- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Quantum electrodynamics of graphene Landau levels in a deep-subwavelength hyperbolic phonon-polariton cavity

Gian Marcello Andolina1,*, Matteo Ceccanti2, Bianca Turini2, Riccardo Riolo3, Marco Polini2,3, Marco Schirò1, and Frank H. L. Koppens2,4

  • *Contact author: gian-marcello.andolina@college-de-france.fr
PDF

Phys. Rev. Research 8, 023044 – Published 14 April, 2026

DOI: https://doi.org/10.1103/69jg-sq8r

Abstract

The confinement of electromagnetic radiation within extremely small volumes offers an effective means to significantly enhance light-matter interactions, to the extent that zero-point quantum vacuum fluctuations can influence and control the properties of materials. Here, we develop a theoretical framework for the quantum electrodynamics of graphene Landau levels embedded in a deep-subwavelength hyperbolic cavity, where light is confined into ultrasmall mode volumes. By studying the spectrum, we discuss the emergence of polaritons, and disentangle the contributions of resonant quantum vacuum effects from those of purely electrostatic interactions. Finally, we study the hybridization between magnetoplasmons and the cavity modes.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (129)

  1. A. H. Compton, A quantum theory of the scattering of x-rays by light elements, Phys. Rev. 21, 483 (1923).
  2. C. V. Raman and K. S. Krishnan, The negative absorption of radiation, Nature (London) 122, 12 (1928).
  3. A. Damascelli, Z. Hussain, and Z.-X. Shen, Angle-resolved photoemission studies of the cuprate superconductors, Rev. Mod. Phys. 75, 473 (2003).
  4. E. M. Purcell, Spontaneous emission probabilities at radio frequencies, Phys. Rev. 69, 681 (1946).
  5. J. M. Raimond, M. Brune, and S. Haroche, Manipulating quantum entanglement with atoms and photons in a cavity, Rev. Mod. Phys. 73, 565 (2001).
  6. S. Haroche, Nobel lecture: Controlling photons in a box and exploring the quantum to classical boundary, Rev. Mod. Phys. 85, 1083 (2013).
  7. F. Schlawin, D. M. Kennes, and M. A. Sentef, Cavity quantum materials, Appl. Phys. Rev. 9, 011312 (2022).
  8. J. Bloch, A. Cavalleri, V. Galitski, M. Hafezi, and A. Rubio, Strongly correlated electron–photon systems, Nature (London) 606, 41 (2022).
  9. H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, Cold atoms in cavity-generated dynamical optical potentials, Rev. Mod. Phys. 85, 553 (2013).
  10. K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, Dicke quantum phase transition with a superfluid gas in an optical cavity, Nature (London) 464, 1301 (2010).
  11. F. J. Garcia-Vidal, C. Ciuti, and T. W. Ebbesen, Manipulating matter by strong coupling to vacuum fields, Science 373, eabd0336 (2021).
  12. G. Jarc, S. Y. Mathengattil, A. Montanaro, F. Giusti, E. M. Rigoni, R. Sergo, F. Fassioli, S. Winnerl, S. Dal Zilio, D. Mihailovic, P. Prelovšek, M. Eckstein, and D. Fausti, Cavity-mediated thermal control of metal-to-insulator transition in 1T-TaS2, Nature (London) 622, 487 (2023).
  13. F. Appugliese, J. Enkner, G. L. Paravicini-Bagliani, M. Beck, C. Reichl, W. Wegscheider, G. Scalari, C. Ciuti, and J. Faist, Breakdown of topological protection by cavity vacuum fields in the integer quantum Hall effect, Science 375, 1030 (2022).
  14. J. Enkner, L. Graziotto, F. Appugliese, V. Rokaj, J. Wang, M. Ruggenthaler, C. Reichl, W. Wegscheider, A. Rubio, and J. Faist, Breakdown of topological protection by cavity vacuum fields in the integer quantum Hall effect, Phys. Rev. X 14, 021038 (2024).
  15. I.-T. Lu, D. Shin, M. K. Svendsen, H. Hübener, U. De Giovannini, S. Latini, M. Ruggenthaler, and A. Rubio, Cavity engineered phonon-mediated superconductivity in MgB2 from first principles quantum electrodynamics, Proc. Natl. Acad. Sci. USA 121, e2415061121 (2024).
  16. M. A. Sentef, M. Ruggenthaler, and A. Rubio, Cavity quantum-electrodynamical polaritonically enhanced electron-phonon coupling and its influence on superconductivity, Sci. Adv. 4, eaau6969 (2018).
  17. F. Schlawin, A. Cavalleri, and D. Jaksch, Cavity-mediated electron-photon superconductivity, Phys. Rev. Lett. 122, 133602 (2019).
  18. D.-P. Nguyen, G. Arwas, Z. Lin, W. Yao, and C. Ciuti, Electron-photon Chern number in cavity-embedded 2D Moiré materials, Phys. Rev. Lett. 131, 176602 (2023).
  19. X. Wang, E. Ronca, and M. A. Sentef, Cavity quantum electrodynamical Chern insulator: Towards light-induced quantized anomalous Hall effect in graphene, Phys. Rev. B 99, 235156 (2019).
  20. H. Gao, F. Schlawin, M. Buzzi, A. Cavalleri, and D. Jaksch, Photoinduced electron pairing in a driven cavity, Phys. Rev. Lett. 125, 053602 (2020).
  21. G. Mazza and A. Georges, Superradiant quantum materials, Phys. Rev. Lett. 122, 017401 (2019).
  22. S. Latini, D. Shin, S. A. Sato, C. Schäfer, U. De Giovannini, H. Hübener, and A. Rubio, The ferroelectric photo ground state of SrTiO3: Cavity materials engineering, Proc. Natl. Acad. Sci. USA 118, e2105618118 (2021).
  23. V. K. Kozin, E. Thingstad, D. Loss, and J. Klinovaja, Cavity-enhanced superconductivity via band engineering, Phys. Rev. B 111, 035410 (2025).
  24. J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, Theory of spontaneous-emission line shape in an ideal cavity, Phys. Rev. Lett. 51, 550 (1983).
  25. G. S. Agarwal, Vacuum-field Rabi splittings in microwave absorption by Rydberg atoms in a cavity, Phys. Rev. Lett. 53, 1732 (1984).
  26. G. S. Agarwal, Vacuum-field Rabi oscillations of atoms in a cavity, J. Opt. Soc. Am. B 2, 480 (1985).
  27. M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, Quantum Rabi oscillation: A direct test of field quantization in a cavity, Phys. Rev. Lett. 76, 1800 (1996).
  28. G. Scalari, C. Maissen, D. Turcinková, D. Hagenmüller, S. De Liberato, C. Ciuti, C. Reichl, D. Schuh, W. Wegscheider, M. Beck, and J. Faist, Ultrastrong coupling of the cyclotron transition of a 2D electron gas to a THz metamaterial, Science 335, 1323 (2012).
  29. O. Bitton, S. N. Gupta, L. Houben, M. Kvapil, V. Křápek, T. Šikola, and G. Haran, Vacuum Rabi splitting of a dark plasmonic cavity mode revealed by fast electrons, Nat. Commun. 11, 487 (2020).
  30. Y. Todorov, A. M. Andrews, I. Sagnes, R. Colombelli, P. Klang, G. Strasser, and C. Sirtori, Strong light-matter coupling in subwavelength metal-dielectric microcavities at Terahertz frequencies, Phys. Rev. Lett. 102, 186402 (2009).
  31. Y. Todorov, A. M. Andrews, R. Colombelli, S. De Liberato, C. Ciuti, P. Klang, G. Strasser, and C. Sirtori, Ultrastrong light-matter coupling regime with polariton dots, Phys. Rev. Lett. 105, 196402 (2010).
  32. G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, Vacuum Rabi splitting in semiconductors, Nat. Phys. 2, 81 (2006).
  33. D. De Bernardis, G. M. Andolina, and I. Carusotto, Light-matter interactions in the vacuum of ultrastrongly coupled systems, Phys. Rev. A 110, 053713 (2024).
  34. J. Dalibard, J. Dupont-Roc, and C. Cohen-Tannoudji, Vacuum fluctuations and radiation reaction: Identification of their respective contributions, J. Phys. France 43, 1617 (1982).
  35. C. Ciuti, G. Bastard, and I. Carusotto, Quantum vacuum properties of the intersubband cavity polariton field, Phys. Rev. B 72, 115303 (2005).
  36. A. F. Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, Ultrastrong coupling between light and matter, Nat. Rev. Phys. 1, 19 (2019).
  37. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, Quantum optics with surface plasmons, Phys. Rev. Lett. 97, 053002 (2006).
  38. M. S. Tame, K. R. McEnery, Ş. K. Özdemir, J. Lee, S. A. Maier, and M. S. Kim, Quantum plasmonics, Nat. Phys. 9, 329 (2013).
  39. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, Channel plasmon subwavelength waveguide components including interferometers and ring resonators, Nature (London) 440, 508 (2006).
  40. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, Plasmon lasers at deep subwavelength scale, Nature (London) 461, 629 (2009).
  41. F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, Graphene plasmonics: A platform for strong light–matter interactions, Nano Lett. 11, 3370 (2011).
  42. J. J. Baumberg, J. Aizpurua, M. H. Mikkelsen, and D. R. Smith, Extreme nanophotonics from ultrathin metallic gaps, Nat. Mater. 18, 668 (2019).
  43. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, New York, 2007).
  44. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, Probing the ultimate limits of plasmonic enhancement, Nat. Mater. 9, 193 (2010).
  45. C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, Probing the ultimate limits of plasmonic enhancement, Science 337, 1072 (2012).
  46. R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, Single-molecule strong coupling at room temperature in plasmonic nanocavities, Nature (London) 535, 127 (2016).
  47. D. Alcaraz Iranzo, S. Nanot, E. J. C. Dias, I. Epstein, C. Peng, D. K. Efetov, M. B. Lundeberg, R. Parret, J. Osmond, J.-Y. Hong, J. Kong, D. R. Englund, N. M. R. Peres, and F. H. L. Koppens, Probing the ultimate plasmon confinement limits with a van der Waals heterostructure, Science 360, 291 (2018).
  48. M. Barra-Burillo, U. Muniain, S. Catalano, M. Autore, F. Casanova, L. E. Hueso, J. Aizpurua, R. Esteban, and R. Hillenbrand, Microcavity phonon polaritons from the weak to the ultrastrong phonon–photon coupling regime, Nat. Commun. 12, 6206 (2021).
  49. C. Maissen, G. Scalari, F. Valmorra, M. Beck, J. Faist, S. Cibella, R. Leoni, C. Reichl, C. Charpentier, and W. Wegscheider, Ultrastrong coupling in the near field of complementary split-ring resonators, Phys. Rev. B 90, 205309 (2014).
  50. S. Dai, Z. Fei, Q. Ma, A. S. Rodin, M. Wagner, A. S. McLeod, M. K. Liu, W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M. Thiemens, G. Dominguez, A. H. Castro Neto, A. Zettl, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, Tunable phonon polaritons in atomically thin van der Waals crystals of boron nitride, Science 343, 1125 (2014).
  51. G. M. Andolina, A. De Pasquale, F. M. D. Pellegrino, I. Torre, F. H. L. Koppens, and M. Polini, Amperean superconductivity cannot be induced by deep subwavelength cavities in a two-dimensional material, Phys. Rev. B 109, 104513 (2024).
  52. I. Amelio, L. Korosec, I. Carusotto, and G. Mazza, Optical dressing of the electronic response of two-dimensional semiconductors in quantum and classical descriptions of cavity electrodynamics, Phys. Rev. B 104, 235120 (2021).
  53. J. Keller, G. Scalari, S. Cibella, C. Maissen, F. Appugliese, E. Giovine, R. Leoni, M. Beck, and J. Faist, Few-electron ultrastrong light-matter coupling at 300 GHz with nanogap hybrid LC microcavities, Nano Lett. 17, 7410 (2017).
  54. G. Arwas and C. Ciuti, Quantum electron transport controlled by cavity vacuum fields, Phys. Rev. B 107, 045425 (2023).
  55. K. Lenk, J. Li, P. Werner, and M. Eckstein, Collective theory for an interacting solid in a single-mode cavity, arXiv:2205.05559.
  56. D. A. Zezyulin, S. A. Kolodny, O. V. Kibis, I. V. Tokatly, and I. V. Iorsh, Dynamical stabilization by vacuum fluctuations in a cavity: Resonant electron scattering in the ultrastrong light-matter coupling regime, Phys. Rev. A 106, 043708 (2022).
  57. F. Pavošević and A. Rubio, Wavefunction embedding for molecular polaritons, J. Chem. Phys. 157, 094101 (2022).
  58. C. B. Dag and V. Rokaj, Engineering topology in graphene with chiral cavities, Phys. Rev. B 110, L121101 (2024).
  59. P.-A. Pantazopoulos, J. Feist, A. Kamra, F. J. García-Vidal, Electrostatic nature of cavity-mediated interactions between low-energy matter excitations, Phys. Rev. B 109, L201408 (2024).
  60. U. Muniain, J. Aizpurua, R. Hillenbrand, L. Martín-Moreno, and R. Esteban, Description of ultrastrong light-matter interaction through coupled harmonic oscillator models and their connection with cavity-QED Hamiltonians, Nanophotonics 14, 2031 (2025).
  61. A. N. Grigorenko, M. Polini, and K. S. Novoselov, Graphene plasmonics, Nat. Photon. 6, 749 (2012).
  62. D. N. Basov, A. Asenjo-Garcia, P. J. Schuck, X. Zhu, and A. Rubio, Polariton panorama, Nanophotonics 10, 549 (2021).
  63. R. Riolo, A. Tomadin, G. Mazza, R. Asgari, A. H. MacDonald, and M. Polini, Tuning Fermi liquids with sub-wavelength cavities, Proc. Natl. Acad. Sci. USA 122, e2407995122 (2025).
  64. J. D. Caldwell, A. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride, Nat. Commun. 5, 5221 (2014).
  65. P. Li, M. Lewin, A. V. Kretinin, J. D. Caldwell, K. S. Novoselov, T. Taniguchi, K. Watanabe, F. Gaussmann, and T. Taubner, Hyperbolic phonon-polaritons in boron nitride for near-field optical imaging and focusing, Nat. Commun. 6, 7507 (2015).
  66. M. Autore, P. Li, I. Dolado, F. J. Alfaro-Mozaz, R. Esteban, A. Atxabal, F. Casanova, L. E. Hueso, P. Alonso-González, J. Aizpurua, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, Boron nitride nanoresonators for phonon-enhanced molecular vibrational spectroscopy at the strong coupling limit, Light Sci. Appl. 7, 17172 (2018).
  67. H. H. Sheinfux, L. Orsini, M. Jung, I. Torre, M. Ceccanti, S. Marconi, R. Maniyara, D. Barcons Ruiz, A. Hötger, R. Bertini, S. Castilla, N. C. H. Hesp, E. Janzen, A. Holleitner, V. Pruneri, J. H. Edgar, G. Shvets, and F. H. L. Koppens, High-quality nanocavities through multimodal confinement of hyperbolic polaritons in hexagonal boron nitride, Nat. Mater. 23, 499 (2024).
  68. The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (Springer-Verlag, New York, 1987).
  69. G. F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, Cambridge, 2005).
  70. D. Hagenmüller, S. De Liberato, and C. Ciuti, Ultrastrong coupling between a cavity resonator and the cyclotron transition of a two-dimensional electron gas in the case of an integer filling factor, Phys. Rev. B 81, 235303 (2010).
  71. D. Hagenmüller and C. Ciuti, Cavity QED of the graphene cyclotron transition, Phys. Rev. Lett. 109, 267403 (2012).
  72. L. Chirolli, M. Polini, V. Giovannetti, and A. H. MacDonald, Drude weight, cyclotron resonance, and the Dicke model of graphene cavity QED, Phys. Rev. Lett. 109, 267404 (2012).
  73. F. M. D. Pellegrino, L. Chirolli, R. Fazio, V. Giovannetti, and M. Polini, Theory of integer quantum Hall polaritons in graphene, Phys. Rev. B 89, 165406 (2014).
  74. Y. Ashida, A. İmamoğlu, and E. Demler, Cavity quantum electrodynamics with hyperbolic van der Waals materials, Phys. Rev. Lett. 130, 216901 (2023).
  75. M. Kamper Svendsen, Y. Kurman, P. Schmidt, F. Koppens, I. Kaminer, and K. S. Thygesen, Combining density functional theory with macroscopic QED for quantum light-matter interactions in 2D materials, Nat. Commun. 12, 2778 (2021).
  76. D. N. Basov, M. M. Fogler, and F. J. García de Abajo, Polaritons in van der Waals materials, Science 354, aag1992 (2016).
  77. J. Enkner, L. Graziotto, D. Boriçi, F. Appugliese, C. Reichl, G. Scalari, N. Regnault, W. Wegscheider, C. Ciuti, and J. Faist, Enhanced fractional quantum Hall gaps in a two-dimensional electron gas coupled to a hovering split-ring resonator, Nature 641, 884 (2025).
  78. C. Ciuti, Cavity-mediated electron hopping in disordered quantum Hall systems, Phys. Rev. B 104, 155307 (2021).
  79. V. Rokaj, J. Wang, J. Sous, M. Penz, M. Ruggenthaler, and A. Rubio, Weakened topological protection of the quantum Hall effect in a cavity, Phys. Rev. Lett. 131, 196602 (2023).
  80. V. Rokaj, M. Penz, M. A. Sentef, M. Ruggenthaler, and A. Rubio, Polaritonic Hofstadter butterfly and cavity control of the quantized Hall conductance, Phys. Rev. B 105, 205424 (2022).
  81. Z. Bacciconi, H. Xavier, I. Carusotto, T. Chanda, and M. Dalmonte, Theory of fractional quantum Hall liquids coupled to quantum light and emergent graviton-polaritons, Phys. Rev. X 15, 021027 (2025).
  82. G. L. Paravicini-Bagliani, F. Appugliese, E. Richter, F. Valmorra, J. Keller, M. Beck, N. Bartolo, C. Rössler, T. Ihn, K. Ensslin, C. Ciuti, G. Scalari, and J. Faist, Magneto-transport controlled by Landau polariton states, Nat. Phys. 15, 186 (2019).
  83. Z. Bacciconi, G. M. Andolina, and C. Mora, Topological protection of Majorana polaritons in a cavity, Phys. Rev. B 109, 165434 (2024).
  84. O. Dmytruk and M. Schirò, Controlling topological phases of matter with quantum light, Commun. Phys. 5, 271 (2022).
  85. O. Dmytruk and M. Schirò, Hybrid light-matter states in topological superconductors coupled to cavity photons, Phys. Rev. B 110, 075416 (2024).
  86. K. Rzażewski, K. Wódkiewicz, and W. Żakowicz, Phase transitions, two-level atoms, and the A2 term, Phys. Rev. Lett. 35, 432 (1975).
  87. P. Nataf and C. Ciuti, No-go theorem for superradiant quantum phase transitions in cavity QED and counter-example in circuit QED, Nat. Commun. 1, 72 (2010).
  88. G. M. Andolina, F. M. D. Pellegrino, A. Mercurio, O. Di Stefano, M. Polini, and S. Savasta, A non-perturbative no-go theorem for photon condensation in approximate models, Eur. Phys. J. Plus 137, 1348 (2022).
  89. G. M. Andolina, F. M. D. Pellegrino, V. Giovannetti, A. H. MacDonald, and M. Polini, Cavity quantum electrodynamics of strongly correlated electron systems: A no-go theorem for photon condensation, Phys. Rev. B 100, 121109(R) (2019).
  90. G. M. Andolina, F. M. D. Pellegrino, V. Giovannetti, A. H. MacDonald, and M. Polini, Theory of photon condensation in a spatially varying electromagnetic field, Phys. Rev. B 102, 125137 (2020).
  91. D. De Bernardis, T. Jaako, and P. Rabl, Cavity quantum electrodynamics in the nonperturbative regime, Phys. Rev. A 97, 043820 (2018).
  92. D. M. Rouse, A. Stokes, and A. Nazir, Theory of photon condensation in an arbitrary-gauge condensed matter cavity model, Phys. Rev. B 107, 205128 (2023).
  93. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).
  94. A. Tomadin, A. Principi, J. C. W. Song, L. S. Levitov, and M. Polini, Accessing phonon polaritons in hyperbolic crystals by angle-resolved photoemission spectroscopy, Phys. Rev. Lett. 115, 087401 (2015).
  95. C.-R. Mann, Tunable Dirac polaritons in cavity-embedded metasurfaces, Ph.D. thesis, University of Exeter, 2021.
  96. C. R. Gubbin and S. De Liberato, Polaritonic quantization in nonlocal polar materials, J. Chem. Phys. 156, 024111 (2022).
  97. T. Ando, A. B. Fowler, and F. Stern, Electronic properties of two-dimensional systems, Rev. Mod. Phys. 54, 437 (1982).
  98. M. O. Goerbig, Electronic properties of graphene in a strong magnetic field, Rev. Mod. Phys. 83, 1193 (2011).
  99. R. Roldán, J.-N. Fuchs, and M. O. Goerbig, Collective modes of doped graphene and a standard two-dimensional electron gas in a strong magnetic field: Linear magnetoplasmons versus magnetoexcitons, Phys. Rev. B 80, 085408 (2009).
  100. R. Roldán, J.-N. Fuchs, and M. O. Goerbig, Spin-flip excitations, spin waves, and magnetoexcitons in graphene Landau levels at integer filling factors, Phys. Rev. B 82, 205418 (2010).
  101. L. Wehmeier et al., Landau-phonon polaritons in Dirac heterostructures, Sci. Adv. 10, eadp3487 (2024).
  102. N. Rivera, I. Kaminer, B. Zhen, J. D. Joannopoulos, and M. Soljačić, Shrinking light to allow forbidden transitions on the atomic scale, Science 353, 263 (2016).
  103. M. Ruggenthaler, D. Sidler, and A. Rubio, Understanding polaritonic chemistry from Ab Initio quantum electrodynamics, Chem. Rev. 123, 11191 (2023).
  104. C. Schäfer, M. Ruggenthaler, V. Rokaj, and A. Rubio, Relevance of the quadratic diamagnetic and self-polarization terms in cavity quantum electrodynamics, ACS Photon. 7, 975 (2020).
  105. D. De Bernardis, P. Pilar, T. Jaako, S. De Liberato, and P. Rabl, Breakdown of gauge invariance in ultrastrong-coupling cavity QED, Phys. Rev. A 98, 053819 (2018).
  106. S. Savasta, O. Di Stefano, A. Settineri, D. Zueco, S. Hughes, and F. Nori, Gauge principle and gauge invariance in two-level systems, Phys. Rev. A 103, 053703 (2021).
  107. O. Di Stefano, A. Settineri, V. Macrì, L. Garziano, R. Stassi, S. Savasta, and F. Nori, Resolution of gauge ambiguities in ultrastrong-coupling cavity quantum electrodynamics, Nat. Phys. 15, 803 (2019).
  108. O. Dmytruk and M. Schiró, Gauge fixing for strongly correlated electrons coupled to quantum light, Phys. Rev. B 103, 075131 (2021).
  109. J. J. Hopfield, Theory of the contribution of excitons to the complex dielectric constant of crystals, Phys. Rev. 112, 1555 (1958).
  110. A. V. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Laussy, Microcavities (Oxford University Press, Oxford, 2011).
  111. Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Experimental observation of the quantum Hall effect and Berry's phase in graphene, Nature (London) 438, 201 (2005).
  112. M. Geiser, F. Castellano, G. Scalari, M. Beck, L. Nevou, and J. Faist, Ultrastrong coupling regime and plasmon polaritons in parabolic semiconductor quantum wells, Phys. Rev. Lett. 108, 106402 (2012).
  113. R. Kuzmin, N. Mehta, N. Grabon, R. Mencia, and V. E. Manucharyan, Superstrong coupling in circuit quantum electrodynamics, npj Quantum Inf. 5, 20 (2019).
  114. I. Shtrichman, C. Metzner, E. Ehrenfreund, D. Gershoni, K. D. Maranowski, and A. C. Gossard, Depolarization shift of the intersubband resonance in a quantum well with an electron-hole plasma, Phys. Rev. B 65, 035310 (2001).
  115. Y. Todorov and C. Sirtori, Intersubband polaritons in the electrical dipole gauge, Phys. Rev. B 85, 045304 (2012).
  116. V. M. Muravev, I. V. Andreev, S. I. Gubarev, V. N. Belyanin, and I. V. Kukushkin, Fine structure of cyclotron resonance in a two-dimensional electron system, Phys. Rev. B 93, 041110(R) (2016).
  117. S. Dai, Q. Ma, M. K. Liu, T. Andersen, Z. Fei, M. Goldflam, M. Wagner, K. Watanabe, T. Taniguchi, M. Thiemens, F. Keilmann, G. C. A. M. Janssen, S.-E. Zhu, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, Graphene on hexagonal boron nitride as a tunable hyperbolic metamaterial, Nat. Nanotechnol. 10, 682 (2015).
  118. A. Kumar, T. Low, K. H. Fung, P. Avouris, and N. X. Fang, Tunable light–matter interaction and the role of hyperbolicity in graphene–hBN system, Nano Lett. 15, 3172 (2015).
  119. X. Li, M. Bamba, Q. Zhang, S. Fallahi, G. C. Gardner, W. Gao, M. Lou, K. Yoshioka, M. J. Manfra, and J. Kono, Vacuum Bloch–Siegert shift in Landau polaritons with ultra-high cooperativity, Nat. Photon. 12, 324 (2018).
  120. L. Knöll, S. Scheel, and D.-G. Welsch, Coherence and Statistics of Photons and Atoms (Wiley, New York, 2001), Chap. 1.
  121. S. Scheel and S. Y. Buhmann, Macroscopic quantum electrodynamics—Concepts and applications, Acta Phys. Slov. 58, 675 (2008).
  122. R. Geick, C. H. Perry, and G. Rupprecht, Normal modes in hexagonal boron nitride, Phys. Rev. 146, 543 (1966).
  123. A. H. Castro Neto, F. Guinea, N. M. R. Peres, A. K. Geim, and K. S. Novoselov, The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009).
  124. M. Dapolito, M. Tsuneto, W. Zheng, L. Wehmeier, S. Xu, X. Chen, J. Sun, Z. Du, Y. Shao, and R. Jing, Infrared nano-imaging of Dirac magnetoexcitons in graphene, Nat. Nanotechnol. 18, 1409 (2023).
  125. Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Correlated insulator at half-filling in magic-angle graphene superlattices, Nature (London) 556, 80 (2018).
  126. A. Thomas, E. Devaux, K. Nagarajan, T. Chervy, M. Seidel, D. Hagenmüller, S. Schütz, J. Schachenmayer, C. Genet, G. Pupillo, and T. W. Ebbesen, Exploring superconductivity under strong coupling with the vacuum electromagnetic field, J. Chem. Phys. 162, 134701 (2025).
  127. T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and A. Amo, Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons, Phys. Rev. Lett. 112, 116402 (2014).
  128. G. M. Andolina, M. Ceccanti, B. Turini, R. Riolo, M. Polini, M. Schiró, and F. H. L. Koppens, Simulation scripts for quantum electrodynamics of graphene Landau levels in a deep-subwavelength hyperbolic phonon polariton cavity, Zenodo, 2026, https://doi.org/10.5281/zenodo.18680601.
  129. G. M. Andolina, M. Ceccanti, B. Turini, R. Riolo, M. Polini, M. Schiró, and F. H. L. Koppens, Figures of quantum electrodynamics of graphene Landau levels in a deep-subwavelength hyperbolic phonon polariton cavity, Figshare, 2026, https://doi.org/10.6084/m9.figshare.31282294.

Outline

Information

Phys. Rev. Research 8, 023044– Published 14 April, 2026
  • Received 31 March 2025
  • Accepted 4 February 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation