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  • Letter
  • Open Access

Sensing quantum fluctuations via forbidden harmonics in quantum materials

Kartik Prabhu1 and Gopal Dixit1,2,3,*

  • *Contact author: gdixit@phy.iitb.ac.in
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Phys. Rev. B 114, L140301 – Published 1 September, 2026

DOI: https://doi.org/10.1103/v886-ypf7

Abstract

High-harmonic generation in solids provides a powerful window into light-driven electron dynamics on attosecond timescales, yet the role of quantum fluctuations of the driving field remains largely unexplored. Here, we show that the quantum state of light offers a nonclassical degree of freedom for controlling strong-field dynamics in quantum materials. Using graphene driven by bicircular quantum light composed of counterrotating ω and 2ω components, we theoretically predict that squeezing-induced fluctuations break the dynamical symmetry of the classical bicircular field and activate otherwise forbidden harmonics. By analyzing the helicity-resolved harmonic spectra, we uncover robust polarization signatures uniquely dictated by the squeezed mode's profile, including a relative-phase-controlled helicity switching in the lowest orders. Crucially, second-order intensity correlations reveal that these forbidden harmonics exhibit strongly enhanced, super-Poissonian photon-number fluctuations, confirming that they inherit and directly transduce the nonclassical statistics of the squeezed driving vacuum. Our findings establish forbidden harmonics as high-fidelity optical probes of the quantum state of light and provide additional opportunities at the intersection of quantum optics, strong-field physics, and lightwave engineering in quantum materials.

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References (86)

  1. F. Krausz and M. Y. Ivanov, Attosecond physics, Rev. Mod. Phys. 81, 163 (2009).
  2. F. Krausz, Nobel lecture: Sub-atomic motions, Rev. Mod. Phys. 96, 030502 (2024).
  3. P. Agostini, Nobel lecture: Genesis and applications of attosecond pulse trains, Rev. Mod. Phys. 96, 030501 (2024).
  4. A. L'Huillier, Nobel lecture: The route to attosecond pulses, Rev. Mod. Phys. 96, 030503 (2024).
  5. 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).
  6. S. Ghimire and D. A. Reis, High-harmonic generation from solids, Nat. Phys. 15, 10 (2019).
  7. E. Goulielmakis and T. Brabec, High harmonic generation in condensed matter, Nat. Photon. 16, 411 (2022).
  8. G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, All-optical reconstruction of crystal band structure, Phys. Rev. Lett. 115, 193603 (2015).
  9. H. Liu, C. Guo, G. Vampa, J. L. Zhang, T. Sarmiento, M. Xiao, P. H. Bucksbaum, J. Vučković, S. Fan, and D. A. Reis, Enhanced high-harmonic generation from an all-dielectric metasurface, Nat. Phys. 14, 1006 (2018).
  10. M. M. S., A. Pattanayak, M. Ivanov, and G. Dixit, Direct numerical observation of real-space recollision in high-order harmonic generation from solids, Phys. Rev. A 100, 043420 (2019).
  11. S. Imai, A. Ono, and S. Ishihara, High harmonic generation in a correlated electron system, Phys. Rev. Lett. 124, 157404 (2020).
  12. L. Yue and M. B. Gaarde, Imperfect recollisions in high-harmonic generation in solids, Phys. Rev. Lett. 124, 153204 (2020).
  13. M. S. Mrudul, N. Tancogne-Dejean, A. Rubio, and G. Dixit, High-harmonic generation from spin-polarised defects in solids, npj Comput. Mater. 6, 10 (2020).
  14. M. Borsch, C. P. Schmid, L. Weigl, S. Schlauderer, N. Hofmann, C. Lange, J. T. Steiner, S. W. Koch, R. Huber, and M. Kira, Super-resolution lightwave tomography of electronic bands in quantum materials, Science 370, 1204 (2020).
  15. C. P. Schmid, L. Weigl, P. Grössing, V. Junk, C. Gorini, S. Schlauderer, S. Ito, M. Meierhofer, N. Hofmann, D. Afanasiev, et al., Tunable non-integer high-harmonic generation in a topological insulator, Nature (London) 593, 385 (2021).
  16. C. Heide, Y. Kobayashi, D. R. Baykusheva, D. Jain, J. A. Sobota, M. Hashimoto, P. S. Kirchmann, S. Oh, T. F. Heinz, D. A. Reis, et al., Probing topological phase transitions using high-harmonic generation, Nat. Photon. 16, 620 (2022).
  17. S. Kaassamani, T. Auguste, N. Tancogne-Dejean, X. Liu, W. Boutu, H. Merdji, and D. Gauthier, Polarization spectroscopy of high-order harmonic generation in gallium arsenide, Opt. Express 30, 40531 (2022).
  18. C. Qian, C. Yu, S. Jiang, T. Zhang, J. Gao, S. Shi, H. Pi, H. Weng, and R. Lu, Role of shift vector in high-harmonic generation from noncentrosymmetric topological insulators under strong laser fields, Phys. Rev. X 12, 021030 (2022).
  19. L. Li, P. Lan, X. Zhu, and P. Lu, High harmonic generation in solids: Particle and wave perspectives, Rep. Prog. Phys. 86, 116401 (2023).
  20. H. K. Avetissian, H. H. Matevosyan, and G. F. Mkrtchian, Berry curvature and shift vector effects at high-order wave mixing in biased bilayer graphene, Phys. Rev. B 111, 045415 (2025).
  21. Y. Murakami, M. Eckstein, and P. Werner, High-harmonic generation in Mott insulators, Phys. Rev. Lett. 121, 057405 (2018).
  22. C. Heide, Y. Kobayashi, S. R. Ul Haque, and S. Ghimire, Ultrafast high-harmonic spectroscopy of solids, Nat. Phys. 20, 1546 (2024).
  23. A. Nayak, D. Rajak, B. Farkas, C. Granados, P. Stammer, J. Rivera-Dean, T. Lamprou, K. Varjú, Y. Mairesse, M. F. Ciappina, M. Lewenstein, and P. Tzallas, Attosecond metrology of vacuum-ultraviolet high-order harmonics generated in semiconductors via laser-dressed photoionization of alkali metals, Nat. Commun. 16, 1428 (2025).
  24. A. Seyen, R. Kohn, U. Bovensiepen, D. von der Linde, and A. Tarasevitch, Toward attosecond pulse synthesis from solids: Spectral shaping, field autocorrelation, and two-color harmonic generation, Phys. Rev. A 99, 033836 (2019).
  25. Z. Nourbakhsh, N. Tancogne-Dejean, H. Merdji, and A. Rubio, High harmonics and isolated attosecond pulses from Mg O, Phys. Rev. Appl. 15, 014013 (2021).
  26. S. Imai and A. Ono, Theory for Fourier-limited attosecond pulse generation in solids, Phys. Rev. B 109, L041303 (2024).
  27. P. Venkat, A. Bharti, and G. Dixit, Attosecond pulse generation from Weyl semimetals, Phys. Rev. B 113, 125144 (2026).
  28. X. Song, X. Bu, X. Zhao, R. Zhang, S. Wang, and F. Dong, Isolated elliptically polarized attosecond pulse generation in gapped graphene driven by linearly polarized laser fields, Phys. Rev. A 112, 043107 (2025).
  29. X. Zhang, S. Hu, M. Guan, and S. Meng, Optimizing attosecond-pulse generation in solids by modulating electronic dynamics with a monochromatic laser field, Phys. Rev. A 111, 023521 (2025).
  30. Z. Chen, M. Levit, Y. Kern, B. Roy, A. Goldner, and M. Krüger, Attosecond pulses from a solid driven by a synthesized two-color field at megahertz repetition rate, ACS Photonics 12, 2819 (2025).
  31. Y. Kim, M. J. Kim, S. Cha, S. Choi, C.-J. Kim, B. J. Kim, M.-H. Jo, J. Kim, and J. Lee, Dephasing dynamics accessed by high harmonic generation: Determination of electron–hole decoherence of Dirac fermions, Nano Lett. 24, 1277 (2024).
  32. S. Cha, M. Kim, Y. Kim, S. Choi, S. Kang, H. Kim, S. Yoon, G. Moon, T. Kim, Y. W. Lee, et al., Gate-tunable quantum pathways of high harmonic generation in graphene, Nat. Commun. 13, 6630 (2022).
  33. N. Yoshikawa, T. Tamaya, and K. Tanaka, High-harmonic generation in graphene enhanced by elliptically polarized light excitation, Science 356, 736 (2017).
  34. N. Rana and G. Dixit, Probing phonon-driven symmetry alterations in graphene via high-order-harmonic spectroscopy, Phys. Rev. A 106, 053116 (2022).
  35. H. K. Avetissian, G. F. Mkrtchian, and A. Knorr, Efficient high-harmonic generation in graphene with two-color laser field at orthogonal polarization, Phys. Rev. B 105, 195405 (2022).
  36. Y. Murakami and M. Schüler, Doping and gap size dependence of high-harmonic generation in graphene: Importance of consistent formulation of light-matter coupling, Phys. Rev. B 106, 035204 (2022).
  37. F. Dong, Q. Xia, and J. Liu, Ellipticity of the harmonic emission from graphene irradiated by a linearly polarized laser, Phys. Rev. A 104, 033119 (2021).
  38. Y. Zhang, L. Li, J. Li, T. Huang, P. Lan, and P. Lu, Orientation dependence of high-order harmonic generation in graphene, Phys. Rev. A 104, 033110 (2021).
  39. R. Boyero-García, A. García-Cabrera, O. Zurrón-Cifuentes, C. Hernández-García, and L. Plaja, Non-classical high harmonic generation in graphene driven by linearly-polarized laser pulses, Opt. Express 30, 15546 (2022).
  40. Z. Guan, Z. Yin, J. You, B. Wang, X. Li, G.-L. Wang, X.-X. Zhou, and C. Jin, Optimal generation and systematic analysis of tunable terahertz emissions from single-layer graphene using two-color laser pulses with different durations, Phys. Rev. A 108, 023515 (2023).
  41. N. Rana, M. S. Mrudul, D. Kartashov, M. Ivanov, and G. Dixit, High-harmonic spectroscopy of coherent lattice dynamics in graphene, Phys. Rev. B 106, 064303 (2022).
  42. Ó. Zurrón-Cifuentes, R. Boyero-García, C. Hernández-García, A. Picón, and L. Plaja, Optical anisotropy of non-perturbative high-order harmonic generation in gapless graphene, Opt. Express 27, 7776 (2019).
  43. M. S. Mrudul, Dependence of high-harmonic generation in twisted bilayer graphene on laser pulse ellipticity, Phys. Rev. B 110, 115415 (2024).
  44. C. Liu, Y. Zheng, Z. Zeng, and R. Li, Driving-laser ellipticity dependence of high-order harmonic generation in graphene, Phys. Rev. A 97, 063412 (2018).
  45. L. A. Chizhova, F. Libisch, and J. Burgdörfer, Nonlinear response of graphene to a few-cycle terahertz laser pulse: Role of doping and disorder, Phys. Rev. B 94, 075412 (2016).
  46. N. Rana, M. S. Mrudul, and G. Dixit, High-harmonic generation from strain-engineered graphene for polarization tailoring, Phys. Rev. B 110, 054103 (2024).
  47. H. K. Avetissian, A. K. Avetissian, G. F. Mkrtchian, and K. V. Sedrakian, Creation of particle-hole superposition states in graphene at multiphoton resonant excitation by laser radiation, Phys. Rev. B 85, 115443 (2012).
  48. H. K. Avetissian and G. F. Mkrtchian, Impact of electron-electron Coulomb interaction on the high harmonic generation process in graphene, Phys. Rev. B 97, 115454 (2018).
  49. M. S. Mrudul and G. Dixit, Controlling valley-polarisation in graphene via tailored light pulses, J. Phys. B: At. Mol. Opt. Phys. 54, 224001 (2021).
  50. N. Saito, P. Xia, F. Lu, T. Kanai, J. Itatani, and N. Ishii, Observation of selection rules for circularly polarized fields in high-harmonic generation from a crystalline solid, Optica 4, 1333 (2017).
  51. M. S. Mrudul, Á. Jiménez-Galán, M. Ivanov, and G. Dixit, Light-induced valleytronics in pristine graphene, Optica 8, 422 (2021).
  52. O. Neufeld, D. Podolsky, and O. Cohen, Floquet group theory and its application to selection rules in harmonic generation. Nat. Commun. 10, 405 (2019).
  53. N. Rana, M. S. Mrudul, and G. Dixit, Generation of circularly polarized high harmonics with identical helicity in two-dimensional materials, Phys. Rev. Appl. 18, 064049 (2022).
  54. 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).
  55. J. Rivera-Dean, L. Petrovic, M. Lewenstein, and P. Stammer, Attosecond quantum optical interferometry, Rep. Prog. Phys. 89, 047901 (2026).
  56. A. Gorlach, O. Neufeld, N. Rivera, O. Cohen, and I. Kaminer, The quantum-optical nature of high harmonic generation, Nat. Commun. 11, 4598 (2020).
  57. 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).
  58. D. Theidel, V. Cotte, P. Heinzel, H. Griguer, M. Weis, R. Sondenheimer, and H. Merdji, Observation of a displaced squeezed state in high-harmonic generation, Phys. Rev. Res. 7, 033223 (2025).
  59. H. Liu, H. Zhang, X. Wang, and J. Yuan, Theory of high-harmonic generation with quantum light, Phys. Rev. Res. 8, 023326 (2026).
  60. M. E. Tzur, M. Birk, A. Gorlach, M. Krüger, I. Kaminer, and O. Cohen, Photon-statistics force in ultrafast electron dynamics, Nat. Photon. 17, 501 (2023).
  61. A. Gorlach, M. E. Tzur, M. Birk, W. P. Schleich, M. Kiffner, H. Pugatch, V. Peano, S. Ronen, and O. Firstenberg, High-harmonic generation driven by quantum light, Nat. Phys. 19, 1689 (2023).
  62. P. Stammer, J. Rivera-Dean, A. Maxwell, T. Lamprou, A. 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).
  63. 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).
  64. P. Stammer, J. Rivera-Dean, P. Tzallas, M. F. Ciappina, and M. Lewenstein, Colloquium: Quantum optics of intense light–matter interaction, arXiv:2510.19045.
  65. S. Wang, S. Yu, X. Lai, and X. Liu, High harmonic generation from an atom in a squeezed-vacuum environment, Phys. Rev. Res. 6, 033010 (2024).
  66. N. Tsatrafyllis, I. K. Kominis, I. A. Gonoskov, and P. Tzallas, High-order harmonics measured by the photon statistics of the infrared driving-field exiting the atomic medium, Nat. Commun. 8, 15170 (2017).
  67. L. Cruz-Rodriguez, D. Dey, A. Freibert, and P. Stammer, Quantum phenomena in attosecond science, Nat. Rev. Phys. 6, 691 (2024).
  68. P. Stammer, J. Rivera-Dean, A. S. Maxwell, T. Lamprou, J. Argüello-Luengo, P. Tzallas, M. F. Ciappina, and M. Lewenstein, Entanglement and squeezing of the optical field modes in high harmonic generation, Phys. Rev. Lett. 132, 143603 (2024).
  69. 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).
  70. R. V. Gothelf, C. S. Lange, and L. B. Madsen, High-order harmonic generation in a crystal driven by quantum light, Phys. Rev. A 111, 063105 (2025).
  71. D. Theidel, V. Cotte, R. Sondenheimer, V. Shiriaeva, M. Froidevaux, V. Severin, A. Merdji-Larue, P. Mosel, S. Fröhlich, K.-A. Weber, et al., Evidence of the quantum optical nature of high-harmonic generation, PRX Quantum 5, 040319 (2024).
  72. 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).
  73. C. S. Lange, T. Hansen, and L. B. Madsen, Excitonic enhancement of squeezed light in quantum-optical high-harmonic generation from a Mott insulator, Phys. Rev. Lett. 135, 043603 (2025).
  74. 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. Photon. 19, 767 (2025).
  75. A. Fleischer, O. Kfir, T. Diskin, P. Sidorenko, and O. Cohen, Spin angular momentum and tunable polarization in high-harmonic generation, Nat. Photon. 8, 543 (2014).
  76. G. Dixit, Á. Jiménez-Galán, L. Medišauskas, and M. Ivanov, Control of the helicity of high-order harmonic radiation using bichromatic circularly polarized laser fields, Phys. Rev. A 98, 053402 (2018).
  77. L. Petrovic, P. Stammer, M. Lewenstein, and J. Rivera-Dean, Generation of circularly polarized high-order harmonics from single-color quantum light, Phys. Rev. A 114, 013127 (2026).
  78. J. Rivera-Dean, P. Stammer, M. F. Ciappina, and M. Lewenstein, Structured squeezed light allows for high-harmonic generation in classically forbidden geometries, Phys. Rev. Lett. 135, 013801 (2025).
  79. A. Bharti and G. Dixit, Photocurrent generation in solids via linearly polarized laser, Phys. Rev. B 109, 104309 (2024).
  80. M. S. Mrudul and G. Dixit, High-harmonic generation from monolayer and bilayer graphene, Phys. Rev. B 103, 094308 (2021).
  81. N. Moiseyev, Photon statistics from non-Hermitian Floquet theory: High-order-harmonic-generation and above-threshold-ionization spectra detected via IR detectors, Phys. Rev. A 110, L051101 (2024).
  82. J. M. González-Monge, F. R. Willemann, and J. Feist, High-harmonic generation driven by temporal-mode quantum states of light, arXiv:2512.06602.
  83. S. de-la-Peña, H. Appel, A. Rubio, and O. Neufeld, Second-quantized numerical simulations of tunable entanglement in quantum high harmonic generation, arXiv:2512.03987.
  84. S. de-la-Peña, O. Neufeld, M. E. Tzur, O. Cohen, H. Appel, and A. Rubio, Quantum electrodynamics in high-harmonic generation: Multitrajectory Ehrenfest and exact quantum analysis, J. Chem. Theory Comput. 21, 283 (2025).
  85. I. N. Ansari, C. Hofmann, L. Medišauskas, M. Lewenstein, M. F. Ciappina, and G. Dixit, Controlling polarization of attosecond pulses with plasmonic-enhanced bichromatic counter-rotating circularly polarized fields, Phys. Rev. A 103, 013104 (2021).
  86. P. Stammer, J. Rivera-Dean, D. Kim, A. Chacón, W. Gao, and C. Granados, Symmetry breaking by quantum light in solid-state high-harmonic generation, arXiv:2605.28236.

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Phys. Rev. B 114, L140301– Published 1 September, 2026
  • Received 6 July 2026
  • Revised 14 August 2026
  • Accepted 17 August 2026
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