Harvesting correlations in Schwarzschild and collapsing shell spacetimes
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Received: July 8, 2020 Accepted: July 28, 2020 Published: August 28, 2020
Harvesting correlations in Schwarzschild and collapsing shell spacetimes
a
Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada b Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada c Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada
E-mail: [email protected], [email protected] 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. Keywords: Black Holes, Field Theories in Lower Dimensions ArXiv ePrint: 2007.02955
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP08(2020)155
JHEP08(2020)155
Erickson Tjoaa,b and Robert B. Manna,c
Contents 1
2 Klein-Gordon field in Schwarzschild spacetime 2.1 Schwarzschild spacetime: geometry 2.2 Schwarzschild spacetime: Klein-Gordon field 2.3 Comment on IR ambiguity and derivative coupling
4 4 6 7
3 Klein-Gordon field in Vaidya spacetime
9
4 Unruh-DeWitt model and entanglement harvesting 4.1 Derivative coupling Unruh-DeWitt model 4.2 The two-point Wightman distributions for derivative coupling 4.3 Comments on switching time and computation of joint density matrix
10 10 14 14
5 Results 5.1 Harvesting entanglement 5.2 Harvesting mutual information 5.3 Communication between detectors:
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