Timelike transitions in an atom by a mirror in light cone and Kruskal-Szekeres regions: a status of quantum equivalence

TL;DR

The paper tests quantum-level equivalence between classically equivalent timelike motions by analyzing atomic excitations and de-excitations of a two-level system coupled to field modes in Minkowski light-cone regions and inside Schwarzschild black holes. Using an Unruh–DeWitt detector framework across 1+1 and 3+1 dimensions, it shows that excitation probabilities acquire thermal factors and depend periodically on atom-mirror separation; equivalence between mirror/atom configurations holds in 1+1 Minkowski light-cone setups for equal atomic and field frequencies but generally fails in 3+1 dimensions and BH interiors. The study also computes excitation-to-de-excitation ratios (EDRs), finding that EDRs can exhibit equivalence only in near-horizon regimes or under specific conditions, and are not purely Boltzmann across backgrounds. These results illuminate the limits of the quantum-epoch equivalence principle and highlight EDR as a finer diagnostic for quantum-level equivalence, with potential implications for experimental analogues and tests of quantum field theory in curved spacetime.

Abstract

We investigate the timelike transitions in a two-level atom in the presence of an infinite reflecting mirror in the future-past light cone regions of a Minkowski spacetime, as well as in the region interior of a $(1+1)$ dimensional Schwarzschild black hole. In particular, when considering the light cone regions, two specific scenarios are dealt with -- $(i)$ a mirror is static in Minkowski spacetime while the atom is attached to a frame confined inside the future light cone region, $(ii)$ an atom is static in Minkowski spacetime, and the mirror is confined inside the future light cone region. For both situations, the atom is interacting with field modes defined in the mirror's frame. Analogous configurations are considered in the black hole spacetime: in one case, the mirror carries field modes represented by the Kruskal time, while the atom follows the Schwarzschild time defined inside the black hole; in the other case, the situations are reversed. The analyses, depending on the frame of the atom, are respectively done within the light cone, Minkowski, Schwarzschild, and Kruskal time-interaction pictures. In all of these scenarios, we observe that the excitation probabilities contain a thermal factor and depend periodically on the separation between the atom and the mirror. At the level of transition probabilities, the aforesaid two scenarios in $(1+1)$ dimensional Minkowski-light cone regions appear to be the same for the equal field and atomic frequencies. However, the same is not true when we consider the $(3+1)$ dimensional Minkowski-light cone or the Schwarzschild interior regions. We also estimate the de-excitation probabilities and encounter similar situations. However, we observe that the excitation to de-excitation ratios (EDRs) corresponding to analogous scenarios are equal for equal atomic and field frequencies.

Figures (3)

• Figure 1: The above diagram depicts the different areas that have been or are being studied in regard to understanding the quantum equivalence between different scenarios with atom-mirror set-ups. Our present work comprises set-ups prepared in the Minkowski-Light cone regions, and also compares the results with set-ups inside the event horizon of a Schwarzschild black hole.
• Figure 2: The above figures provide schematic diagrams depicting different regions in the Minkowski spacetime. The region $RRW$ denotes the right Rindler wedge, while $LRW$ denotes the left Rindler wedge. The top and bottom shaded regions, respectively, denote the future and the past light cone regions ($FLC$ and $PLC$). The above two figures correspond to two different atom and mirror positions. On the left, we have depicted a scenario where the Mirror is kept fixed at $z=z_{0}$, and the atom is in the region $FLC$ at $z=0$. On the right, the mirror is at $z=0$, and the atom is kept fixed at $z=z_{0}$.
• Figure 3: The above figures provide schematic diagrams depicting the Kruskal–Szekeres representation of the Schwarzschild black hole spacetime. The $45^{\circ}$ grey lines denote the event horizon $(r=r_{H})$. The $r=0$ singularity is depicted by curved lines. The shaded regions correspond to the interior of the black hole. The above two figures correspond to two different set-ups for the atom-mirror positions. For instance, on the left, the mirror is free-falling, described by a fixed KS space coordinate $X=X_{0}$. In contrast, on the right, the atom is free-falling, described by $X=X_{0}$.