Detector's response to coherent Rindler and Minkowski photons

We observe that the transition probability in a static two-level quantum detector interacting with a coherent Rindler photon is different from the same of the Rindler detector which is in interaction with a coherent Minkowski photon. Situation does not change in the response of quantum detector for the classical limit of the photon state. This we investigate in $(1+1)$ and $(3+1)$-spacetime dimensions. Interestingly, the transition probabilities of the ``classical’’ detector in the classical limit of the photon state in $(1+1)$-dimensions, for these two scenarios, appear to be identical when the frequencies of photon mode and detector are taken to be same. However, our obtained detector’s transition probabilities in $(3+1)$-dimensions, which are calculated under the large acceleration condition, do not show such signature. The implications of these observations are discussed as well.

💡 Research Summary

The paper investigates how a static two‑level quantum detector (an Unruh‑deWitt atom) responds when it interacts with a single‑mode coherent field that is either a Rindler photon or a Minkowski photon. The authors treat the field as a mass‑less real scalar (“photon”) and consider its coherent state |ψ_c⟩=D(α_k)|0⟩, where D is the displacement operator and α_k is the complex amplitude. The detector couples via H_int=λ φ̂_k m̂(τ), with λ the coupling constant, φ̂_k the field operator, and m̂(τ) the monopole operator of the atom (frequency Ω). The transition probability to first order in perturbation theory splits into a vacuum part P_vac(Ω) and a coherent‑state part P_α(Ω). The vacuum contribution reproduces the standard Unruh‑DeWitt results, while P_α(Ω) is proportional to α_k^2 ℏ_f and encodes the classical field effect.

Two spacetime dimensions are examined: (1+1) and (3+1). In each case two scenarios are compared: (i) an accelerated detector interacting with a static mirror (so the field is expressed in Minkowski modes) and (ii) a static detector interacting with an accelerated mirror (so the field is expressed in Rindler modes). The analysis proceeds by writing the appropriate mode functions, performing the proper‑time integral, and using the change of variables u=e^{-aτ}, ũ=e^{aτ} (or analogous substitutions) together with the Gamma‑function integral ∫_0^∞ x^ν e^{-ax}dx=Γ(ν+1)/a^{ν+1}. The resulting expressions are summarized in Tables I and II.

In (1+1) dimensions with a reflecting boundary, the vacuum contributions are P_vac(Ω)=2λ^2ℏ_fℏ_d (ω/(aΩ)) sin^2(ωz_0+ϕ_1) e^{2πΩ/a}/(e^{2πΩ/a}−1) for the accelerated detector, and the analogous expression with ω↔Ω for the static detector. The coherent‑state contributions are P_α(Ω)=λ^2α_k^2ℏ_fℏ_d (Ω/(ωa)) sinh(πΩ/a)