Abstract
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 ref. [23] 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 ϕ 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 ϕ-hotspot coupling. We compute the thermalization time, and find that it varies inversely with the ϕ-qubit coupling strength in the standard way.
Highlights
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 ref. [23] 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 interacting with a massless quantum field φ coupled to such a hotspot
It was with the view to providing one of these benchmarks that reference [23] proposed a solvable Caldeira-Leggett style [24, 25] model consisting of an external massless quantum field φ interacting with many unseen gapless thermal fields in a very small spatial volume
Summary
The hotspot is taken to contain an observable sector, modelled by a single real scalar field, φ(x), that lives in a spatial region, R+, that represents the exterior of the black hole. The degrees of freedom interior to the black hole is modelled by N real massless scalar fields, χa with a = 1, · · · , N , that reside in a different spatial region R− that is disjoint from the region R+ everywhere except for the surface of a small sphere, Sξ, with radius ξ. Where the radius ξ → 0, in which case Sξ reduces to a single point of contact between R+ and R− (which we situate at the origin x = 0 of both R±) In this limit the couplings between φ and χa are captured by an effective action localized at x = 0. Because we solve for the evolution of φ exactly we need not assume that gbe small
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