Quantum coherence of continuous variables in the black hole quantum atmosphere

TL;DR

The paper investigates the origin of Hawking quanta through the quantum atmosphere by modeling a scalar field near a Schwarzschild black hole as a two-mode Gaussian state and computing CV coherence via Gaussian quantum information methods. It adopts second quantization, Kruskal-like Bogoliubov transformations, and covariance-matrix-based relative-entropy coherence to connect coherence dynamics to the local Hartle--Hawking temperature $T_{HH}$ and atmosphere radius $r_A$. The results show a non-monotonic distance dependence of coherence with a pronounced feature near $r \approx 1.43\,r_h$—the atmosphere's effective radius—where the local temperature peaks, and reveal that higher squeezing $s$ and lower field frequency $w$ enhance detectable coherence while some quantum correlations persist in the quantum atmosphere. The findings bridge black-hole thermodynamics and relativistic quantum information, clarifying the quantum-atmosphere's role in Hawking radiation and suggesting viable CV quantum-information tasks in the near-horizon region.

Abstract

Recently, the concept of quantum atmosphere has been introduced as a potential origin of Hawking quanta. This study investigates the properties of quantum fields by exploring the quantum coherence of a two-mode Gaussian state near a black hole, where Hawking quanta originate from the quantum atmosphere region. It is demonstrated that both physically accessible and inaccessible quantum coherence for continuous variable quantum states distinctly exhibit hallmark features of the quantum atmosphere. Specifically, the quantum coherence for these states varies continuously with changes in the normalized distance; it undergoes rapid decreases (or increases) just outside the event horizon before gradually stabilizing through subsequent increases (or decreases). This behavior contrasts with the behaviors of quantum coherence where originates solely from the black hole's event horizon. The quantum features of the fields distinctly reflect characteristics attributable to the quantum atmosphere, thereby deepening our understanding of the origins of Hawking radiation. We also find that the continuous variable coherence is highly dependent on both the squeezing parameter and field frequency of the prepared state; therefore, appropriately adjusting these values can enhance our ability to detect features within the quantum atmosphere. It is noteworthy to observe that quantum features of fields do not entirely dissipate in the quantum atmosphere region, indicating that tasks related to quantum information processing can still be executed there.

Figures (2)

• Figure 1: (a) Quantum coherence between initially correlated modes as a function of the initial squeezing parameter s and the normalized distance ($r/r_{h}$). The frequency is set to w = 1, and the Hartle--Hawking parameter is set to $D_{HH}=25$. (b) The coherence varies with the frequency w and the normalized distance ($r/r_{h}$). The initial squeezing parameter is set to s = 1, and the Hartle--Hawking parameter is set to $D_{HH}=25$. (c) The physically accessible coherence for different Hartle--Hawking parameters ($D_{HH}$). The remaining parameters are fixed at s = w = 1.
• Figure 2: (a) Density plot of quantum coherence between Bob and anti-Bob as a function of the initial squeezing parameter s and the normalized distance ($r/r_{h}$). The frequency w is fixed at 1 and the Hartle--Hawking parameter $D_{HH}$ is fixed at 25. (b) Plot of physically inaccessible quantum coherence as a function of the frequency w and the normalized distance ($r/r_{h}$). The initial squeezing parameter s is fixed at 1, and the Hartle--Hawking parameter $D_{HH}$ is fixed at 25. (c) Two-dimensional plots showing the quantum coherence of the initially uncorrelated modes as a function of the normalized distance ($r/r_{h}$) for various values of the Hartle--Hawking parameter $D_{HH}$ with the initial squeezing parameter s and the frequency w fixed at 1.