- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

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

Entanglement in the Schwinger effect

Dimitrios Kranas1,2,*, Amaury Marchon3,4,5,†, and Silvia Pla4,‡

  • *Contact author: dimitrioskranas@gmail.com
  • †Contact author: amaury.marchon@lapth.cnrs.fr
  • ‡Contact author: silvia.pla-garcia@tum.de
PDF

Phys. Rev. D 114, 085011 – Published 13 October, 2026

DOI: https://doi.org/10.1103/p4hy-cw4l

Abstract

We analyze entanglement generated by the Schwinger effect using a mode-by-mode formalism for scalar and spinor QED in constant backgrounds. Starting from thermal initial states, we derive compact, closed-form results for bipartite entanglement between particle-antiparticle partners in terms of the Bogoliubov coefficients. For bosons, thermal fluctuations enhance production but suppress quantum correlations: The logarithmic negativity is nonzero only below a (mode-dependent) critical temperature Tc. At fixed T, entanglement appears only above a critical field Eentang. For fermions, we observe a qualitatively different pattern: The fermionic logarithmic negativity is nonvanishing at finite temperature, and is monotonically suppressed by thermal noise. As a function of the electric field, it is nonmonotonic, featuring a temperature-independent optimal field strength E* and decreasing on both sides of the maximum. We give quantitative estimates for analog experiments, where our entanglement criteria convert directly into concrete temperature and electric-field constraints. These findings identify realistic regimes where the quantum character of Schwinger physics may be tested in the laboratory.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (87)

  1. J. S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82, 664 (1951).
  2. L. E. Parker, The Creation of Particles in an Expanding Universe, Ph.D. thesis, Harvard University, Cambridge, Massachusetts, USA, 1966.
  3. L. Parker, Particle creation in expanding universes, Phys. Rev. Lett. 21, 562 (1968).
  4. L. Parker, Quantized fields and particle creation in expanding universes. I, Phys. Rev. 183, 1057 (1969).
  5. S. W. Hawking, Black hole explosions, Nature (London) 248, 30 (1974).
  6. S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975); 46, 206(E) (1976).
  7. G. V. Dunne, New strong-field QED effects at ELI: Nonperturbative vacuum pair production, Eur. Phys. J. D 55, 327 (2009).
  8. F. Karbstein, Probing vacuum polarization effects with high-intensity lasers, Particles 3, 39 (2020).
  9. A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt, H. Taya, and G. Torgrimsson, Advances in QED with intense background fields, Phys. Rep. 1010, 1 (2023).
  10. A. Schmitt et al., Mesoscopic Klein-Schwinger effect in graphene, Nat. Phys. 19, 830 (2023).
  11. M. Hongo, T. Fujimori, T. Misumi, M. Nitta, and N. Sakai, Effective field theory of magnons: Chiral magnets and the Schwinger mechanism, Phys. Rev. B 104, 134403 (2021).
  12. T. C. Adorno, S. P. Gavrilov, and D. M. Gitman, Schwinger mechanism of magnon-antimagnon pair production on magnetic field inhomogeneities and the bosonic Klein effect, Phys. Rev. B 110, 014410 (2024).
  13. S. Grieninger, D. E. Kharzeev, and I. Zahed, Entanglement in a holographic Schwinger pair with confinement, Phys. Rev. D 108, 086030 (2023).
  14. S. Grieninger, D. E. Kharzeev, and I. Zahed, Entanglement entropy in a time-dependent holographic Schwinger pair creation, Phys. Rev. D 108, 126014 (2023).
  15. K. Jensen and A. Karch, Holographic dual of an Einstein-Podolsky-Rosen pair has a wormhole, Phys. Rev. Lett. 111, 211602 (2013).
  16. J. Sonner, Holographic Schwinger effect and the geometry of entanglement, Phys. Rev. Lett. 111, 211603 (2013).
  17. W. Dittrich, Effective Lagrangians at finite temperature, Phys. Rev. D 19, 2385 (1979).
  18. M. Loewe and J. C. Rojas, Thermal effects and the effective action of quantum electrodynamics, Phys. Rev. D 46, 2689 (1992).
  19. H. Gies, QED effective action at finite temperature, Phys. Rev. D 60, 105002 (1999).
  20. L. Medina and M. C. Ogilvie, Schwinger pair production at finite temperature, Phys. Rev. D 95, 056006 (2017).
  21. O. Gould, A. Rajantie, and C. Xie, Worldline sphaleron for thermal Schwinger pair production, Phys. Rev. D 98, 056022 (2018).
  22. M. Korwar and A. M. Thalapillil, Finite temperature Schwinger pair production in coexistent electric and magnetic fields, Phys. Rev. D 98, 076016 (2018).
  23. H. Shapourian, K. Shiozaki, and S. Ryu, Partial time-reversal transformation and entanglement negativity in fermionic systems, Phys. Rev. B 95, 165101 (2017).
  24. H. Shapourian and S. Ryu, Entanglement negativity of fermions: Monotonicity, separability criterion, and classification of few-mode states, Phys. Rev. A 99, 022310 (2019).
  25. Z. Ebadi and B. Mirza, Entanglement generation by electric field background, Ann. Phys. (Amsterdam) 351, 363 (2014).
  26. Y. Li, Y. Dai, and Y. Shi, Pairwise mode entanglement in Schwinger production of particle-antiparticle pairs in an electric field, Phys. Rev. D 95, 036006 (2017).
  27. S. Bhattacharya, S. Chakrabortty, H. Hoshino, and S. Kaushal, Background magnetic field and quantum correlations in the Schwinger effect, Phys. Lett. B 811, 135875 (2020).
  28. S. Kaushal, Fermionic entanglement in the presence of background electric and magnetic fields, Eur. Phys. J. C 85, 75 (2025).
  29. A. Florio and D. E. Kharzeev, Gibbs entropy from entanglement in electric quenches, Phys. Rev. D 104, 056021 (2021).
  30. G. V. Dunne, A. Florio, and D. E. Kharzeev, Entropy suppression through quantum interference in electric pulses, Phys. Rev. D 108, L031901 (2023).
  31. A. Florio, D. Frenklakh, S. Grieninger, D. E. Kharzeev, A. Palermo, and S. Shi, Thermalization from quantum entanglement: Jet simulations in the massive Schwinger model, Phys. Rev. D 112, 094502 (2025).
  32. Y. Afik et al., Quantum information meets high-energy physics: Input to the update of the European strategy for particle physics, Eur. Phys. J. Plus 140, 855 (2025).
  33. J. A. Aguilar-Saavedra and J. A. Casas, Improved tests of entanglement and Bell inequalities with LHC tops, Eur. Phys. J. C 82, 666 (2022).
  34. J. A. Aguilar-Saavedra, A. Bernal, J. A. Casas, and J. M. Moreno, Testing entanglement and Bell inequalities in H→ZZ, Phys. Rev. D 107, 016012 (2023).
  35. C. D. Pueyo, H. Goodhew, C. McCulloch, and E. Pajer, Perturbative unitarity bounds from momentum-space entanglement, J. High Energy Phys. 08 (2025) 047.
  36. A. Bhardwaj, I. Agullo, D. Kranas, J. H. Wilson, and D. E. Sheehy, Entanglement in an expanding toroidal Bose-Einstein condensate, Phys. Rev. A 109, 013305 (2024).
  37. S. K. Manikandan and F. Wilczek, Complementary probes of gravitational radiation states, Phys. Rev. A 112, 043716 (2025).
  38. S. K. Manikandan and F. Wilczek, Probing quantum structure in gravitational radiation, Int. J. Mod. Phys. D 34, 2543001 (2025).
  39. W. Heisenberg and H. Euler, Consequences of Dirac’s theory of positrons, Z. Phys. 98, 714 (1936).
  40. V. Weisskopf, The electrodynamics of the vacuum based on the quantum theory of the electron, Kong. Dan. Vid. Sel. Mat. Fys. Med. 14, 1 (1936).
  41. G. V. Dunne, Heisenberg-Euler effective Lagrangians: Basics and extensions, in From Fields to Strings: Circumnavigating Theoretical Physics. Ian Kogan Memorial Collection (3 Volume Set), edited by M. Shifman, A. Vainshtein, and J. Wheater (2004), pp. 445–522.
  42. C. Schubert, Pair creation in electric fields, EPJ Web Conf. 301, 03004 (2024).
  43. K. D. Lozanov, A. Maleknejad, and E. Komatsu, Schwinger effect by an SU(2) gauge field during inflation, J. High Energy Phys. 02 (2019) 041.
  44. V. Domcke, Y. Ema, and K. Mukaida, Chiral Anomaly, Schwinger effect, Euler-Heisenberg Lagrangian, and application to axion inflation, J. High Energy Phys. 02 (2020) 055.
  45. A. Ferreiro and J. Navarro-Salas, Pair creation in electric fields, anomalies, and renormalization of the electric current, Phys. Rev. D 97, 125012 (2018).
  46. P. Beltrán-Palau, J. Navarro-Salas, and S. Pla, Adiabatic regularization for Dirac fields in time-varying electric backgrounds, Phys. Rev. D 101, 105014 (2020).
  47. A. Álvarez-Domínguez, L. J. Garay, M. Martín-Benito, and R. B. Neves, States of low energy in the Schwinger effect, J. High Energy Phys. 06 (2023) 093.
  48. A. Álvarez-Domínguez, J. A. R. Cembranos, L. J. Garay, M. Martín-Benito, A. Parra-López, and J. M. Sánchez Velázquez, Operational realization of quantum vacuum ambiguities, Phys. Rev. D 108, 065008 (2023).
  49. L. Vicente García-Consuegra and A. Maleknejad, The stochastic Schwinger effect, J. High Energy Phys. 03 (2026) 043.
  50. H. Taya, T. Fujimori, T. Misumi, M. Nitta, and N. Sakai, Exact WKB analysis of the vacuum pair production by time-dependent electric fields, J. High Energy Phys. 03 (2021) 082.
  51. S. P. Gavrilov, D. M. Gitman, and A. A. Shishmarev, Statistical properties of states in QED with unstable vacuum, Phys. Rev. A 91, 052106 (2015).
  52. S. P. Gavrilov, D. M. Gitman, and A. A. Shishmarev, Unitarity and vacuum deformation in QED with critical potential steps, Phys. Rev. D 93, 105040 (2016).
  53. S. P. Gavrilov, D. M. Gitman, and A. A. Shishmarev, States of charged quantum fields and their statistical properties in the presence of critical potential steps, Phys. Rev. A 99, 052116 (2019).
  54. Á. Álvarez-Domínguez, Quantization of charged fields in the presence of intense electromagnetic fields, Ph.D. thesis, Universidad Complutense de Madrid, Madrid, Spain, 2025.
  55. L. E. Parker and D. Toms, Quantum Field Theory in Curved Spacetime: Quantized Field and Gravity, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2009).
  56. T. Padmanabhan, Quantum theory in external electromagnetic and gravitational fields: A comparison of some conceptual issues, Pramana 37, 179 (1991).
  57. A. I. Nikishov, Pair production by a constant external field, Zh. Eksp. Teor. Fiz. 57, 1210 (1969) [Sov. Phys. JETP 30, 660 (1970)].
  58. R. Soldati, Pairs emission in a uniform background field: An algebraic approach, J. Phys. A 44, 305401 (2011).
  59. M. B. Plenio and S. Virmani, An introduction to entanglement measures, Quantum Inf. Comput. 7, 001 (2007).
  60. M. B. Plenio, Logarithmic negativity: A full entanglement monotone that is not convex, Phys. Rev. Lett. 95, 090503 (2005).
  61. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
  62. M. Horodecki, P. Horodecki, and R. Horodecki, Separability of mixed states: Necessary and sufficient conditions, Phys. Lett. A 223, 1 (1996).
  63. G. Adesso, A. Serafini, and F. Illuminati, Multipartite entanglement in three-mode gaussian states of continuous-variable systems: Quantification, sharing structure, and decoherence, Phys. Rev. A 73, 032345 (2006).
  64. A. Florio, Two-fermion negativity and confinement in the Schwinger model, Phys. Rev. D 109, L071501 (2024).
  65. A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods (CRC Press, Boca Raton, FL, 2017).
  66. R. Simon, Peres-horodecki separability criterion for continuous variable systems, Phys. Rev. Lett. 84, 2726 (2000).
  67. W. G. Unruh, Notes on black-hole evaporation, Phys. Rev. D 14, 870 (1976).
  68. L. P. Grishchuk and Y. V. Sidorov, Squeezed quantum states of relic gravitons and primordial density fluctuations, Phys. Rev. D 42, 3413 (1990).
  69. I. Agullo, A. J. Brady, and D. Kranas, Quantum aspects of stimulated hawking radiation in an optical analog white-black hole pair, Phys. Rev. Lett. 128, 091301 (2022).
  70. A. J. Brady, I. Agullo, and D. Kranas, Symplectic circuits, entanglement, and stimulated Hawking radiation in analogue gravity, Phys. Rev. D 106, 105021 (2022).
  71. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 1982).
  72. P. Beltrán-Palau, A. Ferreiro, J. Navarro-Salas, and S. Pla, Breaking of adiabatic invariance in the creation of particles by electromagnetic backgrounds, Phys. Rev. D 100, 085014 (2019).
  73. R. A. Bertlmann, Anomalies in Quantum Field Theory (Oxford University Press, New York, 2000).
  74. M. Reuter and W. Dittrich, Chiral jacobians and two-dimensional qed at finite temperature, Phys. Rev. D 32, 513 (1985).
  75. L. Dolan and R. Jackiw, Symmetry behavior at finite temperature, Phys. Rev. D 9, 3320 (1974).
  76. A. Florio and S. Murciano, Entanglement asymmetry in gauge theories: Chiral anomaly in the finite temperature massless Schwinger model, Phys. Rev. D 113, L091901 (2026).
  77. I. C. E. Turcu et al., High field physics and QED experiments at ELI-NP, Rom. Rep. Phys. 68, S145 (2016).
  78. J. W. Yoon, Y. G. Kim, I. W. Choi, J. H. Sung, H. W. Lee, S. K. Lee, and C. H. Nam, Realization of laser intensity over 1023W/  cm2, Optica 8, 630 (2021).
  79. R. Schutzhold, H. Gies, and G. Dunne, Dynamically assisted Schwinger mechanism, Phys. Rev. Lett. 101, 130404 (2008).
  80. S. S. Bulanov, T. Z. Esirkepov, A. G. R. Thomas, J. K. Koga, and S. V. Bulanov, Schwinger limit attainability with extreme power lasers, Phys. Rev. Lett. 105, 220407 (2010).
  81. I. A. Aleksandrov, D. G. Sevostyanov, and V. M. Shabaev, Schwinger particle production: Rapid switch off of the external field versus dynamical assistance, Phys. Rev. D 111, 016010 (2025).
  82. S. P. Kim and D. N. Page, Schwinger pair production via instantons in strong electric fields, Phys. Rev. D 65, 105002 (2002).
  83. J. Li, S.-Y. Zhu, and G. S. Agarwal, Squeezed states of magnons and phonons in cavity magnomechanics, Phys. Rev. A 99, 021801 (2019).
  84. WOST (WithOut SpaceTime) project, https://withoutspacetime.org.
  85. V. S. Popov, Pair production in a variable external field (quasiclassical approximation), Zh. Eksp. Teor. Fiz. 61, 1334 (1971) [Sov. Phys. JETP 34, 709 (1972)].
  86. L. Hackl and E. Bianchi, Bosonic and fermionic Gaussian states from Kähler structures, SciPost Phys. Core 4, 025 (2021).
  87. P. J. Coles, M. Cerezo, and L. Cincio, Strong bound between trace distance and Hilbert-Schmidt distance for low-rank states, Phys. Rev. A 100, 022103 (2019).

Outline

Information

Phys. Rev. D 114, 085011– Published 13 October, 2026
  • Received 4 March 2026
  • Accepted 19 August 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. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation