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Simulating vacuum birefringence with a diffractive beam propagation code

Aimé Matheron1,2,3,*, Michal Šmíd4,†, Matt Zepf1,2,3,‡, and Felix Karbstein1,2,3,§

  • *Contact author: A.Matheron@hi-jena.gsi.de
  • †Contact author: m.smid@hzdr.de
  • ‡Contact author: m.zepf@hi-jena.gsi.de
  • §Contact author: f.karbstein@hi-jena.gsi.de
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Phys. Rev. D 113, 033005 – Published 18 February, 2026

DOI: https://doi.org/10.1103/qqcw-461k

Abstract

Ninety years after their prediction, quantum vacuum nonlinearities in macroscopic electromagnetic fields still await a direct experimental verification in the laboratory. A particularly promising route towards their first measurement is the collision of counterpropagating laser beams in a pump-probe type experiment. Here, the key challenge is to separate the small quantum vacuum signal at the oscillation frequency of the probe that is mainly emitted in the vicinity of its forward cone from the large probe background. While quantitatively accurate predictions of the associated quantum vacuum signals are available, to date there is no framework that combines these predictions with a diffractive beam propagation code. Such codes are designed to holistically model optical experiments and can reliably account for diffraction and absorption losses of optical devices, such as lenses and apertures. The latter inevitably influence and modify both the induced signal and background components prior to their detection in experiment. The present work addresses this topical issue and reports on the first implementation of a quantum vacuum signals emission module in an established diffractive beam propagation toolkit designed for the realistic modeling of optical experiments.

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References (61)

  1. H. Euler and B. Kockel, Naturwissenschaften 23, 246 (1935).
  2. H. Euler, Ann. Phys. (Berlin) 26, 398 (1936).
  3. W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936).
  4. W. H. Furry, Phys. Rev. 51, 125 (1937).
  5. V. P. Gusynin and I. A. Shovkovy, Can. J. Phys. 74, 282 (1996).
  6. V. P. Gusynin and I. A. Shovkovy, J. Math. Phys. (N.Y.) 40, 5406 (1999).
  7. F. Karbstein, J. High Energy Phys. 09 (2021) 070.
  8. A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, Rev. Mod. Phys. 84, 1177 (2012).
  9. R. Battesti and C. Rizzo, Rep. Prog. Phys. 76, 016401 (2013).
  10. B. King and T. Heinzl, High Power Laser Sci. Eng. 4, e5 (2016).
  11. R. Battesti et al., Phys. Rep. 765–766, 1 (2018).
  12. F. Karbstein, Particles 3, 39 (2020).
  13. A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt, H. Taya, and G. Torgrimsson, Phys. Rep. 1010, 1 (2023).
  14. N. Ahmadiniaz et al., High Power Laser Sci. Eng. 13, e7 (2025).
  15. A. Blinne, H. Gies, F. Karbstein, C. Kohlfürst, and M. Zepf, Phys. Rev. D 99, 016006 (2019).
  16. T. Grismayer, R. Torres, P. Carneiro, F. Cruz, R. Fonseca, and L. O. Silva, New J. Phys. 23, 095005 (2021).
  17. A. Lindner, B. Ölmez, and H. Ruhl, J. Comput. Phys. 17, 100124 (2023).
  18. Z. Zhang et al., Commun. Phys. 8, 224 (2025).
  19. M. Alawashra, J. Benáček, M. Pohl, and M. Medvedev, Phys. Plasmas 32, 113903 (2025).
  20. G. Vdovin, H. van Brug, and F. van Goor, Proc. SPIE Int. Soc. Opt. Eng. 3190, 82 (1997).
  21. G. Vdovin and F. van Goor, LightPipes for python 2.1.5, https://opticspy.github.io/lightpipes/.
  22. M. Šmíd et al., Phys. Rev. A 112, 6 (2025).
  23. A. Laso Garcia et al., High Power Laser Sci. Eng. 9, e59 (2021).
  24. Z. Bialynicka-Birula and I. Bialynicki-Birula, Phys. Rev. D 2, 2341 (1970).
  25. E. Brezin and C. Itzykson, Phys. Rev. D 3, 618 (1971).
  26. F. Karbstein and C. Sundqvist, Phys. Rev. D 94, 013004 (2016).
  27. F. Karbstein and R. Shaisultanov, Phys. Rev. D 91, 085027 (2015).
  28. H. Gies, F. Karbstein, and C. Kohlfürst, Phys. Rev. D 97, 036022 (2018).
  29. B. King, H. Hu, and B. Shen, Phys. Rev. A 98, 023817 (2018).
  30. F. Karbstein, Phys. Rev. D 98, 056010 (2018).
  31. E. A. Mosman and F. Karbstein, Phys. Rev. D 104, 013006 (2021).

  32. Note that the function F˜≡F˜(2zR/T,t0/T,Tprobe/T) can be identically rewritten as F˜=F/(F|zR→∞)=3π4F with F introduced in Eq. (7) of Ref. [30]; this reference uses 1/e2 pulse durations with respect to intensity.

  33. J. S. Toll, Ph.D. thesis, Princeton University, 1952 (unpublished).
  34. J. J. Klein and B. P. Nigam, Phys. Rev. 135, B1279 (1964).
  35. R. Baier and P. Breitenlohner, Acta Phys. Austriaca 25, 212 (1967).
  36. R. Baier and P. Breitenlohner, Nuovo Cimento B 47, 117 (1967).
  37. E. B. Aleksandrov, A. A. Ansel’m, and A. N. Moskalev, Sov. Phys. JETP 62, 680 (1985).
  38. T. Heinzl, B. Liesfeld, K. U. Amthor, H. Schwoerer, R. Sauerbrey, and A. Wipf, Opt. Commun. 267, 318 (2006).
  39. A. Di Piazza, K. Z. Hatsagortsyan, and C. H. Keitel, Phys. Rev. Lett. 97, 083603 (2006).
  40. B. King, A. Di Piazza, and C. H. Keitel, Phys. Rev. A 82, 032114 (2010).
  41. V. Dinu, T. Heinzl, A. Ilderton, M. Marklund, and G. Torgrimsson, Phys. Rev. D 89, 125003 (2014).
  42. V. Dinu, T. Heinzl, A. Ilderton, M. Marklund, and G. Torgrimsson, Phys. Rev. D 90, 045025 (2014).
  43. F. Karbstein, H. Gies, M. Reuter, and M. Zepf, Phys. Rev. D 92, 071301 (2015).
  44. H. P. Schlenvoigt, T. Heinzl, U. Schramm, T. E. Cowan, and R. Sauerbrey, Phys. Scr. 91, 023010 (2016).
  45. B. Shen, Z. Bu, J. Xu, T. Xu, L. Ji, R. Li, and Z. Xu, Plasma Phys. Controlled Fusion 60, 044002 (2018).
  46. B. Shen, Z. Bu, J. Xu, T. Xu, L. Ji, R. Li, and Z. Xu, Nucl. Instrum. Methods Phys. Res., Sect. A 982, 164553 (2020).
  47. N. Ahmadiniaz, T. E. Cowan, J. Grenzer, S. Franchino-Viñas, A. L. Garcia, M. Šmíd, T. Toncian, M. A. Trejo, and R. Schützhold, Phys. Rev. D 108, 076005 (2023).
  48. T. Heinzl, B. King, and D. Liu, Phys. Rev. D 111, 056018 (2025).
  49. S. Ataman and Y. Nakamiya, Phys. Scr. 100, 075537 (2025).
  50. A. Ejlli, F. Della Valle, U. Gastaldi, G. Messineo, R. Pengo, G. Ruoso, and G. Zavattini, Phys. Rep. 871, 1 (2020).
  51. B. Marx et al., Phys. Rev. Lett. 110, 254801 (2013).
  52. F. Karbstein, Habilitation thesis, Faculty of Physics and Astronomy, Friedrich-Schiller-Universität Jena, 2024, 10.22032/dbt.59618.
  53. D. Tommasini and H. Michinel, Phys. Rev. A 82, 011803 (2010).
  54. J. Peatross, J. L. Chaloupka, and D. D. Meyerhofer, Opt. Lett. 19, 942 (1994).
  55. M. Zepf et al., Phys. Rev. E 58, R5253 (1998).
  56. F. Karbstein and E. A. Mosman, Phys. Rev. D 101, 113002 (2020).
  57. F. Karbstein, D. Ullmann, E. A. Mosman, and M. Zepf, Phys. Rev. Lett. 129, 061802 (2022).
  58. K. S. Schulze, APL Photonics 3, 126106 (2018).
  59. A. T. Schmitt et al., Optica 8, 56 (2021).
  60. Q. Yu, D. Xu, B. Shen, T. E. Cowan, and H. P. Schlenvoigt, High Power Laser Sci. Eng. 11, e71 (2023).
  61. Aimé Matheron, amatheron/VIBE: VIBE v1.0.0, Zenodo, 10.5281/zenodo.17979735 (2025).

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Phys. Rev. D 113, 033005– Published 18 February, 2026
  • Received 22 December 2025
  • Accepted 29 January 2026
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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.

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