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 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.

Qubit heating near a hotspot

“…The qubit behaviour we find mirrors similar thermalization behaviour found earlier for qubits in other spactimes with horizons [35][36][37], though with the important difference that unbounded redshift effects near horizons for these other spacetimes generically make thermalization more efficient near the horizon than for a hotspot. Qubit behaviour in the hotspot model is nevertheless both very rich and yet amenable to explicit calculation, and as such provides a useful test of tools that are applied in these other more complicated gravitational settings.…”

“…This agrees with early calculations for Unruh-DeWitt detectors [74], which identified R as the qubit's excitation rate. As we see below, this rate differs from the thermalization rate calculated at late times (these rates also differ for Unruh-DeWitt detectors in nontrivial spacetimes [35][36][37]).…”

“…Our focus is on the late-time thermalization behaviour; a time-scale that varies inversely with the qubit-field coupling, and so lies beyond the reach of naive perturbation theory. Following [35][36][37] we use Open-EFT techniques to access this late-time limit, with the goal of providing a point of comparison for similar calculations in more complicated black-hole and cosmological geometries. We also explore in this simple setting how qubit thermalization depends on an interplay between the strength of its couplings and distance from the hotspot.…”

“…The application of this equation to qubit systems in various environments is studied in detail in [35][36][37] so we here simply quote what the Nakajima-Zwanzig equation gives for (t) in the present instance. The result can be computed explicitly as a series in the coupling λ Q , and when evaluated for t > t 0 to second order in λ Q takes the form…”

Fisher information as a probe of spacetime structure: relativistic quantum metrology in (A)dS

“…We thank Greg Kaplanek for pointing out an additional condition for Markovianity to hold, namely that the detector energy gap needs to be much smaller than the temperature[33][34][35][36].…”