# Radiative Processes of Two Accelerated Entangled Atoms Near Boundaries

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Response of The Two-Atom System Coupled to The Vacuum Massless Scalar Field

## 3. Rates of Transition of a Uniformly Accelerated Two-Atom System Near a Perfectly Reflecting Boundary

## 4. Rates of Transition of The Two-Atom System in Uniform Acceleration in A Cavity

## 5. Summary

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**(

**a**) The contour for the integrations with $\Delta E<0$; (

**b**) the contour for the integrations with $\Delta E>0$.

## References

- Fulling, S.A. Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time. Phys. Rev. D
**1973**, 7, 2850. [Google Scholar] [CrossRef] - Davies, P.C.W. Scalar production in Schwarzschild and Rindler metrics. J. Phys. A
**1975**, 8, 609. [Google Scholar] [CrossRef] - Unruh, W.G. Notes on black-hole evaporation. Phys. Rev. D
**1976**, 14, 870. [Google Scholar] [CrossRef] [Green Version] - D, B.N.; Davies, P.C.W. Quantum Fields in Curved Space; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Audretsch, J.; Müller, R. Spontaneous excitation of an accelerated atom: The contributions of vacuum fluctuations and radiation reaction. Phys. Rev. A
**1994**, 50, 1755. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Audretsch, J.; Müller, R.; Holzmann, M. Generalized Unruh effect and Lamb shift for atoms on arbitrary stationary trajectories. Class. Quant. Grav.
**1995**, 12, 2927. [Google Scholar] [CrossRef] [Green Version] - Yu, H.; Lu, S. Spontaneous excitation of an accelerated atom in a spacetime with a reflecting plane boundary. Phys. Rev. D
**2005**, 72, 064022. [Google Scholar] [CrossRef] [Green Version] - Zhu, Z.; Yu, H.; Lu, S. Spontaneous excitation of an accelerated hydrogen atom coupled with electromagnetic vacuum fluctuations. Phys. Rev. D
**2006**, 73, 107501. [Google Scholar] [CrossRef] [Green Version] - Yu, H.; Zhu, Z. Spontaneous absorption of an accelerated hydrogen atom near a conducting plane in vacuum. Phys. Rev. D
**2006**, 74, 044032. [Google Scholar] [CrossRef] [Green Version] - Lin, S.; Hu, B.L. Accelerated detector-quantum field correlations: From vacuum fluctuations to radiation flux. Phys. Rev. D
**2006**, 73, 124018. [Google Scholar] [CrossRef] [Green Version] - Zhu, Z.; Yu, H. Fulling-Davies-Unruh effect and spontaneous excitation of an accelerated atom interacting with a quantum scalar field. Phys. Lett. B
**2007**, 645, 459. [Google Scholar] [CrossRef] [Green Version] - Zhou, W.; Yu, H. Spontaneous excitation of a uniformly accelerated atom coupled with vacuum Dirac field fluctuations. Phys. Rev. A
**2012**, 86, 033841. [Google Scholar] [CrossRef] [Green Version] - Li, Q.; Yu, H.; Zhou, W. Response of a uniformly accelerated detector to massless Rarita-Schwinger fields in vacuum. Ann. Phys.
**2014**, 348, 144. [Google Scholar] [CrossRef] [Green Version] - Zhou, W. Is the Fulling-Davies-Unruh effect valid for the case of an atom coupled to quantum electromagnetic field? Mod. Phys. Lett. A
**2016**, 31, 1650189. [Google Scholar] [CrossRef] - Noto, A.; Passante, R. van der Waals interaction energy between two atoms moving with uniform acceleration. Phys. Rev. D
**2013**, 88, 025041. [Google Scholar] [CrossRef] [Green Version] - Marino, J.; Noto, A.; Passante, R. Thermal and Nonthermal Signatures of the Unruh Effect in Casimir-Polder Forces. Phys. Rev. Lett.
**2014**, 113, 020403. [Google Scholar] [CrossRef] [Green Version] - Menezes, G.; Svaiter, N.F. Vacuum fluctuations and radiation reaction in radiative processes of entangled states. Phys. Rev. A
**2015**, 92, 062131. [Google Scholar] [CrossRef] [Green Version] - Arias, E.; DueῈas, J.G.; Menezes, G.; Svaiter, N.F. Boundary effects on radiative processes of two entangled atoms. JHEP
**2016**, 1607, 147. [Google Scholar] [CrossRef] [Green Version] - Menezes, G.; Svaiter, N.F. Radiative processes of uniformly accelerated entangled atoms. Phys. Rev. A
**2016**, 93, 052117. [Google Scholar] [CrossRef] [Green Version] - Menezes, G. Radiative processes of two entangled atoms outside a Schwarzschild black hole. Phys. Rev. D
**2016**, 94, 105008. [Google Scholar] [CrossRef] [Green Version] - Rizzuto, L.; Lattuca, M.; Marino, J.; Noto, A.; Spagnolo, S.; Zhou, W.; Passante, R. Nonthermal effects of acceleration in the resonance interaction between two uniformly accelerated atoms. Phys. Rev. A
**2016**, 94, 012121. [Google Scholar] [CrossRef] [Green Version] - Zhou, W.; Passante, R.; Rizzuto, L. Resonance interaction energy between two accelerated identical atoms in a coaccelerated frame and the Unruh effect. Phys. Rev. D
**2016**, 94, 105025. [Google Scholar] [CrossRef] [Green Version] - Tian, Z.; Wang, J.; Jing, J.; Dragan, A. Detecting the Curvature of de Sitter Universe with Two Entangled Atoms. Sci. Rep.
**2016**, 6, 35222. [Google Scholar] [CrossRef] [Green Version] - Liu, X.; Tian, Z.; Wang, J.; Jing, J. Radiative process of two entanglement atoms in de Sitter spacetime. Phys. Rev. D
**2018**, 97, 105030. [Google Scholar] [CrossRef] [Green Version] - Cai, H.; Ren, Z. Radiative processes of two entangled atoms in cosmic string spacetime. Class. Quant. Grav.
**2018**, 35, 025016. [Google Scholar] [CrossRef] - Zhou, W.; Rizzuto, L.; Passante, R. Vacuum fluctuations and radiation reaction contributions to the resonance dipole-dipole interaction between two atoms near a reflecting boundary. Phys. Rev. A
**2018**, 97, 042503. [Google Scholar] [CrossRef] [Green Version] - Zhou, W.; Passante, R.; Rizzuto, L. Resonance Dipole-Dipole Interaction between Two Accelerated Atoms in the Presence of a Reflecting Plane Boundary. Symmetry
**2018**, 10, 185. [Google Scholar] [CrossRef] [Green Version] - Passante, R. Radiative level shifts of an accelerated hydrogen atom and the Unruh effect in quantum electrodynamics. Phys. Rev. A
**1998**, 57, 1590. [Google Scholar] [CrossRef] - Rizzuto, L. Casimir-Polder interaction between an accelerated two-level system and an infinite plate. Phys. Rev. A
**2007**, 76, 062114. [Google Scholar] [CrossRef] - Bennett, C.H.; Wiesner, S.J. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett.
**1992**, 69, 2881. [Google Scholar] [CrossRef] [Green Version] - Bennett, C.H.; Brassard, G.; Popescu, S.; Schumacher, B.; Smolin, J.A.; Wootters, W.K. Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels. Phys. Rev. Lett.
**1996**, 76, 722. [Google Scholar] [CrossRef] [Green Version] - Grinstein, G.; Mukamel, D.; Seidin, R.; Bennett, C.H. Temporally periodic phases and kinetic roughening. Phys. Rev. Lett.
**1993**, 70, 3607. [Google Scholar] [CrossRef] - Benatti, F.; Floreanini, R. Entanglement generation in uniformly accelerating atoms: Reexamination of the Unruh effect. Phys. Rev. A
**2004**, 70, 012112. [Google Scholar] [CrossRef] - Zhang, J.; Yu, H. Unruh effect and entanglement generation for accelerated atoms near a reflecting boundary. Phys. Rev. D
**2007**, 75, 104014. [Google Scholar] [CrossRef] [Green Version] - Lin, S.; Chou, C.; Hu, B.L. Disentanglement of two harmonic oscillators in relativistic motion. Phys. Rev. D
**2008**, 2008 78, 125025. [Google Scholar] [CrossRef] [Green Version] - Landulfo, A.G.S.; Matsas, G.E.A. Sudden death of entanglement and teleportation fidelity loss via the Unruh effect. Phys. Rev. A
**2009**, 80, 032315. [Google Scholar] [CrossRef] [Green Version] - Lin, S.; Hu, B.L. Entanglement creation between two causally disconnected objects. Phys. Rev. D
**2010**, 81, 045019. [Google Scholar] [CrossRef] [Green Version] - Ostapchuk, D.C.M.; Lin, S.; Mann, R.B.; Hu, B.L. Entanglement dynamics between inertial and non-uniformly accelerated detectors. J. High Energy Phys.
**2012**, 7, 72. [Google Scholar] [CrossRef] [Green Version] - Hu, J.; Yu, H. Entanglement dynamics for uniformly accelerated two-level atoms. Phys. Rev. A
**2015**, 91, 012327. [Google Scholar] [CrossRef] [Green Version] - Cheng, S.; Yu, H.; Hu, J. Entanglement dynamics for uniformly accelerated two-level atoms in the presence of a reflecting boundary. Phys. Rev. D
**2018**, 98, 025001. [Google Scholar] [CrossRef] [Green Version] - Yu, T.; Eberly, J.H. Finite-Time Disentanglement Via Spontaneous Emission. Phys. Rev. Lett.
**2004**, 93, 140404. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**The acceleration- and the atom-boundary-separation-dependence of the rates of transition of the two-atom system in the symmetric entangled state and in uniform acceleration near a perfectly reflecting boundary. The atom-boundary separation for both atoms are chosen to be the same, i.e., ${z}_{A}={z}_{B}=z$, and the constant interatomic separation is chosen to be ${\omega}_{0}L=10$. The longitudinal coordinates in (

**a**–

**c**) are in units of $\frac{{\lambda}^{2}{\omega}_{0}}{8\pi}$, while that in (

**d**) is in the unit of $\frac{{\lambda}^{2}{\omega}_{0}}{8\pi}\times {10}^{-10}$. The $a-$axis and the $z-$axis in all the four figures are in units of ${L}^{-1}$.

**Figure 4.**The acceleration- and the relative-location[of the atoms with respect to the cavity]-dependence of the rates of transition of the two-atom system in the symmetric entangled state and uniformly accelerated in a cavity. The constant interatomic separation is chosen to be one half of the width of the cavity, i.e., $L=\frac{D}{2}$, the atomic transition frequency ${\omega}_{0}\sim 3\times {10}^{16}{s}^{-1}$ and the width of the cavity $D=0.1\mathsf{\mu}$m. The longitudinal coordinates in (

**a**–

**c**) are in units of $\frac{{\lambda}^{2}{\omega}_{0}}{8\pi}$, the $a-$axis in (

**b**,

**d**) are in units of ${D}^{-1}$, and the longitudinal coordinate in (

**d**) is in the unit of $\frac{{\lambda}^{2}{\omega}_{0}}{8\pi}\times {10}^{-3}$.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, C.; Zhou, W.
Radiative Processes of Two Accelerated Entangled Atoms Near Boundaries. *Symmetry* **2019**, *11*, 1515.
https://doi.org/10.3390/sym11121515

**AMA Style**

Zhang C, Zhou W.
Radiative Processes of Two Accelerated Entangled Atoms Near Boundaries. *Symmetry*. 2019; 11(12):1515.
https://doi.org/10.3390/sym11121515

**Chicago/Turabian Style**

Zhang, Chi, and Wenting Zhou.
2019. "Radiative Processes of Two Accelerated Entangled Atoms Near Boundaries" *Symmetry* 11, no. 12: 1515.
https://doi.org/10.3390/sym11121515