Qubits on the Horizon: Decoherence and Thermalization near Black Holes

Abstract

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 $g^{2}t/r_s$ (while neglecting order $g^{4}t/r_s$ effects) where $r_s=2GM$ 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, ${\omega }_{\infty }$, between the two qubit states (as measured by a distant observer looking at the detector) satisfies ${\omega }_{\infty }{r}_s\ll 1$ 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 $r_s/g^2$.

24 pages plus appendix, 2 figures v2) now published in JHEP, typos fixed and added subsection on the frame independence of the Markovian limit