We examine the late-time evolution of a qubit (or Unruh-De Witt detector) that hovers very near to the event horizon of a Schwarzschild black hole, while interacting with a free quantum scalar field. The calculation is carried out perturbatively in the dimensionless qubit/field coupling g, but rather than computing the qubit excitation rate due to field interactions (as is often done), we instead use Open EFT techniques to compute the late-time evolution to all orders in g2t/rs (while neglecting order g4t/rs effects) where rs = 2GM is the Schwarzschild radius. We show that for qubits sufficiently close to the horizon the late-time evolution takes a simple universal form that depends only on the near-horizon geometry, assuming only that the quantum field is prepared in a Hadamard-type state (such as the Hartle-Hawking or Unruh vacua). When the redshifted energy difference, ω∞, between the two qubit states (as measured by a distant observer looking at the detector) satisfies ω∞rs ≪ 1 this universal evolution becomes Markovian and describes an exponential approach to equilibrium with the Hawking radiation, with the off-diagonal and diagonal components of the qubit density matrix relaxing to equilibrium with different characteristic times, both of order rs/g2.
S.W. Hawking, Black hole explosions, Nature 248 (1974) 30 [INSPIRE].
S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D 15 (1977) 2738 [INSPIRE].
W. Israel, Thermo field dynamics of black holes, Phys. Lett. A 57 (1976) 107 [INSPIRE].
D.W. Sciama, P. Candelas and D. Deutsch, Quantum field theory, horizons and thermodynamics, Adv. Phys. 30 (1981) 327 [INSPIRE].
N.D. Birrell and P.C.W. Davies, Quantum fields in curved space, Cambridge University Press, Cambridge, U.K. (1982) [INSPIRE].
E.B. Davies, Quantum theory of open systems, Academic Press, London, U.K. (1976).
R. Alicki and K. Lendi, Quantum dynamical semigroups and applications, Springer, Berlin, Heidelberg, Germany (1987).
R. Kubo, M. Toda and N. Hashitsume, Statistical physics II: nonequilibrium statistical mechanics, Springer, Berlin, Heidelberg, Germany (1995).
C.W. Gardiner and P. Zoller, Quantum noise: a handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics, Springer, Berlin, Heidelberg, Germany (2000).
U. Weiss, Quantum dissipative systems, World Scientific, Singapore (2000).
H.P. Breuer and F. Petruccione, The theory of open quantum systems, Oxford University Press, Oxford, U.K. (2002).
A. Rivas and S.F. Huelga, Open quantum systems: an introduction, Springer, Berlin, Heidelberg, Germany (2012).
G. Schaller, Open quantum systems far from equilibrium, Springer, Cham, Switzerland (2014).
A.A. Starobinsky, Stochastic de Sitter (inflationary) stage in the early universe, Lect. Notes Phys. 246 (1986) 107 [INSPIRE].
D.S. Salopek and J.R. Bond, Stochastic inflation and nonlinear gravity, Phys. Rev. D 43 (1991) 1005 [INSPIRE].
C.P. Burgess, R. Holman, G. Tasinato and M. Williams, EFT beyond the horizon: stochastic inflation and how primordial quantum fluctuations go classical, JHEP 03 (2015) 090 [arXiv:1408.5002] [INSPIRE].
C.P. Burgess, Introduction to effective field theory (thinking effectively about hierarchies of scale), Cambridge University Press, Cambridge, U.K. (2020).
L.H. Ford, Quantum instability of de Sitter space-time, Phys. Rev. D 31 (1985) 710 [INSPIRE].
L.H. Ford and A. Vilenkin, Global symmetry breaking in two-dimensional flat space-time and in de Sitter space-time, Phys. Rev. D 33 (1986) 2833 [INSPIRE].
I. Antoniadis, J. Iliopoulos and T.N. Tomaras, Quantum instability of de Sitter space, Phys. Rev. Lett. 56 (1986) 1319 [INSPIRE].
V. Muller, H.J. Schmidt and A.A. Starobinsky, The stability of the de Sitter space-time in fourth order gravity, Phys. Lett. B 202 (1988) 198 [INSPIRE].
I. Antoniadis and E. Mottola, Graviton fluctuations in de Sitter space, J. Math. Phys. 32 (1991) 1037 [INSPIRE].
M. Sasaki, H. Suzuki, K. Yamamoto and J. Yokoyama, Superexpansionary divergence: breakdown of perturbative quantum field theory in space-time with accelerated expansion, Class. Quant. Grav. 10 (1993) L55 [INSPIRE].
C.T. Byrnes, M. Gerstenlauer, A. Hebecker, S. Nurmi and G. Tasinato, Inflationary infrared divergences: geometry of the reheating surface versus δN formalism, JCAP 08 (2010) 006 [arXiv:1005.3307] [INSPIRE].
S.W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14 (1976) 2460 [INSPIRE].
W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870 [INSPIRE].
B.S. DeWitt, Quantum gravity: the new synthesis in General relativity, an Einstein centenary survey, S.W. Hawking and W. Israel eds., Cambridge University Press, Cambridge, U.K. (1979).
J. Hadamard, Lectures on Cauchy’s problem in linear partial differential equations, Yale University Press, New Haven, CT, U.S.A. (1923).
B.S. DeWitt and R.W. Brehme, Radiation damping in a gravitational field, Annals Phys. 9 (1960) 220 [INSPIRE].
S.A. Fulling, M. Sweeny and R.M. Wald, Singularity structure of the two point function in quantum field theory in curved space-time, Commun. Math. Phys. 63 (1978) 257 [INSPIRE].
K. Fredenhagen and R. Haag, On the derivation of Hawking radiation associated with the formation of a black hole, Commun. Math. Phys. 127 (1990) 273 [INSPIRE].
P. Candelas, Vacuum polarization in Schwarzschild space-time, Phys. Rev. D 21 (1980) 2185 [INSPIRE].
K.K. Ng, L. Hodgkinson, J. Louko, R.B. Mann and E. Martin-Martinez, 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].
K.K. Ng, R.B. Mann and E. Martin-Martinez, Over the horizon: distinguishing the Schwarzschild spacetime and the RP3 spacetime using an Unruh-DeWitt detector, Phys. Rev. D 96 (2017) 085004 [arXiv:1706.08978] [INSPIRE].
V.A. Emelyanov, Quantum vacuum near non-rotating compact objects, Class. Quant. Grav. 35 (2018) 155006 [INSPIRE].
S. Nakajima, On quantum theory of transport phenomena, Prog. Theor. Phys. 21 (1959) 659.
R. Zwanzig, Ensemble method in the theory of irreversibility, J. Chem. Phys. 33 (1960) 1338.
G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48 (1976) 119 [INSPIRE].
V. Gorini, A. Frigerio, M. Verri, A. Kossakowski and E.C.G. Sudarshan, Properties of quantum Markovian master equations, Rept. Math. Phys. 13 (1978) 149 [INSPIRE].
H.W. Yu, J. Zhang, H.-W. Yu and J.-L. Zhang, Understanding Hawking radiation in the framework of open quantum systems, Phys. Rev. D 77 (2008) 024031 [Erratum ibid. 77 (2008) 029904] [arXiv:0806.3602] [INSPIRE].
C. Singha, Remarks on distinguishability of Schwarzschild spacetime and thermal Minkowski spacetime using resonance Casimir-Polder interaction, Mod. Phys. Lett. A 35 (2019) 1950356 [arXiv:1808.07041] [INSPIRE].
S. Weinberg, Phenomenological Lagrangians, Physica A 96 (1979) 327 [INSPIRE].
J. Donoghue, Quantum gravity as a low energy effective field theory, Scholarpedia 12 (2017) 32997.
J.L. Synge, Relativity: the general theory, North-Holland, Amsterdam, The Netherlands (1960).
D.N. Page, Thermal stress tensors in static Einstein spaces, Phys. Rev. D 25 (1982) 1499 [INSPIRE].
P. Candelas and K.W. Howard, Vacuum 〈ϕ2〉 in Schwarzschild space-time, Phys. Rev. D 29 (1984) 1618 [INSPIRE].
D.G. Boulware, Quantum field theory in Schwarzschild and Rindler spaces, Phys. Rev. D 11 (1975) 1404 [INSPIRE].
J.B. Hartle and S.W. Hawking, Path integral derivation of black hole radiance, Phys. Rev. D 13 (1976) 2188 [INSPIRE].
B.S. DeWitt, Quantum field theory in curved space-time, Phys. Rept. 19 (1975) 295 [INSPIRE].
S.M. Christensen and S.A. Fulling, Trace anomalies and the Hawking effect, Phys. Rev. D 15 (1977) 2088 [INSPIRE].
B.S. Kay and R.M. Wald, Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on space-times with a bifurcate Killing horizon, Phys. Rept. 207 (1991) 49 [INSPIRE].
R.M. Wald, Quantum field theory in curved space-time and black hole thermodynamics, University of Chicago Press, Chicago, IL, U.S.A. (1994).
M.J. Radzikowski, Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Commun. Math. Phys. 179 (1996) 529 [INSPIRE].
M.J. Radzikowski, A local to global singularity theorem for quantum field theory on curved space-time, Commun. Math. Phys. 180 (1996) 1 [INSPIRE].
T.-P. Hack and V. Moretti, On the stress-energy tensor of quantum fields in curved spacetimes — comparison of different regularization schemes and symmetry of the Hadamard/Seeley-DeWitt coefficients, J. Phys. A 45 (2012) 374019 [arXiv:1202.5107] [INSPIRE].
C. Dappiaggi, V. Moretti and N. Pinamonti, Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime, Adv. Theor. Math. Phys. 15 (2011) 355 [arXiv:0907.1034] [INSPIRE].
F. Olver, D. Lozier, R. Boisvert and C. Clark, NIST handbook of mathematical function, Cambridge University Press, Cambridge, U.K. (2010).
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Kaplanek, G., Burgess, C.P. Qubits on the horizon: decoherence and thermalization near black holes. J. High Energ. Phys. 2021, 98 (2021). https://doi.org/10.1007/JHEP01(2021)098
- Effective Field Theories
- Black Holes
- Renormalization Group
- Renormalization Regularization and Renormalons