Research Article  Open Access
Yuki Kobayashi, Christian Heide, Hamed Koochaki Kelardeh, Amalya Johnson, Fang Liu, Tony F. Heinz, David A. Reis, Shambhu Ghimire, "Polarization Flipping of EvenOrder Harmonics in Monolayer TransitionMetal Dichalcogenides", Ultrafast Science, vol. 2021, Article ID 9820716, 9 pages, 2021. https://doi.org/10.34133/2021/9820716
Polarization Flipping of EvenOrder Harmonics in Monolayer TransitionMetal Dichalcogenides
Abstract
We present a systematic study of the crystalorientation dependence of highharmonic generation in monolayer transitionmetal dichalcogenides, WS_{2} and MoSe_{2}, subjected to intense linearly polarized midinfrared laser fields. The measured spectra consist of both odd and evenorder harmonics, with a highenergy cutoff extending beyond the 15th order for a laserfield strength around ~1 V/nm. In WS_{2}, we find that the polarization direction of the oddorder harmonics smoothly follows that of the laser field irrespective of the crystal orientation, whereas the direction of the evenorder harmonics is fixed by the crystal mirror planes. Furthermore, the polarization of the evenorder harmonics shows a flip in the course of crystal rotation when the laser field lies between two of the crystal mirror planes. By numerically solving the semiconductor Bloch equations for a gappedgraphene model, we qualitatively reproduce these experimental features and find the polarization flipping to be associated with a significant contribution from interband polarization. In contrast, highharmonic signals from MoSe_{2} exhibit deviations from the laserfield following of oddorder harmonics and crystalmirrorplane following of evenorder harmonics. We attribute these differences to the competing roles of the intraband and interband contributions, including the deflection of the electronhole trajectories by nonparabolic crystal bands.
1. Introduction
During the last decade, solidstate highharmonic generation (HHG) has emerged as a novel spectroscopic tool to probe nonperturbative, ultrafast phenomena in a wide range of condensed matter systems [1–25]. The exciting possibilities include alloptical access to valence chargedensity distributions [26], band structure [4, 27], transition dipole moments [28], and Berry curvature [29, 30], as well as to nonequilibrium laserdriven dynamics [31]. These applications benefit from a detailed understanding of the microscopic mechanisms of HHG, which remains the subject of active experimental and theoretical investigation [32].
One of the important factors governing the process of solidstate HHG is crystal symmetry. Experimentally, the effect of broken inversion symmetry can be observed as emission of evenorder harmonics [11, 29]. Physically, it can be described by using a complex transition dipole moment [33] and a nonzero contribution from Berry connection [22]. A series of recent experiments reported nonperturbative HHG from monolayer transitionmetal dichalcogenides (TMDCs), where the roles of Berry curvature and band nesting were considered [11, 29, 34, 35]. These atomically thin semiconductors have a graphenelike hexagonal structure, but with broken inversion symmetry enabling the generation of evenorder harmonics. In particular, the odd and evenorder harmonics are found to exhibit contrasting behavior in their yield and polarization, suggesting that distinct generation mechanisms underlie them [29]. Systematic analysis of the HHG from monolayer TMDCs, accounting both for the harmonic yield and polarization, can help complete the picture of the role of symmetry effects.
Here, we study anisotropic HHG from monolayer TMDCs, tungsten disulfide (WS_{2}) and molybdenum diselenide (MoSe_{2}), and address the role of the crystal symmetry in defining the experimental results. We find that in WS_{2}, the polarization of the oddorder harmonics continuously follows the direction of the driving laser field, whereas the evenorder harmonics are polarized parallel to the crystal mirror planes. There is a sudden flipping of the polarization of the evenorder harmonics from one crystal mirror plane to another as the laser polarization direction is about midway between the two mirror planes. In MoSe_{2}, we find a deviation from these trends, indicating that the symmetry effects are material dependent. The results are compared to quantummechanical simulations based on a tightbinding gappedgraphene model, with the highharmonic signals analyzed by decomposing them into intraband and interband contributions.
2. Materials and Methods
2.1. Experiments
Our femtosecond midinfrared (MIR) beamline for the solid HHG experiments is based on a Ti:sapphire system that is followed by optical parametric amplification (OPA) and differencefrequency generation (DFG). The experimental configuration is depicted schematically in Figure 1(a). Briefly, the output from the Ti:sapphire amplifier (Evolution, Coherent Inc., 6 mJ, 45 fs, 790 nm, and 1 kHz) is used to pump an OPA system (TOPASHE, Light Conversion Inc.). The signal (~1300 nm) and idler (~1900 nm) from the OPA are mixed in a GaSe crystal (Eksma Optics Inc., zcut, 0.5 mm thick) for DFG, and the generated MIR pulse is spectrally filtered by a bandpass filter that is centered at 5.0 μm (Thorlabs Inc., FB5000500). The polarization of the MIR pulse, first cleaned by a BaF_{2} wiregrid polarizer, is controlled by a zeroorder MgF_{2} halfwave plate and focused onto the monolayer TMDC sample by a ZnSe lens ( mm). The transformlimited duration of the MIR pulse is ~70 fs, and the intensity radius is μm. The laserfield strength in the TMDC samples, which takes into account the amplitude reduction from the fused silica substrate, is estimated to be ~1 V/nm ( W/cm^{2}). The measurements are performed in a transmission geometry at normal incidence. The highharmonic signals are collected by a CaF_{2} lens and directed into a spectrometer equipped with a thermoelectrically cooled CCD camera (Princeton Instruments Inc., Pixis 400B). The highharmonic spectra are corrected for the detection efficiencies of the grating and the CCD camera. A glass wiregrid polarizer is used to resolve the polarization of the highharmonic signals. The millimetersized singlecrystal TMDC monolayers (Figures 1(b) and 1(c)) are exfoliated from the corresponding bulk crystals (HQ graphene Inc.) on a fused silica substrate (MTI Corp Inc.) by means of the goldtape exfoliation method [36]. The fused silica substrate has no birefringence and is transparent over the spectral range of the highharmonic signals. It is also confirmed that for the applied laser parameters, no detectable harmonic signals are produced from the bare substrate.
2.2. QuantumMechanical Simulations
The nonlinear electronhole dynamics in monolayer TMDCs and the resulting highharmonic signals are simulated by numerically solving the semiconductor Bloch equations (SBEs) [37],where and are the state labels, is the density matrix of the system, is the energy of the Bloch state, is the dephasing time constant for the electronhole polarization, is the laser electric field, and represents the dipole matrix elements. The Houston basis [38] is used to represent the electron dynamics, and the crystal momentum is time dependent, , where is the fieldfree crystal momentum and is the vector potential of the laser field.
The system Hamiltonian is based on a nearestneighbor tightbinding model for gapped graphene [39, 40],where is the bandgap, is the hopping parameter between the nearest neighboring atoms, and is the lattice constant. The Hamiltonian yields two bands, and the hexagonal crystal structure of the TMDCs as well as the critical role of the complex transition dipoles in the generation of evenorder harmonics is considered [33]. The highharmonic signals are computed from modulus squared of Fourier transform of the electron current, , which consists of intraband and interband contributions,where are the momentum matrix elements. For the results shown here, the SBEs are solved for the entire Brillouin zone including 10980 initial points by applying the fourthorder RungeKutta method with a step size of 60 as and a dephasing time constant of fs. As representative parameters for the TMDCs, the input values of eV, eV, and Å are used. The initial distribution of carriers in the simulations assumes an undoped sample, with a fully occupied valence band and an empty conduction band. The parameters for the laser pulse employed in the simulations are 5 μm center wavelength, 100 fs pulse duration, and 0.6 V/nm peak field amplitude. The smaller laser amplitude compared to the experimental condition is chosen so that the cutoff energy of the highharmonic signals matches with the experimental results.
2.3. StrongField Driving Conditions
To drive the electronhole dynamics throughout the band structure and thus produce highharmonic signals nonperturbatively, the laser parameters are chosen to be in the socalled strongfield regime. The Keldysh adiabaticity parameter for a twoband model is given by [41–44]:where is the driving laser frequency, is the reduced electronhole mass, and is the driving laser amplitude. For an applied laser amplitude of V/nm, a driving wavelength of 5 μm, and a reduced mass of 0.18 ( representing the electron mass) [45], we obtain . Since , the lasermatter interaction can be treated in the strongfield regime, where tunneling excitation becomes important.
To evaluate the intraband dynamics or Bloch oscillation of the electronhole pairs in solids, a ratio between the Bloch frequency and the driving laser frequency is considered [46]:
The present experimental parameters yield , and the condition of corresponds to the regime where Bloch oscillations begin to occur on the subcycle time scale of the HHG process.
3. Results
3.1. CrystalOrientation Dependence of the HighHarmonic Yield
The highharmonic spectra from the monolayer WS_{2} and MoSe_{2} are shown in Figures 1(d) and 1(e), respectively. The present measurements reproduce the main features reported in the previous experiments [11, 29], i.e., the odd and evenorder harmonics are produced over a wide spectral range from 1.2 eV (5th harmonic) to 3.7 eV (15th harmonic), with the evenorder harmonics most prominent around 3.03.5 eV. The appearance of the evenorder harmonics is a direct consequence of the reduced spatial symmetry of the monolayer systems.
Figures 2(a) and 2(b) show the HHG spectra measured for WS_{2} and MoSe_{2}, respectively, while changing the laser polarization angle, . The angle is referenced to one of the crystal mirror planes (Figure 2(a), inset). The measured highharmonic spectra exhibit 6fold symmetric (i.e., 60° periodic) patterns, a result of the 3fold crystal symmetry of the TMDCs probed by a multicycle (inversion symmetric) laser pulse. In WS_{2} (Figure 2(a)), the angledependent modulation is more pronounced in the evenorder harmonics (10th, 12th, and 14th), but the higher oddorder harmonics (11th and 13th) also exhibit discernible amplitude modulation. The periodic modulations of the yield for the odd and evenorder harmonics are not necessarily out of phase with one another; the 13th harmonic shows clear outofphase oscillation with respect to the 10th, 12th, and 14th, but the 11th harmonic is in phase with these evenorder harmonics. Similarly, in MoSe_{2} (Figure 2(b)), the optimum excitation direction varies for different harmonics. These results suggest that the electronhole dynamics are not confined to a minimum bandgap region of the Brillouin zone [47].
To shed light on the measured anisotropy of the highharmonic signals, the experimental results are compared to the SBE simulations. It bears mentioning that the present simulations have a few limitations, notably that the spinorbit coupling and electron correlation are not considered, and that only the highest valence and lowest conduction bands are included. Despite these simplifications, the qualitative features of the experiments are captured by the simulations (Figure 2(c)). Both the odd and evenorder harmonics are produced up to the 15th harmonic (3.72 eV). The polarization dependence of the 10th harmonic is outofphase with that of the 14th harmonic although they are both evenorder harmonics, which also matches with the experimental results of MoSe_{2} (Figure 2(b)). The yield of the lower evenorder harmonics (8th and 10th) is overestimated in the simulations, but a qualitative agreement is obtained and thus the anisotropic HHG observed in the experiments is likely to be correctly captured within this twoband model.
3.2. Symmetry Considerations for HHG
The anisotropic HHG from monolayer TMDCs is further investigated by analyzing the polarization state of the highharmonic radiation. Figures 3(a) – 3(c) show the polarization scans of the 13th (solid circle) and 14th (open circle) harmonics of WS_{2} when the excitation direction is (a) 0°, (b) , and (c) . In the case of the parallel excitation (Figure 3(b)), the 13th and 14th harmonics are polarized parallel to the MIR field, whereas in the case of the perpendicular excitation (Figures 3(a) and 3(c)), the 13th harmonic is parallel to the MIR field and the 14th harmonic is perpendicular to it. These results are in line with the symmetry restrictions required for mirrorplane systems (Table 1) [11, 29]. When the applied laser field is parallel to the mirror plane (top row), highharmonic signals can be produced only along the direction of the driving field, as symmetry requires the polarization to vanish in the perpendicular direction. When excitation is perpendicular to the mirror plane (bottom row), the parallel and perpendicular polarizations cancel out for the even and oddorder harmonics, respectively, because of the mirror symmetry. The present results verify that these symmetry restrictions are obeyed in the nonperturbative process of HHG.

3.3. CrystalOrientation Dependence of the HighHarmonic Polarization
We investigate HHG dynamics beyond the symmetry restrictions by analyzing the polarization of highharmonic spectra while scanning the pump polarization angle from . The results for WS_{2} and MoSe_{2} are shown in Figures 4(a) – 4(d), respectively. The HHG polarization angle is referenced with respect to the crystal mirror plane, the same way as for the MIR polarization angle (Figure 2(a), inset).
In WS_{2} (Figure 4(b)), the polarization angles for the oddorder harmonics (solid circles) follow the MIR polarization angle (dashed line) very closely, within ±5°, irrespective of the harmonic orders. The oddorder harmonics are linearly polarized, with their ellipticity measured to be < (i.e., less than 4% of an offaxis intensity component). This result indicates that the electronhole trajectories in WS_{2} are mostly along the direction of the MIR laser field [26, 47]. The evenorder harmonics exhibit a unique behavior (Figure 4(b), open circles). The yield of the evenorder harmonics is maximized when the MIR excitation is perpendicular to the crystal mirror planes (°), and around this regime, the polarization direction is fixed along the crystal mirror planes (°). This result can be explained by the fact that the spatial symmetry is broken along the direction of the crystal mirror planes, and the evenorder harmonics can only be generated only along those directions. As the MIR excitation changes from the perpendicular to the parallel direction, the polarization of the evenorder harmonics starts to deviate from the crystal mirror planes and flips from one to the other, which can be interpreted as competition between different crystal axes. All in all, the results can be summarized as oddorder harmonics that are polarized in the direction of the MIR field and evenorder harmonics mainly polarized along the direction of the crystal mirror planes.
The same polarization measurements are performed for monolayer MoSe_{2} (Figures 4(c) and 4(d)), and the results exhibit a qualitatively different behavior from that of WS_{2}. The oddorder harmonics follow the MIR polarization direction only for the lower orders (5th, 7th, and 9th), whereas the higher orders (11th and 13th) show noticeable deviation. These results indicate that the electronhole trajectories in MoSe_{2} are highly deflected by the periodic crystal potentials, a similar dynamics to that shown in a previous study of HHG in MgO, where the electronhole trajectories were found to depend on the dispersion of the bands [26, 47]. The smaller bandgap of MoSe_{2} (1.56 eV) compared to that of (2.01 eV) further suggests that stronger effects of nonparabolic band in amplify the deflection.
The polarization direction of the evenorder harmonics also shows complex behavior: the 12th and 14th harmonics are polarized along the crystal axis relatively well when the MIR excitation field is perpendicular to the mirror planes (°), while the 10th harmonic shows a continuous variation. A clear contrast is also present in the harmonic yield (Figure 4(c)): the 12th and 14th harmonics are maximized for perpendicular excitation (0°), but the 10th harmonic is maximized for parallel excitation (), at which point the polarization is parallel to the driving MIR field.
The dynamic polarization response from the TMDCs is further investigated in the SBE simulations by decomposing the highharmonic signals into the interband and intraband contributions (Equations (4) and (5)). Figure 5(a) shows the simulated HHG spectra that are averaged for the excitation direction. The trend can be summarized as follows: (i) the intraband contribution is dominant for the lower oddorder harmonics, and (ii) the interband contribution is dominant for all the evenorder harmonics and the higher oddorder harmonics. This result is in line with a previous study [11], where the evenorder harmonics from the TMDCs were attributed to interband polarization.
Figures 5(b) – 5(d) show the highharmonic polarization response for the intraband, interband, and total contributions, respectively. Overall, the total contribution result (Figure 5(d)) reproduces the qualitative features of the experiments: the symmetry restrictions are satisfied, the oddorder harmonics largely follow the MIR polarization direction, and the evenorder harmonics are mostly polarized along the crystal axes. The intraband versus interband analysis for the oddorder harmonics (Figures 5(b) and 5(c), filled circles) shows that the intraband contribution follows the MIR direction with little deviation, while the interband contribution shows a noticeable modulation. Experimentally for MoSe_{2} (Figure 4(d)), the 11th and 13th harmonics are found to show deviation from the MIR polarization direction, and the SBE simulations suggest that this result can be understood as reflecting significant contributions from the interband polarization. The other harmonic signals that follow the direction of the MIR polarization may contain both the intraband and interband contributions.
The same analysis for the evenorder harmonics (Figures 5(b) and 5(c), open circles) reveals that the polarization of the intraband contribution (Figure 5(b)) is fixed along the crystal mirror planes for the parallel excitation (). On the other hand, the interband contribution shows a fixed polarization for the perpendicular excitation (0°), and the total results (Figure 5(d)) replicate the behavior of the interband contribution as expected from its dominant ratio for the evenorder harmonics (Figure 5(a)). The experimental results (Figures 4(b) and 4(d)) show similar trends to that of the interband contribution, indicating its significant role in the evenorder highharmonic generation in monolayer TMDCs.
4. Discussion
We have provided a comprehensive analysis for the crystalorientation dependence of anisotropic HHG from the two materials of the TMDC family. In WS_{2}, we find that the polarization direction of the oddorder harmonics smoothly follows that of the driving laser field irrespective of crystal orientation. Polarization characteristics of the evenorder harmonics are more complex and are influenced significantly by the crystal symmetry. In particular, they show a flip during the course of the crystal rotation when the pump polarization is between the crystal mirror planes. These results are reproduced qualitatively by numerical simulations that employ a tightbinding gappedgraphene model, showing that key features of the experimentally observed behavior can be explained by a minimum twoband system within a singleactive electron picture. The highharmonic signals from MoSe_{2} exhibit noticeable deviations from these trends. In particular, the 10th harmonic is maximized for parallel excitation and its polarization exhibits a clear contrast between the other evenorder harmonics. The combined results of the experiments and simulations presented in this work suggest the capabilities of the polarizationresolved HHG measurements to reveal the roles of the intraband and interband contributions as well as the deflection of the electronhole trajectories by nonparabolic bands in the crystal.
Data Availability
The experimental and simulation data presented in this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this article.
Authors’ Contributions
Y.K. and C.H. performed the experiments and analyzed the data. A.J. and F.L. fabricated the monolayer samples. H.K.K., Y.K., and C.H. performed the simulations. T.F.H., D.A.R., and S.G. supervised the project. All authors contributed to the writing of the manuscript. Y.K. and C.H. contributed equally to this work.
Acknowledgments
This work was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division through the AMOS program. F.L. was supported by a Terman Fellowship and startup funds from the Department of Chemistry at Stanford University. Y.K. acknowledges support from the UrbanekChodorow Fellowship from Stanford University. C.H. acknowledges support from the W. M. Keck Foundation and a Humboldt Research Fellowship. We thank Dr. S. Azar Oliaei Motlagh for the fruitful discussion on the simulations.
References
 S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of highorder harmonic generation in a bulk crystal,” Nature Physics, vol. 7, no. 2, pp. 138–141, 2011. View at: Publisher Site  Google Scholar
 G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of highharmonic generation in solids,” Physical Review Letters, vol. 113, no. 7, artice 073901, 2014. View at: Publisher Site  Google Scholar
 O. Schubert, M. Hohenleutner, F. Langer et al., “Subcycle control of terahertz highharmonic generation by dynamical Bloch oscillations,” Nature Photonics, vol. 8, no. 2, pp. 119–123, 2014. View at: Publisher Site  Google Scholar
 T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, “Extreme ultraviolet highharmonic spectroscopy of solids,” Nature, vol. 521, no. 7553, pp. 498–502, 2015. View at: Publisher Site  Google Scholar
 M. Hohenleutner, F. Langer, O. Schubert et al., “Realtime observation of interfering crystal electrons in highharmonic generation,” Nature, vol. 523, no. 7562, pp. 572–575, 2015. View at: Publisher Site  Google Scholar
 G. Ndabashimiye, S. Ghimire, M. Wu et al., “Solidstate harmonics beyond the atomic limit,” Nature, vol. 534, no. 7608, pp. 520–523, 2016. View at: Publisher Site  Google Scholar
 N. Saito, P. Xia, F. Lu, T. Kanai, J. Itatani, and N. Ishii, “Observation of selection rules for circularly polarized fields in highharmonic generation from a crystalline solid,” Optica, vol. 4, no. 11, pp. 1333–1336, 2017. View at: Publisher Site  Google Scholar
 Z. Wang, H. Park, Y. H. Lai et al., “The roles of photocarrier doping and driving wavelength in high harmonic generation from a semiconductor,” Nature Communications, vol. 8, no. 1, p. 1686, 2017. View at: Publisher Site  Google Scholar
 N. Yoshikawa, T. Tamaya, and K. Tanaka, “Highharmonic generation in graphene enhanced by elliptically polarized light excitation,” Science, vol. 356, no. 6339, pp. 736–738, 2017. View at: Publisher Site  Google Scholar
 N. TancogneDejean and A. Rubio, “Atomiclike highharmonic generation from twodimensional materials,” Science Advances, vol. 4, no. 2, p. eaao5207, 2018. View at: Publisher Site  Google Scholar
 N. Yoshikawa, K. Nagai, K. Uchida et al., “Interband resonant highharmonic generation by valley polarized electron hole pairs,” Nature Communications, vol. 10, no. 1, p. 3709, 2019. View at: Publisher Site  Google Scholar
 N. Klemke, N. TancogneDejean, G. M. Rossi et al., “Polarizationstateresolved highharmonic spectroscopy of solids,” Nature Communications, vol. 10, no. 1, p. 1319, 2019. View at: Publisher Site  Google Scholar
 L. Yue and M. B. Gaarde, “Structure gauges and laser gauges for the semiconductor Bloch equations in highorder harmonic generation in solids,” Physical Review A, vol. 101, no. 5, 2020. View at: Publisher Site  Google Scholar
 F. Navarrete and U. Thumm, “Twocolordriven enhanced highorder harmonic generation in solids,” Physical Review A, vol. 102, no. 6, article 063123, 2020. View at: Publisher Site  Google Scholar
 P. Jürgens, B. Liewehr, B. Kruse et al., “Origin of strongfieldinduced loworder harmonic generation in amorphous quartz,” Nature Physics, vol. 16, no. 10, pp. 1035–1039, 2020. View at: Publisher Site  Google Scholar
 A. J. Uzan, G. Orenstein, Á. JiménezGalán et al., “Attosecond spectral singularities in solidstate highharmonic generation,” Nature Photonics, vol. 14, no. 3, pp. 183–187, 2020. View at: Publisher Site  Google Scholar
 R. Hollinger, D. Hoff, P. Wustelt et al., “Carrierenvelopephase measurement of fewcycle midinfrared laser pulses using high harmonic generation in ZnO,” Optics Express, vol. 28, no. 5, pp. 7314–7322, 2020. View at: Publisher Site  Google Scholar
 A. Chacón, D. Kim, W. Zhu et al., “Circular dichroism in higherorder harmonic generation: heralding topological phases and transitions in Chern insulators,” Physical Review B, vol. 102, no. 13, article 134115, 2020. View at: Publisher Site  Google Scholar
 Y. Bai, F. Fei, S. Wang et al., “Highharmonic generation from topological surface states,” Nature Physics, vol. 17, no. 3, pp. 311–315, 2021. View at: Publisher Site  Google Scholar
 M. S. Mrudul, Á. JiménezGalán, M. Ivanov, and G. Dixit, “Lightinduced valleytronics in pristine graphene,” Optica, vol. 8, no. 3, pp. 422–427, 2021. View at: Publisher Site  Google Scholar
 Y. Morimoto, Y. Shinohara, M. Tani, B.H. Chen, K. L. Ishikawa, and P. Baum, “Asymmetric singlecycle control of valence electron motion in polar chemical bonds,” Optica, vol. 8, no. 3, pp. 382–387, 2021. View at: Publisher Site  Google Scholar
 D. Baykusheva, A. Chacón, D. Kim, D. E. Kim, D. A. Reis, and S. Ghimire, “Strongfield physics in threedimensional topological insulators,” Physical Review A, vol. 103, no. 2, article 023101, 2021. View at: Publisher Site  Google Scholar
 C. P. Schmid, L. Weigl, P. Grössing et al., “Tunable noninteger highharmonic generation in a topological insulator,” Nature, vol. 593, no. 7859, pp. 385–390, 2021. View at: Publisher Site  Google Scholar
 L. Yue and M. B. Gaarde, “Expanded view of electronhole recollisions in solidstate highorder harmonic generation: fullBrillouinzone tunneling and imperfect recollisions,” Physical Review A, vol. 103, no. 6, article 063105, 2021. View at: Publisher Site  Google Scholar
 G. Wang and T.Y. Du, “Quantum decoherence in highorder harmonic generation from solids,” Physical Review A, vol. 103, no. 6, article 063109, 2021. View at: Publisher Site  Google Scholar
 Y. S. You, D. Reis, and S. Ghimire, “Anisotropic highharmonic generation in bulk crystals,” Nature Physics, vol. 13, no. 4, pp. 345–349, 2017. View at: Publisher Site  Google Scholar
 G. Vampa, T. J. Hammond, N. Thiré et al., “Alloptical reconstruction of crystal band structure,” Physical Review Letters, vol. 115, no. 19, 2015. View at: Publisher Site  Google Scholar
 K. Uchida, V. Pareek, K. Nagai, K. M. Dani, and K. Tanaka, “Visualization of twodimensional transition dipole moment texture in momentum space using highharmonic generation spectroscopy,” Physical Review B, vol. 103, no. 16, article L161406, 2021. View at: Publisher Site  Google Scholar
 H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, “Highharmonic generation from an atomically thin semiconductor,” Nature Physics, vol. 13, no. 3, pp. 262–265, 2017. View at: Publisher Site  Google Scholar
 T. T. Luu and H. J. Wörner, “Measurement of the Berry curvature of solids using highharmonic spectroscopy,” Nature Communications, vol. 9, no. 1, p. 916, 2018. View at: Publisher Site  Google Scholar
 R. E. F. Silva, Á. JiménezGalán, B. Amorim, O. Smirnova, and M. Ivanov, “Topological strongfield physics on sublasercycle timescale,” Nature Photonics, vol. 13, no. 12, pp. 849–854, 2019. View at: Publisher Site  Google Scholar
 S. Ghimire and D. A. Reis, “Highharmonic generation from solids,” Nature Physics, vol. 15, no. 1, pp. 10–16, 2019. View at: Publisher Site  Google Scholar
 S. Jiang, J. Chen, H. Wei, C. Yu, R. Lu, and C. D. Lin, “Role of the transition dipole amplitude and phase on the generation of odd and even highorder harmonics in crystals,” Physical Review Letters, vol. 120, no. 25, article 253201, 2018. View at: Publisher Site  Google Scholar
 Z. Lou, Y. Zheng, C. Liu et al., “Ellipticity dependence of nonperturbative harmonic generation in fewlayer MoS_{2},” Optics Communications, vol. 469, p. 125769, 2020. View at: Publisher Site  Google Scholar
 C. Liu, Y. Zheng, Z. Zeng, and R. Li, “Polarizationresolved analysis of highorder harmonic generation in monolayer MoS_{2},” New Journal of Physics, vol. 22, no. 7, article 073046, 2020. View at: Publisher Site  Google Scholar
 F. Liu, W. Wu, Y. Bai et al., “Disassembling 2d van der waals crystals into macroscopic monolayers and reassembling into artificial lattices,” Science, vol. 367, no. 6480, pp. 903–906, 2020. View at: Publisher Site  Google Scholar
 J. Li, X. Zhang, S. Fu, Y. Feng, B. Hu, and H. Du, “Phase invariance of the semiconductor Bloch equations,” Physical Review A, vol. 100, no. 4, article 043404, 2019. View at: Publisher Site  Google Scholar
 M. Wu, S. Ghimire, D. A. Reis, K. J. Schafer, and M. B. Gaarde, “Highharmonic generation from Bloch electrons in solids,” Physical Review A, vol. 91, no. 4, article 043839, 2015. View at: Publisher Site  Google Scholar
 S. A. Oliaei Motlagh, F. Nematollahi, A. Mitra, A. J. Zafar, V. Apalkov, and M. I. Stockman, “Ultrafast optical currents in gapped graphene,” Journal of Physics: Condensed Matter, vol. 32, no. 6, p. 65305, 2020. View at: Publisher Site  Google Scholar
 H. K. Kelardeh, “Ultrafast and strongfield physics in graphenelike crystals: Bloch band topology and highharmonic generation,” 2021, http://arxiv.org/abs/2101.03635. View at: Google Scholar
 L. Keldysh, “Ionization in the field of a strong electromagnetic wave,” JETP, vol. 20, no. 5, 1965. View at: Google Scholar
 H. R. Reiss, “Effect of an intense electromagnetic field on a weakly bound system,” Physical Review A, vol. 22, no. 5, pp. 1786–1813, 1980. View at: Publisher Site  Google Scholar
 S. Y. Kruchinin, F. Krausz, and V. S. Yakovlev, “Colloquium: strongfield phenomena in periodic systems,” Reviews of Modern Physics, vol. 90, no. 2, article 021002, 2018. View at: Publisher Site  Google Scholar
 C. Heide, T. Boolakee, T. Higuchi, and P. Hommelho, “Adiabaticity parameters for the categorization of lightmatter interaction  from weak to strong driving,” 2021, http://arxiv.org/abs/2104.10112. View at: Google Scholar
 Z. Jin, X. Li, J. T. Mullen, and K. W. Kim, “Intrinsic transport properties of electrons and holes in monolayer transitionmetal dichalcogenides,” Physical Review B, vol. 90, no. 4, article 045422, 2014. View at: Publisher Site  Google Scholar
 S. Ghimire, G. Ndabashimiye, A. D. DiChiara et al., “Strongfield and attosecond physics in solids,” Journal of Physics B: Atomic, Molecular and Optical Physics, vol. 47, no. 20, p. 204030, 2014. View at: Publisher Site  Google Scholar
 Y. S. You, J. Lu, E. F. Cunningham, C. Roedel, and S. Ghimire, “Crystal orientationdependent polarization state of highorder harmonics,” Optics Letters, vol. 44, no. 3, pp. 530–533, 2019. View at: Publisher Site  Google Scholar
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