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Harvesting correlations in Schwarzschild and collapsing shell spacetimes

A preprint version of the article is available at arXiv.

Abstract

We study the harvesting of correlations by two Unruh-DeWitt static detectors from the vacuum state of a massless scalar field in a background Vaidya spacetime consisting of a collapsing null shell that forms a Schwarzschild black hole (hereafter Vaidya spacetime for brevity), and we compare the results with those associated with the three preferred vacua (Boulware, Unruh, Hartle-Hawking-Israel vacua) of the eternal Schwarzschild black hole spacetime. To do this we make use of the explicit Wightman functions for a massless scalar field available in (1+1)-dimensional models of the collapsing spacetime and Schwarzschild spacetimes, and the detectors couple to the proper time derivative of the field. First we find that, with respect to the harvesting protocol, the Unruh vacuum agrees very well with the Vaidya vacuum near the horizon even for finite-time interactions. Second, all four vacua have different capacities for creating correlations between the detectors, with the Vaidya vacuum interpolating between the Unruh vacuum near the horizon and the Boulware vacuum far from the horizon. Third, we show that the black hole horizon inhibits any correlations, not just entanglement. Finally, we show that the efficiency of the harvesting protocol depend strongly on the signalling ability of the detectors, which is highly non-trivial in presence of curvature. We provide an asymptotic analysis of the Vaidya vacuum to clarify the relationship between the Boulware/Unruh interpolation and the near/far from horizon and early/late-time limits. We demonstrate a straightforward implementation of numerical contour integration to perform all the calculations.

References

  1. S.J. Summers and R. Werner, The vacuum violates Bell’s inequalities, Phys. Lett. A 110 (1985) 257.

    ADS  MathSciNet  Google Scholar 

  2. S.J. Summers and R. Werner, Bell’s inequalities and quantum field theory. 1. General setting, J. Math. Phys. 28 (1987) 2440 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  3. S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].

  4. A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  5. D. Marolf, The black hole information problem: past, present, and future, Rept. Prog. Phys. 80 (2017) 092001.

    ADS  Google Scholar 

  6. S. Haco, S.W. Hawking, M.J. Perry and A. Strominger, Black hole entropy and soft hair, JHEP 12 (2018) 098 [arXiv:1810.01847] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  7. M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge, U.K. (2009).

    MATH  Google Scholar 

  8. E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys. 91 (2019) 025001.

    ADS  MathSciNet  Google Scholar 

  9. F.G. Brandão and G. Gour, Reversible framework for quantum resource theories, Phys. Rev. Lett. 115 (2015) 070503.

    ADS  MathSciNet  Google Scholar 

  10. A. Valentini, Non-local correlations in quantum electrodynamics, Phys. Lett. A 153 (1991) 321.

    ADS  Google Scholar 

  11. B. Reznik, Entanglement from the vacuum, Found. Phys. 33 (2003) 167 [quant-ph/0212044] [INSPIRE].

  12. B. Reznik, A. Retzker and J. Silman, Violating Bell’s inequalities in the vacuum, Phys. Rev. A 71 (2005) 042104 [quant-ph/0310058] [INSPIRE].

  13. G. Salton, R.B. Mann and N.C. Menicucci, Acceleration-assisted entanglement harvesting and rangefinding, New J. Phys. 17 (2015) 035001 [arXiv:1408.1395] [INSPIRE].

    ADS  Google Scholar 

  14. A. Pozas-Kerstjens and E. Martín-Martínez, Harvesting correlations from the quantum vacuum, Phys. Rev. D 92 (2015) 064042 [arXiv:1506.03081] [INSPIRE].

    ADS  Google Scholar 

  15. G.L. Ver Steeg and N.C. Menicucci, Entangling power of an expanding universe, Phys. Rev. D 79 (2009) 044027 [arXiv:0711.3066] [INSPIRE].

    ADS  Google Scholar 

  16. S. Kukita and Y. Nambu, Harvesting large scale entanglement in de Sitter space with multiple detectors, Entropy 19 (2017) 449 [arXiv:1708.01359] [INSPIRE].

    MATH  Google Scholar 

  17. L.J. Henderson, R.A. Hennigar, R.B. Mann, A.R.H. Smith and J. Zhang, Harvesting entanglement from the black hole vacuum, Class. Quant. Grav. 35 (2018) 21LT02.

    MathSciNet  MATH  Google Scholar 

  18. K.K. Ng, R.B. Mann and E. Martín-Martínez, Unruh-DeWitt detectors and entanglement: the anti-de Sitter space, Phys. Rev. D 98 (2018) 125005 [arXiv:1809.06878] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  19. E. Martín-Martínez, A.R.H. Smith and D.R. Terno, Spacetime structure and vacuum entanglement, Phys. Rev. D 93 (2016) 044001 [arXiv:1507.02688] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  20. L.J. Henderson, R.A. Hennigar, R.B. Mann, A.R.H. Smith and J. Zhang, Entangling detectors in anti-de Sitter space, JHEP 05 (2019) 178 [arXiv:1809.06862] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  21. W. Cong, E. Tjoa and R.B. Mann, Entanglement harvesting with moving mirrors, JHEP 06 (2019) 021 [Erratum ibid. 07 (2019) 051] [arXiv:1810.07359] [INSPIRE].

  22. L.J. Henderson et al., Quantum temporal superposition: the case of QFT, arXiv:2002.06208 [INSPIRE].

  23. J. Foo, S. Onoe and M. Zych, Unruh-DeWitt detectors in quantum superpositions of trajectories, arXiv:2003.12774 [INSPIRE].

  24. J. Foo, S. Onoe, R.B. Mann and M. Zych, Thermality, causality and the quantum-controlled Unruh-DeWitt detector, arXiv:2005.03914 [INSPIRE].

  25. S. Olson and T.C. Ralph, Entanglement between the future and past in the quantum vacuum, Phys. Rev. Lett. 106 (2011) 110404 [arXiv:1003.0720] [INSPIRE].

    ADS  Google Scholar 

  26. S. Olson and T.C. Ralph, Extraction of timelike entanglement from the quantum vacuum, Phys. Rev. A 85 (2012) 012306 [arXiv:1101.2565] [INSPIRE].

    ADS  Google Scholar 

  27. C. Sabín, J.J. García-Ripoll, E. Solano and J. Leon, Dynamics of entanglement via propagating microwave photons, Phys. Rev. B 81 (2010) 184501 [arXiv:0912.3459] [INSPIRE].

    ADS  Google Scholar 

  28. C. Sabín, B. Peropadre, M. del Rey and E. Martín-Martínez, Extracting past-future vacuum correlations using circuit QED, Phys. Rev. Lett. 109 (2012) 033602 [arXiv:1202.1230] [INSPIRE].

    ADS  Google Scholar 

  29. E. Martín-Martínez, E.G. Brown, W. Donnelly and A. Kempf, Sustainable entanglement production from a quantum field, Phys. Rev. A 88 (2013) 052310 [arXiv:1309.1090] [INSPIRE].

    ADS  Google Scholar 

  30. J.S. Ardenghi, Entanglement harvesting in double-layer graphene by vacuum fluctuations in a microcavity, Phys. Rev. D 98 (2018) 045006 [arXiv:1808.03990] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  31. C. Bény, C.T. Chubb, T. Farrelly and T.J. Osborne, Energy cost of entanglement extraction in complex quantum systems, Nature Commun. 9 (2018) 3792.

    ADS  Google Scholar 

  32. W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870 [INSPIRE].

    ADS  Google Scholar 

  33. B.S. Dewitt, Quantum gravity: the new synthesis, in General relativity: an Einstein centenary survey, S.W. Hawking and W. Israel eds., (1979), pg. 680.

  34. A. Pozas-Kerstjens and E. Martín-Martínez, Entanglement harvesting from the electromagnetic vacuum with hydrogenlike atoms, Phys. Rev. D 94 (2016) 064074 [arXiv:1605.07180] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  35. A.M. Sachs, R.B. Mann and E. Martín-Martínez, Entanglement harvesting from multiple massless scalar fields and divergences in Unruh-DeWitt detector models, arXiv:1808.05980 [INSPIRE].

  36. B.A. Juárez-Aubry and J. Louko, Onset and decay of the 1 + 1 Hawking-Unruh effect: what the derivative-coupling detector saw, Class. Quant. Grav. 31 (2014) 245007.

    ADS  MathSciNet  MATH  Google Scholar 

  37. L. Hodgkinson and J. Louko, Static, stationary and inertial Unruh-DeWitt detectors on the BTZ black hole, Phys. Rev. D 86 (2012) 064031 [arXiv:1206.2055] [INSPIRE].

    ADS  Google Scholar 

  38. Q. Xu, S.A. Ahmad and A.R.H. Smith, Gravitational waves affect vacuum entanglement, arXiv:2006.11301 [INSPIRE].

  39. B.A. Juárez-Aubry and J. Louko, Quantum fields during black hole formation: how good an approximation is the Unruh state?, JHEP 05 (2018) 140 [arXiv:1804.01228] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  40. L. Hodgkinson, Particle detectors in curved spacetime quantum field theory, Ph.D. thesis, University of Nottingham, Nottingham, U.K. (2013) [arXiv:1309.7281] [INSPIRE].

  41. NIST digital library of mathematical functions, release 1.0.27, http://dlmf.nist.gov/, 15 June 2020.

  42. N. Birrell, N. Birrell and P. Davies, Quantum fields in curved space, Cambridge University Press, Cambridge, U.K. (1984).

    MATH  Google Scholar 

  43. E. Martín-Martínez and J. Louko, Particle detectors and the zero mode of a quantum field, Phys. Rev. D 90 (2014) 024015 [arXiv:1404.5621] [INSPIRE].

    ADS  Google Scholar 

  44. D. Marolf and A.C. Wall, State-dependent divergences in the entanglement entropy, JHEP 10 (2016) 109 [arXiv:1607.01246] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  45. E. Martín-Martínez, T.R. Perche and B. de S.L. Torres, General relativistic quantum optics: finite-size particle detector models in curved spacetimes, Phys. Rev. D 101 (2020) 045017 [arXiv:2001.10010] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  46. E. Martín-Martínez, T.R. Perche and B.d.S.L. Torres, Broken covariance of particle detector models in relativistic quantum information, arXiv:2006.12514 [INSPIRE].

  47. W.K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80 (1998) 2245 [quant-ph/9709029] [INSPIRE].

  48. G. Vidal and R.F. Werner, Computable measure of entanglement, Phys. Rev. A 65 (2002) 032314 [quant-ph/0102117] [INSPIRE].

  49. P. Simidzija and E. Martín-Martínez, Harvesting correlations from thermal and squeezed coherent states, Phys. Rev. D 98 (2018) 085007 [arXiv:1809.05547] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  50. H. Ollivier and W.H. Zurek, Introducing quantum discord, Phys. Rev. Lett. 88 (2001) 017901 [quant-ph/0105072] [INSPIRE].

  51. L. Henderson and V. Vedral, Classical, quantum and total correlations, J. Phys. A 34 (2001) 6899.

    ADS  MathSciNet  MATH  Google Scholar 

  52. W. Cong, C. Qian, M.R.R. Good and R.B. Mann, Effects of horizons on entanglement harvesting, arXiv:2006.01720 [INSPIRE].

  53. K.K. Ng, L. Hodgkinson, J. Louko, R.B. Mann and E. Martín-Martínez, Unruh-DeWitt detector response along static and circular geodesic trajectories for Schwarzschild-AdS black holes, Phys. Rev. D 90 (2014) 064003 [arXiv:1406.2688] [INSPIRE].

    ADS  Google Scholar 

  54. R.H. Jonsson, D.Q. Aruquipa, M. Casals, A. Kempf and E. Martín-Martínez, Communication through quantum fields near a black hole, Phys. Rev. D 101 (2020) 125005 [arXiv:2002.05482] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  55. J. Louko and A. Satz, How often does the Unruh-DeWitt detector click? Regularisation by a spatial profile, Class. Quant. Grav. 23 (2006) 6321 [gr-qc/0606067] [INSPIRE].

  56. J. Louko and A. Satz, Transition rate of the Unruh-DeWitt detector in curved spacetime, Class. Quant. Grav. 25 (2008) 055012 [arXiv:0710.5671] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  57. J. Daicic and N. Frankel, Relativistic spin-one boson plasma, Prog. Theor. Phys. 88 (1992) 1.

    ADS  Google Scholar 

  58. E. Martín-Martínez, Causality issues of particle detector models in QFT and quantum optics, Phys. Rev. D 92 (2015) 104019 [arXiv:1509.07864] [INSPIRE].

    ADS  Google Scholar 

  59. R. Wald, General relativity, University of Chicago Press, Chicago, IL, U.S.A. (2010).

    MATH  Google Scholar 

  60. H.A. Weldon, Thermal Green functions in coordinate space for massless particles of any spin, Phys. Rev. D 62 (2000) 056010 [hep-ph/0007138] [INSPIRE].

  61. R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Jpn. 12 (1957) 570.

    ADS  MathSciNet  Google Scholar 

  62. P.C. Martin and J.S. Schwinger, Theory of many particle systems. I, Phys. Rev. 115 (1959) 1342 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  63. R.C. Tolman, On the weight of heat and thermal equilibrium in general relativity, Phys. Rev. 35 (1930) 904 [INSPIRE].

    ADS  MATH  Google Scholar 

  64. R. Tolman and P. Ehrenfest, Temperature equilibrium in a static gravitational field, Phys. Rev. 36 (1930) 1791 [INSPIRE].

    ADS  Google Scholar 

  65. R. Carballo-Rubio, L.J. Garay, E. Martín-Martínez and J. De Ramón, Unruh effect without thermality, Phys. Rev. Lett. 123 (2019) 041601 [arXiv:1804.00685] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  66. W. Brenna, R.B. Mann and E. Martín-Martínez, Anti-Unruh phenomena, Phys. Lett. B 757 (2016) 307.

    ADS  Google Scholar 

  67. L.J. Garay, E. Martín-Martínez and J. de Ramón, Thermalization of particle detectors: the Unruh effect and its reverse, Phys. Rev. D 94 (2016) 104048 [arXiv:1607.05287] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  68. L.J. Henderson, R.A. Hennigar, R.B. Mann, A.R.H. Smith and J. Zhang, The BTZ black hole exhibits anti-Hawking phenomena, arXiv:1911.02977 [INSPIRE].

  69. G. Compère, J. Long and M. Riegler, Invariance of Unruh and Hawking radiation under matter-induced supertranslations, JHEP 05 (2019) 053 [arXiv:1903.01812] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  70. S. Kolekar and J. Louko, Gravitational memory for uniformly accelerated observers, Phys. Rev. D 96 (2017) 024054 [arXiv:1703.10619] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  71. S. Kolekar and J. Louko, Quantum memory for Rindler supertranslations, Phys. Rev. D 97 (2018) 085012 [arXiv:1709.07355] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  72. R.H. Jonsson, D.Q. Aruquipa, M. Casals, A. Kempf and E. Martín-Martínez, Communication through quantum fields near a black hole, Phys. Rev. D 101 (2020) 125005 [arXiv:2002.05482] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  73. E. Tjoa, Aspects of quantum field theory with boundary conditions, MSc. thesis, University of Waterloo, Waterloo, ON, Canada (2019).

    Google Scholar 

  74. V. Mukhanov and S. Winitzki, Introduction to quantum effects in gravity, Cambridge University Press, Cambridge, U.K. (2007).

    MATH  Google Scholar 

  75. L. Sriramkumar and T. Padmanabhan, Response of finite time particle detectors in noninertial frames and curved space-time, Class. Quant. Grav. 13 (1996) 2061 [gr-qc/9408037] [INSPIRE].

  76. Wolfram research inc., Mathematica, version 12.0, U.S.A. (2019).

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Tjoa, E., Mann, R.B. Harvesting correlations in Schwarzschild and collapsing shell spacetimes. J. High Energ. Phys. 2020, 155 (2020). https://doi.org/10.1007/JHEP08(2020)155

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Keywords

  • Black Holes
  • Field Theories in Lower Dimensions