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 g2t/rs (while neglecting order g4t/rs effects) where rs = 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, ω∞, between the two qubit states (as measured by a distant observer looking at the detector) satisfies ω∞rs ≪ 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 rs/g2.
Highlights
In this paper we explore similar late-time issues for interacting quantum systems moving in gravitational fields
We examine the late-time evolution of a qubit that hovers very near to the event horizon of a Schwarzschild black hole, while interacting with a free quantum scalar field
Eq (1.1) applies well if the quantum field is prepared in either the Hartle-Hawking or Unruh vacua, and is independent of the scalar-field mass in the mass range m rs 1. It has been known for some time that Hadamard behaviour suffices for deriving the steady-state Hawking flux around Schwarzschild black holes [70], and our results extend this conclusion to the approach to equilibrium for quantum probes
Summary
This section sets up the framework — a qubit/field system and the spacetime through which the qubit moves — that is used to perform the calculations to follow. In the absence of couplings (λ = g = 0) the scalar field is quantized in the usual fashion for a static curved space [6]. The scalar field hamiltonian (including self-interactions) is computed in the presence of any spacetime metric of the form ds2 = −f dt2 + γab dxa dxb (2.13). We can assemble everything to identify the total field/qubit hamiltonian, leading to the following sum:. In what follows we compute the implications of this interaction out to second order in g The nature of this perturbation theory depends on the relative size of ω and the O(g2) corrections to the qubit energy levels, and for simplicity we work in the regime where these corrections are much smaller than the qubit’s zeroth-order level splitting, a restriction that eventually leads to the parameter conditions summarized in table 1
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