Orbital perspective on high-harmonic generation from solids

High-harmonic generation in solids allows probing and controlling electron dynamics in crystals on few femtosecond timescales, paving the way to lightwave electronics. In the spatial domain, recent advances in the real-space interpretation of high-harmonic emission in solids allows imaging the field-free, static, potential of the valence electrons with picometer resolution. The combination of such extreme spatial and temporal resolutions to measure and control strong-field dynamics in solids at the atomic scale is poised to unlock a new frontier of lightwave electronics. Here, we report a strong intensity-dependent anisotropy in the high-harmonic generation from ReS2 that we attribute to angle-dependent interference of currents from the different atoms in the unit cell. Furthermore, we demonstrate how the laser parameters control the relative contribution of these atoms to the high-harmonic emission. Our findings provide an unprecedented atomic perspective on strong-field dynamics in crystals, revealing key factors to consider in the route towards developing efficient harmonic emitters.

lution.The combination of such extreme spatial and temporal resolutions to measure and control strong-field dynamics in solids at the atomic scale is poised to unlock a new frontier of lightwave electronics.Here, we report a strong intensity-dependent anisotropy in the high-harmonic generation from ReS 2 that we attribute to angle-dependent interference of currents from the different atoms in the unit cell.Furthermore, we demonstrate how the 1 arXiv:2309.06290v1[physics.optics]12 Sep 2023 laser parameters control the relative contribution of these atoms to the high-harmonic emission.Our findings provide an unprecedented atomic perspective on strong-field dynamics in crystals and suggest that crystals with a large number of atoms in the unit cell are not necessarily more efficient harmonic emitters than those with fewer atoms.
The foundational concept underpinning attosecond physics, and high-harmonic generation in gas-phase atoms and molecules in particular, is the energetic recollision of an electron ionized and accelerated by a strong laser field with the parent ion [1][2][3][4] .This dynamic real-space framework is instrumental to link the characteristics of the emitted harmonic radiation (amplitudes, phases and polarization) to sub-laser-cycle dynamics of atomic and molecular orbitals [5][6][7][8][9] .In solids, highharmonic generation (HHG) is understood using a similar framework, albeit exchanging the realspace perspective for one in reciprocal space, where electron-hole pairs accelerate and recombine across energy bands in the Brillouin zone of the crystal [10][11][12][13] .This reciprocal-space approach has been paramount in virtually all investigations of solid-state high-harmonics: from revealing the role of electron-hole recollisions in the emission process 14 , to reconstructing the band structure of a ZnO crystal 15 , to explaining the multiple plateaus observed in the HHG spectrum 16 and to map regions of crystal momenta where the electron-hole velocity vanishes 17 , among others [18][19][20][21][22][23][24] .
Despite the success of the reciprocal-space picture, a real-space approach offers a more intuitive framework, in particular in complex materials with many narrowly spaced and overlapping bands.The advantages of using a real-space perspective to understand HHG from solids are quickly starting to become apparent [25][26][27] , for example, in interpreting spatially-displaced electron-hole recollision processes 27,28 or as means to directly reconstruct the field-free (static) valence electron potential at the picometer scale 29 .The possibility to link features of the high-harmonic spectrum to dynamics occurring at specific orbitals in the lattice remains, however, largely unexplored.Here, we demonstrate this possibility through angle-resolved measurements of HHG in ReS 2 .We measure a strong, intensity-dependent anisotropy of the HHG emission and trace it back to the interplay between the currents generated by each individual atom in the unit cell.Simulating the laser-matter interaction using a basis constructed from maximally-localized Wannier orbitals, we show that by changing the laser parameters (intensity and polarization), one can activate or suppress the contribution of specific atoms to the HHG emission and interfere the atomic currents differently, increasing or decreasing the high-harmonic emission efficiency.
Figure 1a illustrates the unit cell of the monolayer, formed by 4 rhenium atoms and 8 sulfur atoms.
The 4 Re clusters are linked in a chain oriented along θ = 120 ○ (see panel a).While the anisotropy of the crystal structure is clear (the crystal symmetry group is P-1), the band structure is similar along different angles and is very dense (see Figure 1b and Supplementary Note 1), and with a density of states near the Fermi energy significantly higher than other prototypical materials used in HHG spectroscopy, such as MgO or ZnO 35 .Going from the monolayer limit to bulk, these features remain, and the band structure changes only slightly 30,31 .In such dense band diagram, associating an individual harmonic with reciprocal space trajectories of charge carriers in a particular set of bands, according to the reciprocal-space method, is hardly straightforward (see circular markers in Fig. 1b-d), and is unlikely to provide much insight into the carrier dynamics.On the other hand, inversion symmetric and display a very similar electronic band structure, with a nearly identical direct band gap of 1.4eV at the Γ point 30,31 .
the small bandwidth indicates that the electrons are very localized in the individual atoms of the lattice, making it ideally suited for a real-space or orbital-based framework.
The first question we want to address is if high harmonics generated from ReS 2 reflect the strong anisotropy apparent in real space or rather the weak angular dependence of its band structure.We generate high harmonics from bulk ReS 2 with a linearly-polarized mid-infrared pulse with a duration of 80 fs and a center wavelength of 3.5 µm (see Methods).Figure 2a shows the highharmonic spectrum measured for a laser intensity of 0.64 TW/cm 2 and polarization along θ = 120 ○ (see inset).We observe odd harmonics extending up to the 13 th order, while even harmonics are absent as expected from the inversion symmetry of ReS 2 .We measure the orientation dependence of the harmonics by rotating the polarization of the linear pulse with respect to the crystal.The results, shown in Figure 2b-d, display a clear anisotropy for all harmonic orders.Furthermore, the anisotropy depends strongly on the laser intensity.
In order to understand the origin of this anisotropy, we perform time-dependent simulations in a basis constructed from 44 maximally-localized Wannier orbitals (see Methods for details) 36 .
The similarity between the monolayer and bulk forms in the case of ReS 2 30,31 , allows us to reduce the computational complexity and simulate the monolayer system.The orientation dependence of H9 and H11 obtained from the numerical simulations is shown in Figure 3a observed in the experiment.As a result, the simulations in monolayer ReS 2 can provide valuable insight for the origin of this effect.
The high-harmonic spectrum is given by the Fourier components of the time-dependent current that is generated by the laser-induced oscillating dipole of the medium (see Methods), where J α (t) is the total current along direction α = (∥, ⊥), corresponding to the components parallel and perpendicular to the electric field, respectively.The total current can be expressed as a sum of currents from all the orbitals in the lattice, J α (t) = ∑ in the Wannier gauge and, even if they are not observable, provide a unique real-space perspective into the HHG process.Expressed in terms of the individual orbital currents, the high-harmonic spectrum is where A n,α and φ n,α are, respectively, the spectral amplitude and phase of the current of orbital n along direction α.
Equation 2 allows us to distinguish features that arise from the interference of different orbital currents.The incoherent sum of the individual currents, , will be absent of such interference.In Figure 3c we compare the angle-dependent harmonic yield of H11 for I incoh α (solid lines) and the observable signal I α (faint dashed lines).A similar analysis for H9 is made in Supplementary Note 2. The angular variation is stronger for I α , with near-complete suppression of various secondary maxima that are present in I incoh α (most notably near 60°), strongly modifying the orientation dependence.Thus, orbital phase interference is an important factor determining the orientation dependence.
Since the electrons are well localized on each atomic site (see Extended Data), we can group together the currents of the m orbitals belonging to the same atom A into an atomic current, m,α (t).Furthermore, due to the inversion symmetry of ReS 2 , each atom is related to one other by an inversion operation, for example, Re 1 and Re 3 or S 1 and S 6 (see Fig. 1a).
Both of the atoms in the pair give rise to the same Fourier amplitudes and phases, so that the total harmonic spectrum in Eq. 2 reduces to the sum of the Fourier amplitudes and phases of six atomic (inversion-related) pairs.The results shown are for the harmonic polarization α that is parallel to the electric field.
Figure 3b (blue curve).At θ = 100 ○ , the atomic pair Re 2 -Re 4 , which contributes the most to H11 at low intensity, is largely suppressed at large intensity (compare size of orange circle in Figure 4a,b).
This analysis shows that atoms that do not contribute to the generation of a particular harmonic order for one driver intensity, can be activated for other intensities, and vice versa, suggesting that laser intensity could be used as a mechanism to control the relative weight of atomic orbitals in HHG.An analogous analysis can be made for the rest of harmonic orders, along both α =∥, ⊥ directions (see Supplementary Note 2), where we observe a larger spread of the Fourier phases for increasing harmonic orders.This leads to sharper changes in the angle-resolved spectrum for higher orders, as also seen in the experiment.
In conclusion, we identify how the nonlinear currents residing on each of the twelve atoms in the unit cell of a ReS 2 crystal are responsible for the strongly anisotropic and intensity-dependent emission of high-order harmonics.Our orbital analysis based on maximally localized Wannier functions reveals that each atomic contribution depends strongly on the polarization angle and intensity of the driving field, paving the way to characterizing and controlling electron dynamics at the picometer-scale in solids on sub-laser-cycle timescales.Moreover, we show that interference between atoms in the unit cell of a crystal is key to determine the macroscopic high-harmonic emission, a critical factor to consider in the route towards developing efficient harmonic emitters.

Numerical methods
The field-free Hamiltonian and dipole couplings of monolayer ReS 2 were calculated with the electronic structure code Quantum Espresso 37 on a Monkhorst-Pack (MP) grid of 12x12x1 points using a norm-conserving Perdew-Burke-Ernzerhof (PBE) exchange correlation functional.The field-free Hamiltonian used in the time-dependent propagation was constructed by projecting the Bloch states onto a set of maximally-localized Wannier functions using the Wan-nier90 code 36 .In particular, we projected onto the d orbitals of the four rhenium atoms and the p orbitals of the six sulfur atoms, totalling 44 bands.The Hamiltonian in the basis constructed from Wannier functions was then propagated in the presence of the electric field using the density matrix formalism with the code described in Ref. 26 .The large size of the unit cell allowed us to obtain convergence with a modest MP grid of 50x50 k-points along the b 1 and b 2 reciprocal lattice vectors.The time step was set to 0.2 a.u. and the dephasing time was chosen to be T 2 = 10 fs.
The time-dependent current along direction α, used to extract the high harmonic spectrum, is defined as Above, e is the electron charge, N k is the number of crystal momenta included in the calculation, v is the velocity operator, and ρ is the density matrix.In the Wannier gauge, the density matrix ρ (W) contains the orbital populations and coherences in its diagonal and off-diagonal terms, respectively.
In the Wannier gauge, we may define a (real) current from an individual orbital n along direction α as such that the sum of the currents from all orbitals equals the total current, For clarity, we give an example for a two-orbital model, although we point out that our analysis is only relevant for multi-orbital crystals as the one presented in this work.In the twoorbital case, where the subscripts 1 and 2 identify the orbital and α =∥, ⊥ the direction of current emission.
Since both the velocity and density matrices are hermitian, the current of an individual orbital is composed of a term associated to the population change of that orbital, plus exactly half of the contribution of the coherence between that orbital and the rest.Thus, this approach offers a way of quantifying the contribution of individual orbitals, and their interference, to the high-harmonic generation.

FiguresFigure 4 :
Figures 4a,b show the Fourier amplitudes and phases of the six atomic pairs, indicated with