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Quantum treatment of black hole superradiance

Lingyun Fu1,2,*, Hidetoshi Omiya3,†, Takahiro Tanaka3,4,‡, Xi Tong2,§, Yi Wang5,6,∥, and Hui-Yu Zhu7,¶

  • *Contact author: lf556@cam.ac.uk
  • †Contact author: omiya@tap.scphys.kyoto-u.ac.jp
  • ‡Contact author: t.tanaka@tap.scphys.kyoto-u.ac.jp
  • §Contact author: xt246@cam.ac.uk
  • ∥Contact author: phyw@ust.hk
  • Contact author: hzhuav@ibs.re.kr
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Phys. Rev. D 114, 024026 – Published 10 July, 2026

DOI: https://doi.org/10.1103/jsgc-7b39

Abstract

Rotating black holes can form dense boson clouds through superradiant instability, making Kerr black holes a powerful probe of ultralight massive bosons. Previous studies of black hole superradiance have often treated bosonic fields classically, leaving open questions about how particles are produced and how the clouds grow over time. In this work, we canonically quantize a massive scalar field around a Kerr black hole, providing a fully quantum description of black hole superradiance. We show that the evolution of the particle number in the cloud, as well as the energy and angular momentum of the scalar field, can be consistently explained within the standard framework of quantum field theory in curved spacetime. Furthermore, we prove that the growth of the cloud occurs independently of the choice of initial state. We also explore several phenomena related to a massive scalar field in a rotating black hole spacetime, including Hawking radiation, adiabatic backreaction on the black hole spin, and the direction of level transitions in the presence of self-interactions of the field. Our analysis provides a consistent quantum-mechanical perspective that includes all these phenomena.

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References (66)

  1. W. H. Press, B. S. Ryden, and D. N. Spergel, Single mechanism for generating large scale structure and providing dark missing matter, Phys. Rev. Lett. 64, 1084 (1990).
  2. W. Hu, R. Barkana, and A. Gruzinov, Cold and fuzzy dark matter, Phys. Rev. Lett. 85, 1158 (2000).
  3. J. Goodman, Repulsive dark matter, New Astron. 5, 103 (2000).
  4. D. J. E. Marsh, Axion cosmology, Phys. Rep. 643, 1 (2016).
  5. L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, Ultralight scalars as cosmological dark matter, Phys. Rev. D 95, 043541 (2017).
  6. Y. B. Zel’Dovich, Generation of waves by a rotating body, JETP Lett. 14, 180 (1971).
  7. I. Zel’dovich, Amplification of cylindrical electromagnetic waves reflected from a rotating body, Sov. Phys. JETP 35, 1085 (1972).
  8. A. A. Starobinskii, Amplification of waves during reflection from a rotating “black hole”, Sov. Phys. JETP 37, 28 (1973).
  9. T. Damour, N. Deruelle, and R. Ruffini, On quantum resonances in stationary geometries, Lett. Nuovo Cimento 15, 257 (1976).
  10. T. J. M. Zouros and D. M. Eardley, Instabilities of massive scalar perturbations of a rotating black hole, Ann. Phys. (Amsterdam) 118, 139 (1979).
  11. S. L. Detweiler, Klein-Gordon equation and rotating black holes, Phys. Rev. D 22, 2323 (1980).
  12. S. R. Dolan, Instability of the massive Klein-Gordon field on the Kerr spacetime, Phys. Rev. D 76, 084001 (2007).
  13. R. Brito, V. Cardoso, and P. Pani, Superradiance: New frontiers in black hole physics, Lect. Notes Phys. 906, 1 (2015).
  14. A. Arvanitaki and S. Dubovsky, Exploring the string axiverse with precision black hole physics, Phys. Rev. D 83, 044026 (2011).
  15. H. Yoshino and H. Kodama, Gravitational radiation from an axion cloud around a black hole: Superradiant phase, Prog. Theor. Exp. Phys. 2014, 043E02 (2014).
  16. K. H. M. Chan and O. A. Hannuksela, Extracting ultralight boson properties from Boson clouds around postmerger remnants, Phys. Rev. D 109, 023009 (2024).
  17. I. K. Banerjee, S. Bonthu, and U. K. Dey, Multi-messenger approach to ultra-light scalars, Phys. Rev. D 112, 063012 (2025).
  18. N. Siemonsen, T. May, and W. E. East, Modeling the black hole superradiance gravitational waveform, Phys. Rev. D 107, 104003 (2023).
  19. S. Collaviti, L. Sun, M. Galanis, and M. Baryakhtar, Observational prospects of self-interacting scalar superradiance with next-generation gravitational-wave detectors, Classical Quantum Gravity 42, 025006 (2025).
  20. H. Omiya, T. Takahashi, T. Tanaka, and H. Yoshino, Deci-Hz gravitational waves from the self-interacting axion cloud around a rotating stellar mass black hole, Phys. Rev. D 110, 044002 (2024).
  21. R. Brito, S. Ghosh, E. Barausse, E. Berti, V. Cardoso, I. Dvorkin, A. Klein, and P. Pani, Gravitational wave searches for ultralight bosons with LIGO and LISA, Phys. Rev. D 96, 064050 (2017).
  22. R. Brito, S. Ghosh, E. Barausse, E. Berti, V. Cardoso, I. Dvorkin, A. Klein, and P. Pani, Stochastic and resolvable gravitational waves from ultralight bosons, Phys. Rev. Lett. 119, 131101 (2017).
  23. M. Isi, L. Sun, R. Brito, and A. Melatos, Directed searches for gravitational waves from ultralight bosons, Phys. Rev. D 99, 084042 (2019); 102, 049901(E) (2020).
  24. S. Ghosh, E. Berti, R. Brito, and M. Richartz, Follow-up signals from superradiant instabilities of black hole merger remnants, Phys. Rev. D 99, 104030 (2019).
  25. L. Tsukada, T. Callister, A. Matas, and P. Meyers, First search for a stochastic gravitational-wave background from ultralight bosons, Phys. Rev. D 99, 103015 (2019).
  26. N. Afshordi et al. (LISA Consortium Waveform Working Group), Waveform modelling for the laser interferometer space antenna, Living Rev. Relativity 28, 9 (2025).
  27. D. Baumann, H. S. Chia, R. A. Porto, and J. Stout, Gravitational collider physics, Phys. Rev. D 101, 083019 (2020).
  28. Q. Ding, X. Tong, and Y. Wang, Gravitational collider physics via pulsar-black hole binaries, Astrophys. J. 908, 78 (2021).
  29. T. Takahashi, H. Omiya, and T. Tanaka, Axion cloud evaporation during inspiral of black hole binaries: The effects of backreaction and radiation, Prog. Theor. Exp. Phys. 2022, 043E01 (2022).
  30. X. Tong, Y. Wang, and H.-Y. Zhu, Gravitational collider physics via pulsar–black hole binaries II: Fine and hyperfine structures are favored, Astrophys. J. 924, 99 (2022).
  31. X. Tong, Y. Wang, and H.-Y. Zhu, Termination of superradiance from a binary companion, Phys. Rev. D 106, 043002 (2022).
  32. D. Baumann, G. Bertone, J. Stout, and G. M. Tomaselli, Ionization of gravitational atoms, Phys. Rev. D 105, 115036 (2022).
  33. P. S. Cole, G. Bertone, A. Coogan, D. Gaggero, T. Karydas, B. J. Kavanagh, T. F. M. Spieksma, and G. M. Tomaselli, Distinguishing environmental effects on binary black hole gravitational waveforms, Nat. Astron. 7, 943 (2023).
  34. T. Takahashi, H. Omiya, and T. Tanaka, Evolution of binary systems accompanying axion clouds in extreme mass ratio inspirals, Phys. Rev. D 107, 103020 (2023).
  35. G. M. Tomaselli, T. F. M. Spieksma, and G. Bertone, Dynamical friction in gravitational atoms, J. Cosmol. Astropart. Phys. 07 (2023) 070.
  36. K. Fan, X. Tong, Y. Wang, and H.-Y. Zhu, Modulating binary dynamics via the termination of black hole superradiance, Phys. Rev. D 109, 024059 (2024).
  37. T. F. M. Spieksma, V. Cardoso, G. Carullo, M. Della Rocca, and F. Duque, Black hole spectroscopy in environments: Detectability prospects, Phys. Rev. Lett. 134, 081402 (2025).
  38. H.-Y. Zhu, X. Tong, G. Manzoni, and Y. Ma, Survival of the fittest: Testing superradiance termination with simulated binary black hole statistics, Astrophys. J. 981, 165 (2025).
  39. G. M. Tomaselli, T. F. M. Spieksma, and G. Bertone, Legacy of Boson clouds on black hole binaries, Phys. Rev. Lett. 133, 121402 (2024).
  40. Q. Ding, M. He, V. Takhistov, and H.-Y. Zhu, Superradiant Bosons driving supermassive black hole mergers, Phys. Rev. D 112, 103051 (2025).
  41. C. Dyson, T. F. M. Spieksma, R. Brito, M. van de Meent, and S. Dolan, Environmental effects in extreme-mass-ratio inspirals: Perturbations to the environment in Kerr spacetimes, Phys. Rev. Lett. 134, 211403 (2025).
  42. G. M. Tomaselli, Smooth binary evolution from wide resonances in Boson clouds, Phys. Rev. D 112, 063033 (2025).
  43. Y. Guo, Z. Zhong, Y. Chen, V. Cardoso, T. Ikeda, and L. Zhou, Ultralight Boson ionization from comparable-mass binary black holes, arXiv:2509.09643.
  44. Q. Ding, M. He, and H.-Y. Zhu, Extracting properties of dark dense environment around black holes from gravitational waves, Phys. Rev. D 113, 083011 (2026).
  45. W. G. Unruh, Second quantization in the Kerr metric, Phys. Rev. D 10, 3194 (1974).
  46. L. H. Ford, Quantization of a scalar field in the Kerr space-time, Phys. Rev. D 12, 2963 (1975); 14, 658(E) (1976).
  47. B. S. DeWitt, Quantum field theory in curved space-time, Phys. Rep. 19, 295 (1975).
  48. I. Agullo, A. J. Brady, A. Delhom, and D. Kranas, Entanglement from rotating black holes in thermal baths, Phys. Rev. D 110, 025021 (2024).
  49. D.-C. Dai and D. Stojkovic, Separating the superradiant emission from the Hawking radiation from a rotating black hole, Phys. Lett. B 843, 138056 (2023).
  50. G. Kang, Quantum aspects of Ergoregion instability, Phys. Rev. D 55, 7563 (1997).
  51. E. Cannizzaro, L. Sberna, S. R. Green, and S. Hollands, Relativistic perturbation theory for black-hole Boson clouds, Phys. Rev. Lett. 132, 051401 (2024).
  52. S. A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations, Astrophys. J. 185, 635 (1973).
  53. D. R. Brill, P. L. Chrzanowski, C. Martin Pereira, E. D. Fackerell, and J. R. Ipser, Solution of the scalar wave equation in a Kerr background by separation of variables, Phys. Rev. D 5, 1913 (1972).
  54. V. Balakumar, R. P. Bernar, and E. Winstanley, Quantization of a charged scalar field on a charged black hole background, Phys. Rev. D 106, 125013 (2022).
  55. Black Hole Physics: Basic Concepts and New Developments, edited by V. P. Frolov and I. D. Novikov (Springer Science & Business Media, 1998).
  56. E. Berti, V. Cardoso, and A. O. Starinets, Quasinormal modes of black holes and black branes, Classical Quantum Gravity 26, 163001 (2009).
  57. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, England, 1997).
  58. P. Candelas, Vacuum polarization in Schwarzschild space-time, Phys. Rev. D 21, 2185 (1980).
  59. W. G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14, 870 (1976).
  60. J. B. Hartle and S. W. Hawking, Path integral derivation of black hole radiance, Phys. Rev. D 13, 2188 (1976).
  61. S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975); 46, 206(E) (1976).
  62. A. C. Ottewill and E. Winstanley, The Renormalized stress tensor in Kerr space-time: General results, Phys. Rev. D 62, 084018 (2000).
  63. R. Casadio and G. Venturi, Black holes and the adiabatic phase, Classical Quantum Gravity 12, 1267 (1995).
  64. M. Baryakhtar, M. Galanis, R. Lasenby, and O. Simon, Black hole superradiance of self-interacting scalar fields, Phys. Rev. D 103, 095019 (2021).
  65. H. Omiya, T. Takahashi, T. Tanaka, and H. Yoshino, Impact of multiple modes on the evolution of self-interacting axion condensate around rotating black holes, J. Cosmol. Astropart. Phys. 06 (2023) 016.
  66. S. J. Witte and A. Mummery, Stepping up superradiance constraints on axions, Phys. Rev. D 111, 083044 (2025).

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Phys. Rev. D 114, 024026– Published 10 July, 2026
  • Received 27 January 2026
  • Accepted 21 May 2026
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