  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 $r_s/g^2$.
