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Near-bandgap harmonic generation in solids from multiple scattering revealed by complex quantum trajectories

Xunqin Huo1,2, Xiwang Liu1, Shidong Yang1,2, Xiaohong Song1,*, Torsten Meier3, and Weifeng Yang1,4,†

  • 1School of Physics and Optoelectronic Engineering, Hainan University, Haikou 570228, China
  • 2School of Information and Communication Engineering, Hainan University, Haikou 570228, China
  • 3Department of Physics and Institute for Photonic Quantum Systems (PhoQS), Paderborn University, Warburger Straße 100, D-33098 Paderborn, Germany
  • 4Center for Theoretical Physics, Hainan University, Haikou 570228, China

  • *Contact author: song_xiaohong@hainanu.edu.cn
  • †Contact author: wfyang@hainanu.edu.cn
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Phys. Rev. Research 8, 033303 – Published 11 September, 2026

DOI: https://doi.org/10.1103/l939-jglf

Abstract

High-harmonic generation in solids enables probing of strong-field electron dynamics in condensed matter and the development of compact ultrafast light sources. Whereas most studies have focused on above-bandgap high-order harmonics, which are usually interpreted through short electron-hole trajectories that recombine within one optical cycle, the microscopic origin of low-order emission near the bandgap remains poorly understood. Here, we investigate near-bandgap harmonic generation in ZnO by combining semiconductor Bloch equations with a complex-time quantum-trajectory analysis. Our results indicate that these harmonics are not a continuation of the short trajectory. Rather, they originate from multiple-scattering trajectories, in which the electron-hole pair undergoes repeated Bragg scattering at the Brillouin-zone boundary and electron-hole reencounter near the Γ point before the actual recombination occurs. We identify that the imaginary part of the electron-hole displacement provides a practical diagnostic of the residual tunneling memory. Real-space electron-hole overlap alone does not imply coherent recombination unless the full complex displacement vanishes. These results clarify the microscopic picture of near-bandgap harmonic emission in ZnO and offer a framework for interpreting phase-resolved strong-field dynamics in solids.

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References (49)

  1. F. Krausz and M. Ivanov, Attosecond physics, Rev. Mod. Phys. 81, 163 (2009).
  2. R. Weissenbilder, S. Carlström, L. Rego, C. Guo, C. M. Heyl, P. Smorenburg, E. Constant, C. L. Arnold, and A. L’ Huillier, How to optimize high-order harmonic generation in gases, Nat. Rev. Phys. 4, 713 (2022).
  3. M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Attosecond metrology, Nature (London) 414, 509 (2001).
  4. P.-M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, Observation of a train of attosecond pulses from high harmonic generation, Science 292, 1689 (2001).
  5. P. B. Corkum, Plasma perspective on strong-field multiphoton ionization, Phys. Rev. Lett. 71, 1994 (1993).
  6. S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, Observation of high-order harmonic generation in a bulk crystal, Nat. Phys. 7, 138 (2011).
  7. S. Ghimire and D. A. Reis, High-harmonic generation from solids, Nat. Phys. 15, 10 (2019).
  8. G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, and P. B. Corkum, Linking high harmonics from gases and solids, Nature (London) 522, 462 (2015).
  9. F. Langer, M. Hohenleutner, U. Huttner, S. W. Koch, M. Kira, and R. Huber, Symmetry-controlled temporal structure of high-harmonic carrier fields from a bulk crystal, Nat. Photon. 11, 227 (2017).
  10. S. M. Cavaletto, K. M. Kowalczyk, F. O. Navarrete, and J. Rivera-Dean, The attoscience of strong-field-driven solids, Nat. Rev. Phys. 7, 38 (2025).
  11. C. Heide, Y. Kobayashi, S. R. U. Haque, and S. Ghimire, Ultrafast high-harmonic spectroscopy of solids, Nat. Phys. 20, 1546 (2024).
  12. A. Jiménez-Galán, C. Bossaer, G. Ernotte, A. M. Parks, R. E. F. Silva, D. M. Villeneuve, A. Staudte, T. Brabec, A. Luican-Mayer, and G. Vampa, Orbital perspective on high-harmonic generation from solids, Nat. Commun. 14, 8421 (2023).
  13. T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. Th. Hassan, and E. Goulielmakis, Extreme ultraviolet high-harmonic spectroscopy of solids, Nature (London) 521, 498 (2015).
  14. G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, Theoretical analysis of high-harmonic generation in solids, Phys. Rev. Lett. 113, 073901 (2014).
  15. G. Vampa, C. R. McDonald, G. Orlando, P. B. Corkum, and T. Brabec, Semiclassical analysis of high harmonic generation in bulk crystals, Phys. Rev. B 91, 064302 (2015).
  16. T. Y. Du and X. B. Bian, Quasi-classical analysis of the dynamics of the high-order harmonic generation from solids, Opt. Express 25, 151 (2017).
  17. Y. W. Kim, T. J. Shao, H. Kim, S. Han, S. Kim, M. Ciappina, X. B. Bian, and S. W. Kim, Spectral interference in high harmonic generation from solids, ACS Photon. 6, 851 (2019).
  18. W. H. Xiong, J. W. Geng, J. Y. Tang, L. Y. Peng, and Q. Gong, Mechanisms of below-threshold harmonic generation in atoms, Phys. Rev. Lett. 112, 233001 (2014).
  19. J. A. Hostetter, J. L. Tate, K. J. Schafer, and M. B. Gaarde, Semiclassical approaches to below-threshold harmonics, Phys. Rev. A 82, 023401 (2010).
  20. J. Heslar and S. I. Chu, Probing multirescattering dynamics and electron quantum paths for below- and near-threshold harmonic generation of H2 in an intense laser field, Phys. Rev. A 95, 043414 (2017).
  21. P. C. Li, Y. L. Sheu, C. Laughlin, and S. I. Chu, Dynamical origin of near- and below-threshold harmonic generation of Cs in an intense mid-infrared laser field, Nat. Commun. 6, 7178 (2015).
  22. P. C. Li, Y. L. Sheu, C. Laughlin, and S. I. Chu, Role of laser-driven electron-multirescattering in resonance-enhanced below-threshold harmonic generation in He atoms, Phys. Rev. A 90, 041401(R) (2014).
  23. B. Wang, Y. Zhang, P. Lan, C. Zhai, M. Li, X. Zhu, J. Chen, P. Lu, and C. D. Lin, Anomalous ellipticity dependence of the generation of near-threshold harmonics in noble gases, Phys. Rev. A 103, 053119 (2021).
  24. P. C. Li, Z. B. Wang, and S. I. Chu, Multichannel-resolved dynamics in resonance-enhanced below-threshold harmonic generation of H2+ molecular ions, Phys. Rev. A 103, 043113 (2021).
  25. P. C. Li, Y. L. Sheu, H. Z. Jooya, X. X. Zhou, and S. I. Chu, Exploration of laser-driven electron-multirescattering dynamics in high-order harmonic generation, Sci. Rep. 6, 32763 (2016).
  26. E. A. Bleda, I. Yavuz, Z. Altun, and T. Topcu, High-order harmonic generation from Rydberg states at fixed Keldysh parameter, Phys. Rev. A 88, 043417 (2013).
  27. H. Soifer, P. Botheron, D. Shafir, A. Diner, O. Raz, B. D. Bruner, Y. Mairesse, B. Pons, and N. Dudovich, Near-threshold high-order harmonic spectroscopy with aligned molecules, Phys. Rev. Lett. 105, 143904 (2010).
  28. N. Mayer, S. Beaulieu, Á. Jiménez-Galán, S. Patchkovskii, O. Kornilov, D. Descamps, S. Petit, O. Smirnova, Y. Mairesse, and M. Y. Ivanov, Role of spin-orbit coupling in high-order harmonic generation revealed by supercycle Rydberg trajectories, Phys. Rev. Lett. 129, 173202 (2022).
  29. G. Vampa and T. Brabec, Merge of high harmonic generation from gases and solids and its implications for attosecond science, J. Phys. B: At. Mol. Opt. Phys. 50, 083001 (2017).
  30. P. Jürgens, B. Liewehr, B. Kruse, C. Peltz, D. Engel, A. Husakou, T. Witting, M. Ivanov, M. J. J. Vrakking, T. Fennel, and A. Mermillod-Blondin, Origin of strong-field-induced low-order harmonie generation in amorphous quartz, Nat. Phys. 16, 1035 (2020).
  31. A. Nayak, M. Dumergue, S. Kühn, S. Mondal, T. Csizmadia, N. G. Harshitha, M. Füle, M. U. Kahaly, B. Farkas, B. Major, et al., Saddle point approaches in strong field physics and generation of attosecond pulses, Phys. Rep. 833, 1 (2019).
  32. T. T. Luu and H. J. Wörner, Observing broken inversion symmetry in solids using two-color high-order harmonic spectroscopy, Phys. Rev. A 98, 041802(R) (2018).
  33. O. Kneller, D. Azoury, Y. Federman, M. Krüger, A. J. Uzan, G. Orenstein, B. D. Bruner, O. Smirnova, S. Patchkovskii, M. Ivanov, and N. Dudovich, A look under the tunnelling barrier via attosecond-gated interferometry, Nat. Photon. 16, 304 (2022).
  34. A. J. Uzan-Narovlansky, A. Jiménez-Galán, G. Orenstein, R. E. F. Silva, T. Arusi-Parpar, S. Shames, B. D. Bruner, B. Yan, O. Smirnova, M. Ivanov, and N. Dudovich, Observation of light-driven band structure via multiband high-harmonic spectroscopy, Nat. Photon. 16, 428 (2022).
  35. M. Krüger and N. Dudovich, Attosecond interferometry, in Ultrafast Electronic and Structural Dynamics, edited by K. Ueda (Springer, Singapore, 2024), pp. 45–71.
  36. T. Helk, E. Berger, S. Jamnuch, L. Hoffmann, A. Kabacinski, J. Gautier, F. Tissandier, J.-P. Goddet, H.-T. Chang, J. Oh, C. D. Pemmaraju, T. A. Pascal, S. Sebban, C. Spielmann, and M. Zuerch, Table-top extreme ultraviolet second harmonic generation, Sci. Adv. 7, eabe2265 (2021).
  37. O. Pedatzur, G. Orenstein, V. Serbinenko, H. Soifer, B. D. Bruner, A. J. Uzan, D. S. Brambila, A. G. Harvey, L. Torlina, F. Morales, O. Smirnova, and N. Dudovich, Attosecond tunnelling interferometry, Nat. Phys. 11, 815 (2015).
  38. G. Vampa, J. Lu, Y. S. You, D. R. Baykusheva, M. Wu, H. Liu, K. J. Schafer, M. B. Gaarde, D. A. Reis, and S. Ghimire, Attosecond synchronization of extreme ultraviolet high harmonics from crystals, J. Phys. B: At. Mol. Opt. Phys. 53, 144003 (2020).
  39. R. Zuo, X. Song, S. Ben, T. Meier, and W. Yang, Revealing the nonadiabatic tunneling dynamics in solid-state high harmonic generation, Phys. Rev. Res. 5, L022040 (2023).
  40. O. Smirnova and M. Ivanov, Multielectron high harmonic generation: Simple man on a complex plane, in Attosecond, XUV Physics: Ultrafast Dynamics, Spectroscopy, edited by T. Schultz and M. Vrakking (Wiley-VCH, Weinheim, 2014), Chap. 7, pp. 201–256.
  41. D. Golde, T. Meier, and S. W. Koch, High harmonics generated in semiconductor nanostructures by the coupled dynamics of optical inter- and intraband excitations, Phys. Rev. B 77, 075330 (2008).
  42. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th ed. (World Scientific, Singapore, 2009).
  43. Z. Guan, X. X. Zhou, and X. B. Bian, High-order-harmonic generation from periodic potentials driven by few-cycle laser pulses, Phys. Rev. A 93, 033852 (2016).
  44. G. Wang, Y.-X. Chen, M. Li, and T.-Y. Du, High-order harmonic spectroscopy for probing intraband and interband dynamics, Phys. Rev. A 109, 063511 (2024).
  45. Z. Wang, H. Park, Y. H. Lai, J. Xu, C. I. Blaga, F. Yang, P. Agostini, and L. F. DiMauro, The roles of photo-carrier doping and driving wavelength in high harmonic generation from a semiconductor, Nat. Commun. 8, 1686 (2017).
  46. J. Liu, S. Jiang, J. Chen, L. He, J. Wang, and X. Zhao, Complex saddle point analysis of solid high order harmonic generation and reconstruction of dephasing time, APL Comput. Phys. 1, 026104 (2025).
  47. L. Yue and M. B. Gaarde, Imperfect recollisions in high-harmonic generation in solids, Phys. Rev. Lett. 124, 153204 (2020).
  48. L. Yue and M. B. Gaarde, Expanded view of electron-hole recollisions in solid-state high-order harmonic generation: Full-Brillouin-zone tunneling and imperfect recollisions, Phys. Rev. A 103, 063105 (2021).
  49. C. M. Wang, N. Tancogne-Dejean, M. Altarelli, A. Rubio, and S. A. Sato, Role of electron scattering on the high-order harmonic generation from solids, Phys. Rev. Res. 2, 033333 (2020).

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Phys. Rev. Research 8, 033303– Published 11 September, 2026
  • Received 17 May 2026
  • Accepted 27 August 2026
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