reflectivity To time-resolve the strong-field-induced reflectivity enhancement of SiO 2, we adapted a concentric mirror setup similar to Ref. 23 of the main text. The incident laser pulses are split into two replica by a 5 mm diameter inner mirror (partially reflective, reflectance ~10%) and a silver-coated outer mirror (reflectance 100%). Both beams (the inner one employed as the probe pulse and the outer, stronger one as the pump pulse) propagate collinearly towards a focusing mirror with 30 cm focal length. By moving the inner mirror along the beam axis with a piezo actuated translation stage the weak probe pulse is delayed by t with respect to the strong pump. The experiment is again set up in vacuum. We first verified the ultrashort pulse duration and the feasibility of fringe-resolved pump-probe measurements with this apparatus by generating the second harmonic of the NIR pulse in a 20-µm-thick BBO crystal (placed in the focus of the, this time attenuated, beams) and recording this signal as a function of t. A highpass filter was used to reject the residual fundamental radiation. The resulting autocorrelation trace is shown in Fig. S2(a), yielding a sub-4-fs pulse duration. For varying delays the varying concentric interference redistributes the energy between a high power concentration near the 8

9 RESEARCH optical axis (r~0) caused by constructive interference at (r~0), see upper image in Fig. S2, and an annular distribution with a minimum on the beam axis observed for destructive interference at the beam axis, see lower image in Fig. S2. The same setup was then used to record nonlinear pump-probe reflectance traces from the surface of a thin SiO 2 wedge inserted again at Brewster s angle. The peak electric field strength resulting from the positive superposition of the pump and probe fields near zero delay was set to be close to (slightly below) the threshold for optical breakdown. The piezo translation stage adjusting the time delay between the two pulses was scanning a range of ±30 fs with a frequency of up to 8 Hz. The reflected probe signal was detected with a fast photodiode and a boxcar-integrator and recorded with a fast sampling oscilloscope. The laser pulses carry a small (< 0.5%) fraction of s-polarized light due to polarization modification of higher order modes generated in the hollow-core fiber used to spectrally broaden the light. We used a Glan- Thompson polarizer in front of the photodiode to suppress the reflected light in the low-intensity limit for improving the contrast of the nonlinear reflectivity measurement by using p-polarized probe (as well as pump) light. The achieved linear reflectance of ~10-4 is dominated by the variation of the Brewster s angle within the bandwidth of the laser. Fig. S2(b)-(d) show 3 subsequently recorded reflectivity autocorrelation functions. After every trace recorded we blocked the laser beam for several seconds to avoid significant heating of the sample. The data presented in Fig. 3(b) of the main text show the average of 10 individual scans selected from a set of 32 subsequent scans based on their peak signal strength at t 0 : all those traces exhibiting a maximum signal strength within ~25% of the most intense trace were selected. Several of the scans outside this selection range showed a significantly reduced strength, which we attribute to extreme sensitivity of the signal to even small fluctuations of the laser intensity. This sensitivity is revealed by the data in Fig. 3(a) as well as by the scan shown in Fig. S2(e), which plots the results of the same measurement repeated with a peak laser intensity reduced by 8%. The almost complete disappearance of the induced nonlinear reflectivity corroborates the highly nonlinear nature of the underlying electron processes. Our time-resolved measurement, in its current implementation, does not reveal the absolute magnitude of the induced reflectivity due to the mismatch between the radial sizes of the pump and the probe beams in the focus. In fact, the FWHM diameter of the pump pulse inducing the nonlinear effect amounts to only 30 % of the that of the probe pulse. As a result, a large fraction of the probe light does not benefit from the reflectivity increase induced by the pump beam near the optical axis. 9

10 RESEARCH SUPPLEMENTARY INFORMATION Figure S2. (a) Collinear second-order cross-correlation trace recorded with the setup sketched in Fig. 3(a) of the main text. Note that the intensity contrast is impeded by the substantially weaker probe field as compared to the pump field. False-colour plots display the overall radial intensity distribution of the beams in the focal plane (imaged to a CCD camera) for constructive (upper panel) and destructivee (lower panel) interference on the optical axis. (b-d) Representative nonlinear reflectivity pump-probe traces recorded in individual scans. (e) The same nonlinear reflectance trace as shown in panel (b-d) but with the intensity of the pump beam reduced by 8% %. 10

11 RESEARCH References 1 Slater, J. C. & Koster, G. F. Simplified LCAO method for the periodic potential problem. Phys. Rev. 94, , (1954). 2 Frauenheim, T. et al. A self-consistent charge density-functional based tight-binding method for predictive materials simulations in physics, chemistry and biology. Phys Status Solidi B 217, 41-62, (2000). 3 Dunlap, D. H. & Kenkre, V. M. Dynamic localization of a charged particle moving under the influence of an electric field. Phys. Rev. B 34, , (1986). 4 Kunz, K. S. & Luebbers, R. J. The finite difference time domain method for electromagnetics. 7 edn, (CRC Press, 1993). 5 Taflove, A. Computational electrodynamics: The finite-difference time-domain method. 3 edn, (Artech House, 2005). 6 Brivio, G. P. & Trioni, M. I. The adiabatic molecule-metal surface interaction: Theoretical approaches. Reviews of Modern Physics 71, , (1999). 7 Jackson, J. D. Classical electrodynamics. 2nd edn, (Wiley, 1975). 8 Mero M., Sabbah A.J., Zeller J. and Rudolph W. Femtosecond dynamics of dielectric films in the pre-ablation regime Appl. Phys. A 81, (2005) 11