  Effective theories describing black hole exteriors contain many open-system
features due to the large number of gapless degrees of freedom that lie beyond
reach across the horizon. A simple solvable Caldeira-Leggett type model of a
quantum field interacting within a small area with many unmeasured thermal
degrees of freedom was recently proposed in arXiv:2106.09854 to provide a toy
model of this kind of dynamics against which more complete black hole
calculations might be compared. We here compute the response of a simple
Unruh-DeWitt detector (or qubit) interacting with a massless quantum field
$\phi$ coupled to such a hotspot. Our treatment differs from traditional
treatments of Unruh-DeWitt detectors by using Open-EFT tools to reliably
calculate the qubit's late-time behaviour. We use these tools to determine the
efficiency with which the qubit thermalizes as a function of its proximity to
the hotspot. We identify a Markovian regime in which thermalization does occur,
though only for qubits closer to the hotspot than a characteristic distance
scale set by the $\phi$-hotspot coupling. We compute the thermalization time,
and find that it varies inversely with the $\phi$-qubit coupling strength in
the standard way.
