Spontaneous Radiation of a Two-Level System Confined in a Reflective Spherical Shell Quantum Dot

Spontaneous Radiation of a Two-Level System in a Spherical Shell
  • F. N. LimaEmail author
  • R. P. A. Lima
  • M. L. Lyra
Atomic Physics


Using a first-order time-dependent perturbation theory, we calculate the spontaneous emission rate of a two-level system trapped between perfectly reflecting concentric spheres. The emitter is represented by a two-level monopole coupled to a Hermitian massless scalar field satisfying Dirichlet boundary conditions in such quantum-confined low-dimensional structure. We obtained the appropriate Green’s function evaluated in worldline of the atom which incorporates contributions from an infinite set of variable image charges. We provide an analytical expression for the decay rate to investigate the radiation process of the trapped atomic system. We perform a broad analysis of the dependence of the decay rate for different relations between the radii of spheres and the emitted radiation energy. We unveil regimes of strong suppression of the spontaneous emission rate as well as the development of irregular oscillations as a function of the quantum of emitted energy.


Spontaneous radiation Cavity quantum electrodynamics Perturbation theory Scalar field theory 


Funding Information

This study is partially and financiall supported by the Brazilian research agencies CNPq and CAPES, as well as from the Alagoas State research agency FAPEAL. FNL was supported by the program CAPES/DINTER/IFPI/UFAL/23038.000902/2016-90.


  1. 1.
    E. Betzig, R.J. Chichester, Single molecules observed by near-field scanning optical microscopy, Science (5138). (1422)
  2. 2.
    T.H. Taminiau, F.D. Stefani, F.B. Segerink, N.F. van Hulst, Optical antennas direct single-molecule emission, Nat. Photonics (4), 234.
  3. 3.
    S. Kühn, U. Håkanson, L. Rogobete, V. Sandoghdar, Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna, Phys. Rev. Lett. (1), 017402.
  4. 4.
    M. Khajavikhan, A. Simic, M. Katz, J.H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, Y. Fainman, Thresholdless nanoscale coaxial lasers, Nature (7384), 204.
  5. 5.
    M. Frimmer, A.F. Koenderink, Spontaneous emission control in a tunable hybrid photonic system, Phys. Rev. Lett. (21), 217405.
  6. 6.
    E.M. Purcell, Spontaneous emission probabilities at radio frequencies, Phys. Rev. (11-12), 681.
  7. 7.
    J. Zhang, M. Wubs, P. Ginzburg, G. Wurtz, A.V. Zayats, Transformation quantum optics: designing spontaneous emission using coordinate transformations. J. Opt. 18, 044029 (2016). ADSCrossRefGoogle Scholar
  8. 8.
    P. Stehle, Atomic radiation in a cavity, Phys. Rev. A (1), 102.
  9. 9.
    G. Barton, Quantum electrodynamics of spinless particles between conducting plates, Proceedings of the Royal Society A: Mathematical, Phys. Eng. Sci. (1541), 251.
  10. 10.
    M.R. Philpott, Fluorescence from molecules between mirrors, Chem. Phys. Lett. (3), 435.
  11. 11.
    P. Milonni, P. Knight, Spontaneous emission between mirrors, Opt. Commun. (2), 119.
  12. 12.
    P. Goy, J.M. Raimond, M. Gross, S. Haroche, Observation of cavity-enhanced single-atom spontaneous emission, Phys. Rev. Lett. (24), 1903.
  13. 13.
    R.G. Hulet, E.S. Hilfer, D. Kleppner, Inhibited spontaneous emission by a Rydberg atom, Phys. Rev. Lett. (20), 2137.
  14. 14.
    W. Jhe, A. Anderson, E. A. Hinds, D. Meschede, L. Moi, S. Haroche, Suppression of spontaneous decay at optical frequencies: test of vacuum-field anisotropy in confined space, Phys. Rev. Lett. (14), 1497.
  15. 15.
    H.M. França, T.W. Marshall, E. Santos, Spontaneous emission in confined space according to stochastic electrodynamics, Phys. Rev. A (9), 6436.
  16. 16.
    L.H. Ford, N.F. Svaiter, M.L. Lyra, Radiative properties of a two-level system in the presence of mirrors. Phys. Rev. A. 49(2), 1378 (1994). ADSCrossRefGoogle Scholar
  17. 17.
    P.S. Davids, P.B. Lerner, Suppression of atomic radiation in a cylindrical nanocavity, Zeitschrift fr Physik D Atoms, Mol. Clusters (3), 203.
  18. 18.
    M. Kauranen, Y. Van Rompaey, J.J. Maki, A. Persoons, Nonvanishing field between a dipole oscillator and a reflecting boundary during suppression of dipole radiation, Phys. Rev. Lett. (5), 952.
  19. 19.
    M.J.A. de Dood, L.H. Slooff, A. Polman, A. Moroz, A. van Blaaderen, Modified spontaneous emission in erbium-doped SiO[sub 2] spherical colloids, Appl. Phys. Lett. (22), 3585.
  20. 20.
    H.T. Dung, L. Knöll, D.G. Welsch, Decay of an excited atom near an absorbing microsphere, Phys. Rev. A (1), 013804.
  21. 21.
    K.J. Vahala, Optical microcavities, Nature (6950), 839.
  22. 22.
    V.V. Klimov, M. Ducloy, V.S. Letokhov, Spontaneous emission rate and level shift of an atom inside a dielectric microsphere, J. Mod. Opt. (3), 549.
  23. 23.
    A. Moroz, Spectroscopic properties of a two-level atom interacting with a complex spherical nanoshell, Chem. Phys. (1), 1.
  24. 24.
    R. Carminati, J.J. Greffet, C. Henkel, J. Vigoureux, Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle, Opt. Commun. (2), 368.
  25. 25.
    H. Walther, B.T.H. Varcoe, B.G. Englert, T. Becker, Cavity quantum electrodynamics, Rep. Prog. Phys. (5), 1325.
  26. 26.
    V. Yannopapas, N.V. Vitanov, Spontaneous emission of a two-level atom placed within clusters of metallic nanoparticles, J. Phys. Condens. Matter (9), 096210.
  27. 27.
    K.K. Pukhov, T.T. Basiev, Y.V. Orlovskii, Spontaneous emission in dielectric nanoparticles, JETP Lett. (1), 12.
  28. 28.
    L. Ford, T.A. Roman, Effects of vacuum fluctuation suppression on atomic decay rates, Ann. Phys. (8), 2294.
  29. 29.
    F.A. Inam, T. Gaebel, C. Bradac, L. Stewart, M.J. Withford, J.M. Dawes, J.R. Rabeau, M.J. Steel, Modification of spontaneous emission from nanodiamond colour centres on a structured surface, New J. Phys. (7), 073012.
  30. 30.
    C. Sauvan, J.P. Hugonin, I.S. Maksymov, P. Lalanne, Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators, Phys. Rev. Lett. (23), 237401.
  31. 31.
    S. Haroche, Controlling photons in a box and exploring the quantum to classical boundary, Ann. Phys. (10-11), 753.
  32. 32.
    Z. Mohammadi, F. Kheirandish, Energy-level shifts and the decay rate of an atom in the presence of a conducting wedge, Phys. Rev. A (6), 062118.
  33. 33.
    A. Bienfait, J.J. Pla, Y. Kubo, X. Zhou, M. Stern, C.C. Lo, C.D. Weis, T. Schenkel, D. Vion, D. Esteve, J.J.L. Morton, P. Bertet, Controlling spin relaxation with a cavity, Nature (7592), 74.
  34. 34.
    H. Xiong, M. Scully, M. Zubairy, Correlated spontaneous emission laser as an entanglement amplifier, Phys. Rev. Lett. (2), 023601.
  35. 35.
    E. Arias, J.G. Dueñas, G. Menezes, N.F. Svaiter, Boundary effects on radiative processes of two entangled atoms, J. High Energy Phys. (7), 147.
  36. 36.
    Y. Yang, J. Hu, H. Yu, Entanglement dynamics for uniformly accelerated two-level atoms coupled with electromagnetic vacuum fluctuations, Phys. Rev. A (3), 032337.
  37. 37.
    G. Menezes, N.F. Svaiter, Radiative processes of uniformly accelerated entangled atoms, Phys. Rev. A (5), 052117.
  38. 38.
    R.P.A. Lima, F.N. Lima, M.L. Lyra, Spontaneous decay of a two-level system close to a perfectly reflecting sphere. Ann. Phys. 372, 162 (2017). ADSCrossRefGoogle Scholar
  39. 39.
    L. Dong, A. Sugunan, J. Hu, S. Zhou, S. Li, S. Popov, M.S. Toprak, A.T. Friberg, M. Muhammed, Photoluminescence from quasi-type-II spherical CdSe-CdS core-shell quantum dots. Appl. Opt. 52(1), 105 (2013). ADSCrossRefGoogle Scholar
  40. 40.
    C. Liao, K. Fan, R. Xu, H. Zhang, C. Lu, Y. Cui, J. Zhang, Laser-annealing-made amplified spontaneous emission of “giant” CdSe/CdS core/shell nanocrystals transferred from bulk-like shell to quantum-confined core. Photon. Res. 3(5), 200 (2015). CrossRefGoogle Scholar
  41. 41.
    M.S.R. Miltão, Casimir energy for a double spherical shell: a global mode sum approach, Phys. Rev. D (6), 065023.
  42. 42.
    W.G. Unruh, Notes on black-hole evaporation, Phys. Rev. D (4), 870.
  43. 43.
    B. DeWitt, in . General relativity: an Einstein centenary survey, ed. by S. Hawking, W. Israel. (Cambridge University Press), chap. Quantum gr, p. 944, (1979)Google Scholar
  44. 44.
    P.C.W. Davies, Z.X. Liu, A.C. Ottewill, Particle detectors in the presence of boundaries, Classical and Quantum Gravity (7), 1041.
  45. 45.
    B.F. Svaiter, N.F. Svaiter, Inertial and noninertial particle detectors and vacuum fluctuations, Phys. Rev. D (12), 5267.
  46. 46.
    B.F. Svaiter, N.F. Svaiter, Quantum processes: stimulated and spontaneous emission near cosmic strings, Classical and Quantum Gravity (2), 347.
  47. 47.
    F. Benatti, R. Floreanini, Entanglement generation in uniformly accelerating atoms: Reexamination of the Unruh effect, Phys. Rev. A (1), 012112.
  48. 48.
    J. Zhang, H. Yu, Unruh effect and entanglement generation for accelerated atoms near a reflecting boundary, Phys. Rev. D (10), 104014.
  49. 49.
    A.G.S. Landulfo, G.E.A. Matsas, Sudden death of entanglement and teleportation fidelity loss via the Unruh effect, Phys. Rev. A (3), 032315.
  50. 50.
    J. Doukas, B. Carson, Entanglement of two qubits in a relativistic orbit, Phys. Rev. A (6), 062320.
  51. 51.
    D.C.M. Ostapchuk, S.Y. Lin, R.B. Mann, B.L. Hu, Entanglement dynamics between inertial and non-uniformly accelerated detectors, J. High Energy Phys. (7), 72.
  52. 52.
    X. Liu, Z. Tian, J. Wang, J. Jing, Radiative process of two entanglement atoms in de Sitter spacetime, Phys. Rev. D (10), 105030.
  53. 53.
    J.T. Chen, H.C. Shieh, J.J. Tsai, J.W. Lee, Equivalence between Trefftz method and method of fundamental solutions for the Green’s function of concentric spheres using the addition theorem and image concept. WIT Transactions on Modelling and Simulation. 49, 23 (2009). MathSciNetCrossRefGoogle Scholar
  54. 54.
    P. Langlois, Causal particle detectors and topology, Ann. Phys. pp. 2027–2070.
  55. 55.
    P.C.W. Davies, A.C. Ottewill, Detection of negative energy: 4-dimensional examples, Phys. Rev. D (1), 104014.
  56. 56.
    N. Megier, D. Chruściński, J. Piilo, W.T. Strunz, Eternal non-Markovianity: from random unitary to Markov chain realisations, Sci. Rep. (1), 6379.
  57. 57.
    D. Moustos, C. Anastopoulos, Non-Markovian time evolution of an accelerated qubit, Phys. Rev. D (2), 025020.
  58. 58.
    G. Menezes, N.F. Svaiter, Vacuum fluctuations and radiation reaction in radiative processes of entangled states, Phys. Rev. A (6), 062131.

Copyright information

© Sociedade Brasileira de Física 2019

Authors and Affiliations

  1. 1.Instituto Federal do PiauíSão Raimundo NonatoBrazil
  2. 2.GFTC and GISC, Instituto de FísicaUniversidade Federal de AlagoasMaceióBrazil
  3. 3.GFTC, Instituto de FísicaUniversidade Federal de AlagoasMaceióBrazil

Personalised recommendations