Abstract
Intense light–matter interactions have revolutionized our ability to probe and manipulate quantum systems at subfemtosecond timescales^{1}, opening routes to the alloptical control of electronic currents in solids at petahertz rates^{2,3,4,5,6,7}. Such control typically requires electricfield amplitudes in the range of almost volts per angstrom, when the voltage drop across a lattice site becomes comparable to the characteristic bandgap energies. In this regime, intense light–matter interaction induces notable modifications to the electronic and optical properties^{8,9,10}, dramatically modifying the crystal band structure. Yet, identifying and characterizing such modifications remain an outstanding problem. As the oscillating electric field changes within the driving field’s cycle, does the band structure follow and how can it be defined? Here we address this fundamental question, proposing alloptical spectroscopy to probe the laserinduced closing of the bandgap between adjacent conduction bands. Our work reveals the link between nonlinear light–matter interactions in strongly driven crystals and the subcycle modifications in their effective band structure.
Main
The ability of nearresonant light fields to modify the energetics and dynamics of an atom (or a molecule) is central to many phenomena such as laser cooling, trapping, quantum optics of atoms in a cavity^{11,12} or extremely efficient generation of hyperRaman lines^{13}. The combined states of matter and nearresonant light are generally studied over many laser cycles using the Floquet formalism, which takes advantage of the periodicity of light oscillations. In solids, cycleaveraged modification of the hopping rates between the neighbouring sites leads to, for example, the coherent destruction of tunnelling^{14} and modified local interaction potentials^{10}.
The situation changes in light fields with frequencies well below resonance but with intensities sufficiently high to induce electronvoltscale voltages across a lattice site. As the oscillating electric field of the lightwave changes from zero to its maximum value within a quarter cycle, rapidly changing voltages can lead to subcycle modifications of the macroscopic properties, such as transmittance^{4,6} or conductance^{7}. In this regime, a cycleaveraged frequencydomain Floquet perspective is hardly satisfying.
Here we describe how the effective band structure can be introduced in such an interaction regime within a timedomain perspective. We experimentally demonstrate alloptical spectroscopy of a strongly driven crystal, revealing an anomalous spectral intensity response, using highharmonic generation (HHG). Our theoretical study links these observations to laserinduced closing of the gap between adjacent conduction bands (Fig. 1a).
HHG in solids involves optical tunnelling^{15} across the gap between the valence and conduction bands. This transition initiates harmonic generation associated with both intraband currents^{10,16,17} and electron–hole recombination, the latter leading to higherorder harmonic emission^{18}. At higher light intensities and/or lower frequencies, one enters a new dynamical regime. In many systems, as the electron–hole wavepacket approaches the edges of the Brillouin zone, Landau–Dykhnetype transitions^{19} can promote electrons to higher conduction bands, as reflected in the harmonic spectra^{20,21} (Fig. 1a). However, can the imprint of these transitions on highharmonic emission be used to observe lightinduced modifications of the bands?
One can show that lightinduced modifications of the bands are directly linked to the subcycle Landau–Dykhnetype transitions between them (Methods). In the lowfrequency limit, one begins with the adiabatic approximation, which treats the phase ωt of the electricfield oscillations as a parameter, and finds the adiabatic band structure ϵ(k, ωt) in the presence of a quasistatic electric field. When a nonadiabatic Landau–Dykhnetype transition between a pair of bands occurs, typically as the electron is laser driven across the minimal bandgap, it modifies the gap as follows:
where Δϵ_{eff} is the effective bandgap, Δϵ_{ad} is the bandgap in the adiabatic approximation and w_{LD} is the probability of the subcycle Landau–Dykhnetype transition. The result is intuitive: the effective bandgap is closed by the laser field when the subcycle transition approaches unity, that is, w_{LD}(t) ≃ 1. Given the exponential sensitivity of w_{LD}(t) to the laserfield strength, the effective bandgap closes soon after the subcycle excitation probability becomes appreciable.
To experimentally investigate this phenomenon, we use twocolour HHG spectroscopy^{22,23} (Fig. 1b), augmenting the strong fundamental driver with a weak secondharmonic (SH) field and controlling their subcycle delay τ. Electron–hole trajectories responsible for different harmonics^{24} are perturbed by the weak field: each trajectory acquires an additional complex phase σ(τ), which is accumulated along the entire trajectory, serving as a sensitive label of its temporal properties. If the fundamental field generates only odd harmonics, the SH field breaks the symmetry of interaction, as σ(τ) changes sign between two consecutive halfcycles (Supplementary Information). This phase is mapped into the harmonic intensity as
Scanning τ modifies σ(τ) in a periodic manner, modulating the harmonic spectrum. The modulation phase and contrast encode the dynamical properties of electron trajectories associated with each harmonic order and allow their reconstruction with attosecond precision. In previous studies this method resolved the interband contribution to highharmonic emission^{24} and provided an alloptical reconstruction of the fieldfree band structure on exciting a single conduction band^{25,26}. In this paper, we use this scheme to study the underlying dynamics of driven multiband currents, probing the dressed band structure.
Experiments were performed on MgO (ref. ^{27}) using λ = 1.3 μm laser field at intensities of ~10^{13} W cm^{–2} and a weak SH field, polarized parallel to the fundamental field. We have measured the HHG modulations with the twocolour delay and extracted the oscillation phase Φ_{N} associated with each harmonic order N. Figure 2 presents Φ_{N} as a function of the harmonic order for orientation angles of 0° (ΓX) and 45° (ΓK) with respect to the fundamental field’s polarization. For harmonics N = 11−15 (10.5–14.5 eV), which are associated with electron–hole recombination from the first conduction band, we measure a gradual slope of Φ_{N} with N. The slope reflects the evolution of the trajectory length with harmonic order^{24}, supported by our semiclassical calculations (Supplementary Information). As we approach the edge of the Brillouin zone, represented by harmonic 17 (H17; 16 eV) for the ΓK direction and harmonic 19 (H19) for the Γ–X direction, this simple description fails. A clear phase jump appears in the measurements, associated with the edge of the zone and changing the band curvature. At this point, the mapping between momentum and energy becomes singular, leading to the appearance of spectral caustic^{21}. Thus, the phase measurement serves as a sensitive probe of the bandgap and allows its accurate identification.
Beyond the cutoff energy of the first conduction band, the electron dynamics involve multiple bands^{20,28}. The harmonic emission is dictated by electron currents originating from higher conduction bands, determined by their structure and coupling. Importantly, these parameters depend on the crystal orientation, with each orientation offering a new onedimensional slice of the band structure^{27,29}. We can, thus, track the dependence of electron dynamics on the band structure and laserinduced couplings by resolving the oscillation phase of each harmonic as a function of crystal orientation.
Figure 3a presents the HHG oscillation phase, for various crystal orientations. The oscillation phases of the lower harmonics (H11, H13 and H15) emitted far from the edge of the Brillouin zone remain unchanged with the crystal orientation. Indeed, the bottom of the first conduction band, as well as its distance from the valence band (ϵ_{c1,v} = ϵ_{c1} − ϵ_{v}), is approximately isotropic (Fig. 3c).
For higher harmonics (N = 19−29), a strong variation in the oscillation phase with crystal orientation is observed (Fig. 3a). Figure 3b presents the oscillation phase as a function of crystal orientation for H21. The phase shows a sharp increase by more than π as the orientation angle changes towards 45°. The origin of this sharp variation can be understood by looking at the angular dependence of the bands. Figure 3c shows the energy difference between the valence, first and second conduction bands (ϵ_{c1,v}, ϵ_{c2,v}) for different crystal orientations, together with the energy contours for H21. The crystal momentum at the emission point associated with H21 quickly changes with the crystal orientation (Fig. 3d). The strong orientationdependent modifications of the bandgap, ϵ_{c2,v}, lead to notable angular changes of the corresponding electron trajectories, as captured by the phase measurement. In contrast, the emission points for H11–H15 are almost constant with the crystal orientation (Fig. 3d). Such nearly isotropic response is captured by the phase measurement, too (Fig. 3b).
We now turn to resolving the lightinduced dressing of the band structure. Although in large bandgap materials, such as MgO, observing the bandgap modifications requires high intensities, and the gap between the conduction bands is relatively small. Therefore, it can be notably modified at moderate field intensities. We focus on harmonic emission associated with an energy gap between the first and second conduction bands. In Fig. 4a, we plot the fieldfree band structure, which provides a good description of the system at low laser intensities. At 0° orientation, the minimum energy gap between the two conduction bands, ϵ_{c2} − ϵ_{c1}, is around 3 eV; H20 is located inside this gap. As we rotate the crystal, the energy gap rapidly reduces to zero, and H20 is emitted from the second conduction band. In Fig. 4b, we plot the oscillation phase of H20 (Φ_{20}) as a function of crystal orientation for different laser intensities. At low intensity (light green), Φ_{20} shows a dramatic angular dependence, mainly between 0° and 25° as a consequence of the strong angular dependence of ϵ_{c2} − ϵ_{c1}. Importantly, as we increase the field intensity, the angular dependence of Φ_{20} decreases and then notably flattens (dark green). These experimental results are confirmed by our numerical simulations (Fig. 4c). Numerically, the flattening of the angular dependence of the oscillation phase coincides with the onset of strong subcycle Landau–Dykhnetype transitions, with the probability approaching 50% (Supplementary Information).
Our theoretical and numerical results link these intensitydependent observations with the closing of the effective bandgap by the laser field. According to equation (2), the oscillation phase is dictated by the additional complex phase σ induced by the SH field^{23}. The imaginary component, Im(σ), is associated with perturbations of tunnelling and recombination probabilities, whereas the real component Re(σ) reflects subtle modifications in the electron trajectory as it propagates within the band. Since the odd and even harmonics represent constructive and destructive interference of two subcycle emissions, Re(σ) leads to an oscillation phase difference. However, Im(σ) simultaneously affects both even and odd harmonics; therefore, their oscillation phases coincide (Supplementary Information).
In Fig. 4d, we plot the relative oscillation phase of H19 and H20, namely, Φ_{20} − Φ_{19}, at 0° orientation for different fundamental field intensities. At a sufficiently low laser intensity, this phase difference vanishes, reflecting the dominant role of the imaginary component. The origin of this imaginary component could be associated with an anomalous emission event of these harmonics, emitted from an energy gap. At higher field intensities, the band structure is strongly dressed so that the energy gap between the bands becomes negligible. Therefore, the associated imaginary component is reduced and the perturbation is dominated by Re(σ), leading to a phase difference between the even and odd harmonics (Supplementary Information).
In summary, our study establishes the alloptical spectroscopy of a strongly driven crystal, revealing a laserinduced modification of the band structure. We identify the dynamical transitions between several conduction bands as well as probe their structural dependence. Importantly, we resolve the clear signature of harmonic emission from the energy gap between the two conduction bands, probing its modification by the laser field. This study provides a general framework for resolving and interpreting attosecond electronic response phenomena in strongly driven solids. Looking forward, twocolour HHG spectroscopy opens a window into the observation of a broad range of electronic phenomena—from subcycle phase transitions to ultrafast dynamics in correlated systems; some of these have been theoretically predicted decades ago, whereas others are still hotly debated.
Methods
Dressed bands in strongly driven solids
We consider the response of a twoband solid driven by a strong lowfrequency field, and we show how an effective band structure can be introduced in this case on the subcycle timescale, which is relevant for lowfrequency drivers. In addition to a simple result that averages the instantaneous adiabatic energies over the current wavefunction, we introduce an extension of the Floquettype analysis to the subcycle timescale.
Floquettype analysis on the subcycle timescale: effective bandgap in a strong lowfrequency field
Let us start with some welldefined eigenstate, with wavefunction ψ_{κ} and energies E_{g}, which satisfies the stationary Schrödinger equation, namely, \(\hat{H}{\psi }_{\mathbf{\upkappa}}={E}_{\mathrm{g}}{\psi }_{\mathbf{\upkappa}}\). Here κ collects the quantum numbers that label the state. In a solid, κ labels the crystal momentum k together with band index n.
Let the Hamiltonian depend on some external parameter β, that is, \(\hat{H}(\beta )\). If we change β slowly, our eigenstate will slowly evolve following the stationary Schrödinger equation
In the lowfrequency field, where the driver has frequency ω substantially lower than the energy gap, such parameter β is β = ωt. From now on, we will simply refer to time t. In the lowfrequency field, the adiabatic states are solutions of the stationary Schrödinger equation with time treated as a parameter:
where V(t) is the interaction with the lowfrequency field. The adiabatic eigenstates ϕ_{κ}(t) form a complete basis set. Each state has an associated timedependent state Ψ_{κ} that incorporates the standard energy phase factor as follows:
Putting this back into the timedependent Schrödinger equation, we see that the equation for Ψ_{κ} contains an extra term proportional to the derivative of the eigenstate:
As long as the nonadiabatic Landau–Dykhnetype transitions (between the adiabatic states) caused by this term are small, their instantaneous energies ϵ_{κ}(t) offer a very good approximation for the effective instantaneous (and hence the subcycle) energies of the driven system. However, in the presence of strong nonadiabatic transitions between the adiabatic states, these concepts require corrections.
Consider now the specific case of a solid, with bands ϵ_{n}(k) and Bloch wavefunctions ϕ_{n,k}, interacting with a lowfrequency laser field. We note that the analysis below is not, in fact, limited to such a lowfrequency case, but the lowfrequency case presents the most natural physical situation where our analysis and its conclusions are physically transparent.
With the light–solid interaction treated in the dipole approximation and in the length gauge, the initial crystal momentum k becomes a function of time. We label this timedependent momentum κ(t) = k + A(t), where A(t) is the field vector potential.
It is very useful then to use the Houston states to analyse the interaction. In this basis, Bloch wavefunctions ϕ_{n,k} and band energies ϵ_{n}(k) also follow the vector potential, replacing k with κ(t) = k + A(t). The fieldfree energy ϵ_{n}(k) goes into ϵ_{n}(κ(t)) ≡ ϵ_{n}(k + A(t)), so that the band energy ‘slides’ with instantaneous momentum κ(t). The Bloch wavefunction also ‘slides’ with this instantaneous momentum, becoming ϕ_{n,κ(t)}.
In this basis, each timedependent crystal momentum κ(t) = k + A(t) ‘traces’ its own multilevel system, with its states labelled with band index n as well as timedependent energies and couplings. The multilevel system with crystal momentum k is decoupled from other multilevel systems with momenta k′.
Beyond adiabatic evolution
Consider a twoband solid, with band indexes n = 1, 2, driven by a lowfrequency field. Owing to the low frequency, the adiabatic evolution would be a good approximation for most part of the Brillouin zone. The adiabatic evolution will break down to the greatest extent in regions of the smallest bandgap, with exponential dependence on the bandgap.
Different k values reach these regions at different times and hence with different instantaneous values of field F(t). Suppose we start in state k, turn on the field, and at a moment t_{i}, we arrive at some momentum κ in one of the adiabatic states (say with label 1, κ〉), approaching a region where the two bands are coming close to each other.
We shall now look at the propagator across the region of interest. The analysis is general, but for the physical interpretation to be clear, the time interval should be enough to go through the region. For compactness, we will drop the crystal momentum index for the moment.
The propagator, written in the basis of the adiabatic states Ψ^{(1)}〉 and Ψ^{(2)}〉 with energies ϵ^{(1)}(t) and ϵ^{(2)}(t), respectively, has the following form:
This form of the propagator is general and meets the key requirements.
The last point also ensures that the wavefunctions remain orthogonal during passage, that is, \(\langle {{{\varPsi }}}^{(2)}{\hat{U}}^{{\dagger} }\hat{U}{{{\varPsi }}}^{(1)}\rangle =0\).
The meaning of the matrix elements in this propagator is as follows.

The phases λ^{(1,2)}(t) are associated with the adiabatic energies plus, in general, the geometrical Berry phase, γ_{i}.
$${\lambda }^{(i)}(t)=\int\nolimits_{{t}_{i}}^{t}{\mathrm{d}}t^{\prime} {\epsilon }^{(i)}(t^{\prime} )+{\gamma }_{i}$$(9)The reason we want to complete the passage across the region of interest is that we want the geometrical phase, associated with this passage, to accumulate fully. But, if we treat the problem fully numerically, then, of course, such a requirement is not necessary: the phases λ^{(i)}(t) are simply found numerically.

The overall factor in front of the propagator sets the zeroenergy level as
$$\langle \epsilon \rangle =\frac{1}{2}\left[{\epsilon }^{(2)}+{\epsilon }^{(1)}\right]$$(10)through the phases associated with the adiabatic energies.

Irrespective of how the phases λ^{(i)}(t) are obtained, their physical interpretation remains the same: their time derivatives have to be associated with the adiabatic energies (which are viable and meaningful in the absence of nonadiabatic transitions, that is, when sinα ≪ 1).
$${\tilde{\epsilon }}^{(i)}(t)=\frac{\partial {\lambda }^{(i)}(t)}{\partial t}$$(11)The reason to add the ‘tilde’ above the adiabatic energy is to stress that the adiabatic energies \({\tilde{\epsilon }}^{(i)}(t)\) obtained in such a way may not always coincide with the adiabatic energies obtained by diagonalizing the adiabatic Hamiltonian, because of the presence of the geometric phase. Again, we stress that the phases associated with \({\tilde{\epsilon }}^{(i)}(t)\) can be numerically extracted. For the physical interpretation, we will need their derivatives.

The offdiagonal elements describe the Landau–Zener–Dykhne nonadiabatic transitions between the two adiabatic states.

Phases ϕ of these offdiagonal elements are determined by the landscape of the bands and the Berry connections (couplings). As we shall see below, the effective band structure is independent of ɸ.

The probability of staying in the adiabatic state is cos^{2}α and the probability of making the transition is sin^{2}α.
We can now develop the subcycle version of the Floquet analysis. To this end, we return to the propagator, dropping the common zeroenergylevel phase factor for compactness, as
and look for orthogonal states Ψ^{µ}(t_{i})〉 with timedependent quasienergies ϵ^{μ}(t, t_{i}), which depend on crystal momentum k. We want these states to behave as if they were the Floquet states for this propagator.
The positive sign in the phase of the exponent is for convenience because we shall start with the ‘lower’ state, which has negative energy. The minus sign in the equation for the quasienergy is related to the positive sign in the exponent in the first equation.
These quasienergies and the associated states are as close as one can get to the effective bands and effective eigenstates in a strongly driven system with nonadiabatic transitions. As we shall see below, in the absence of nonadiabatic transitions, they—of course—coincide with the adiabatic energies and states.
In principle, one can try to find such states for any time interval after t_{i}, but a meaningful time interval is an interval that is sufficient to cross the transition region. Once the region is crossed, each component of the wavefunction, projected on the adiabatic states, will mostly evolve on the associated adiabatic bands; in the absence of nonadiabatic transitions, these bands are fine and, as mentioned above, coincide with the quasienergies discussed below.
The timedependent ‘eigenstates’ of the propagator have two components corresponding to the amplitudes in the two adiabatic states ɸ^{(}^{i}^{)}(t)〉.
The analysis is straightforward. We consider that the determinant of matrix \(\hat{U}{{\mathrm{e}}}^{{\mathrm{i}}\mu }\hat{1}\) is equal to zero, and the solution is found to be
There are two solutions of this equation, namely, μ_{1} = μ and μ_{2} = −μ, and the quasienergies \({\epsilon }^{{\mu }_{1}}\) and \({\epsilon }^{{\mu }_{2}}\) are obtained by differentiating μ_{1} and μ_{2} with respect to time, respectively.
The first observation is that in the absence of nonadiabatic transitions, when cosα = 1, cosµ = cosλ, μ_{1} = −λ and μ_{2} = λ, and the quasienergies coincide with the adiabatic states.
Equation (15) is already sufficient to find the effective bandgap, which is equal to
We differentiate the two sides of equation (15) with respect to time and find
where we have used that
Using the relationship cosµ = cosαcosλ, we can rewrite
and hence
Finally, we can simplify this expression by taking into account that if the action λ(t) is large, which is usually the case in a strong, lowfrequency field, then sin^{2}λ(t, t_{i}) is a fastoscillating function of t − t_{i}. Replacing it with its average (sin^{2}λ(t, t_{i}) => 1/2) and introducing the notation sin^{2}α = w_{LD}, we get the final result for the bandgap between the two quasienergies as
This result shows that the bandgap collapses when the nonadiabatic Landau–Dykhnetype transition approaches unity, that is, w_{LD} ≃ 1.
We note that the formalism based on the subcycle analogue of the quasienergy states of a strongly driven system is especially attractive because it naturally merges into the Floquet analysis when t − t_{i} is equal to one period.
One can also find the quasieigenenergies in a different way by only using the relationship cosµ = cosαcosλ. Namely, one can use this relationship to solve the equations for the amplitudes a^{μ} and b^{μ}, and then average the full Hamiltonian over the eigenvectors obtained in such a way from equation (14). The result is exactly the same, as expected.
Data availability
The data and datasets that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
Code availability
The custom code used for the current study has been described in previous publications, and parts of it can be made available from the corresponding author on reasonable request.
References
Corkum, P. á & Krausz, F. Attosecond science. Nat. Phys. 3, 381–387 (2007).
Garg, M. et al. Multipetahertz electronic metrology. Nature 538, 359–363 (2016).
Langer, F. et al. Lightwave valleytronics in a monolayer of tungsten diselenide. Nature 557, 76–80 (2018).
Lucchini, M. et al. Attosecond dynamical FranzKeldysh effect in polycrystalline diamond. Science 353, 916–919 (2016).
Schultze, M. et al. Attosecond bandgap dynamics in silicon. Science 346, 1348–1352 (2014).
Schiffrin, A. et al. Opticalfieldinduced current in dielectrics. Nature 493, 70–74 (2013).
Schultze, M. et al. Controlling dielectrics with the electric field of light. Nature 493, 75–78 (2013).
Silva, R., JiménezGalán, Á., Amorim, B., Smirnova, O. & Ivanov, M. Topological strongfield physics on sublasercycle timescale. Nat. Photon. 13, 849–854 (2019).
JiménezGalán, Á., Silva, R., Smirnova, O. & Ivanov, M. Lightwave control of topological properties in 2D materials for subcycle and nonresonant valley manipulation. Nat. Photon. 14, 728–732 (2020).
Lakhotia, H. et al. Laser picoscopy of valence electrons in solids. Nature 583, 55–59 (2020).
CohenTannoudji, C. N. Nobel lecture: manipulating atoms with photons. Rev. Mod. Phys. 70, 707 (1998).
Haroche, S. Nobel lecture: controlling photons in a box and exploring the quantum to classical boundary. Rev. Mod. Phys. 85, 1083 (2013).
Sokolov, A. V., Walker, D. R., Yavuz, D. D., Yin, G. Y. & Harris, S. E. Raman generation by phased and antiphased molecular states. Phys. Rev. Lett. 85, 562 (2000).
Jiangbin, G., MoralesMolina, L. & Hänggi, P. Manybody coherent destruction of tunneling. Phys. Rev. Lett. 103, 133002 (2009).
Keldysh, L. et al. Ionization in the field of a strong electromagnetic wave. Sov. Phys. JETP 20, 1307–1314 (1965).
Jürgens, P. et al. Origin of strongfieldinduced loworder harmonic generation in amorphous quartz. Nat. Phys. 16, 1035–1039 (2020).
Ghimire, S. et al. Observation of highorder harmonic generation in a bulk crystal. Nat. Phys. 7, 138–141 (2011).
Vampa, G. et al. Theoretical analysis of highharmonic generation in solids. Phys. Rev. Lett. 113, 073901 (2014).
Hawkins, P. G. & Ivanov, M. Y. Role of subcycle transition dynamics in highorderharmonic generation in periodic structures. Phys. Rev. A 87, 063842 (2013).
You, Y. S. et al. Laser waveform control of extreme ultraviolet high harmonics from solids. Opt. Lett. 42, 1816–1819 (2017).
Uzan, A. J. et al. Attosecond spectral singularities in solidstate highharmonic generation. Nat. Photon. 14, 183–187 (2020).
Dudovich, N. et al. Measuring and controlling the birth of attosecond XUV pulses. Nat. Phys. 2, 781–786 (2006).
Pedatzur, O. et al. Attosecond tunnelling interferometry. Nat. Phys. 11, 815–819 (2015).
Vampa, G. et al. Linking high harmonics from gases and solids. Nature 522, 462–464 (2015).
Vampa, G. et al. Alloptical reconstruction of crystal band structure. Phys. Rev. Lett. 115, 193603 (2015).
Vampa, G. et al. Attosecond synchronization of extreme ultraviolet high harmonics from crystals. J. Phys. B: At. Mol. Opt. Phys. 53, 144003 (2020).
You, Y. S., Reis, D. A. & Ghimire, S. Anisotropic highharmonic generation in bulk crystals. Nat. Phys. 13, 345–349 (2017).
Schubert, O. et al. Subcycle control of terahertz highharmonic generation by dynamical Bloch oscillations. Nat. Photon. 8, 119–123 (2014).
Wu, M. et al. Orientation dependence of temporal and spectral properties of highorder harmonics in solids. Phys. Rev. A 96, 063412 (2017).
Acknowledgements
N.D. is the incumbent Robin Chemers Neustein Professorial Chair. N.D. acknowledges the Minerva Foundation, the Israeli Science Foundation and the European Research Council for financial support. A.J.U.N. acknowledges financial support by the Rothschild Foundation and the Zuckerman Foundation. M.I. acknowledges funding from the DFG QUTIF grant IV152/62. Á.J.G. acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie grant agreement no. 101028938. Á.J.G. and M.I. acknowledge funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 899794. R.E.F.S. acknowledges support from the fellowship LCF/BQ/PR21/11840008 from ‘La Caixa’ Foundation (ID 100010434) and from the European Union’s Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie grant agreement no. 847648.
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N.D. and M.I. supervised the study. M.I. supervised the theoretical work and developed the theoretical model. A.J.U.N. and G.O. conceived and planned the experiments. Á.J.G. and R.S. performed the theoretical study and numerical analysis. A.J.U.N., T.A.P., G.O. and B.D.B performed the measurements. A.J.U.N., G.O. and S.S. analysed the data. B.Y. performed the DFT calculations. All the authors discussed the results and contributed to writing the manuscript.
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UzanNarovlansky, A.J., JiménezGalán, Á., Orenstein, G. et al. Observation of lightdriven band structure via multiband highharmonic spectroscopy. Nat. Photon. 16, 428–432 (2022). https://doi.org/10.1038/s41566022010101
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DOI: https://doi.org/10.1038/s41566022010101
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