Abstract

It is often remarked that an explanation of spontaneous emission and the Lamb shift requires quantization of the electromagnetic field. Here these two quantities are derived in a semiclassical formalism by use of second-order perturbation theory. The purpose of the present paper is not to argue the validity of QED but rather to develop a semiclassical approximation to QED that may nonetheless have certain computational advantages over QED. To this end, the vacuum of QED is simulated with a classical zero-point field (ZPF), and as a consequence the resulting theory is entitled semiclassical random electrodynamics (SRED). In the theory, the atom is coupled to the ZPF and to its own radiation-reaction field through an electric dipole interaction. These two interactions add to produce exponential decay of excited states while they cancel each other to prevent spontaneous excitation of the ground state; the Lamb shift appears in the theory as an ac Stark shift induced by the ZPF. The spontaneous decay rate of an excited-state derived in SRED is equal to the Einstein A coefficient for that state, and the Lamb shift agrees with that of nonrelativistic QED. Moreover, SRED is shown to be useful for the numerical simulation of spontaneous decay.

© 1999 Optical Society of America

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References

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  1. P. Meystre and M. Sargent III, Elements of Quantum Optics (Springer-Verlag, Berlin, 1991).
  2. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).
  3. C. P. Slichter, Principles of Magnetic Resonance (Harper & Row, New York, 1963).
  4. We do not consider here the absorber theory of radiation, which posits the existence of advanced radiation fields from distant absorbers as the seat of radiation reaction. For a review of absorber theory see D. T. Pegg, “Absorber theory of radiation,” Rep. Prog. Phys. 38, 1339–1383 (1975).
    [CrossRef]
  5. J. N. Dodd, Atoms and Light: Interactions (Plenum, New York, 1991), Chap. 10.
  6. P. W. Milonni, “Semiclassical and quantum-electrodynamical approaches in nonrelativistic radiation theory,” Phys. Rep. 25, 1–81 (1976).
    [CrossRef]
  7. See, for example, G. W. Series, “A long journey with classical fields,” Phys. Scr. T12, 5–13 (1986); J. N. Dodd, “Spontaneous emission and the vacuum field,” Phys. Scr. T70, 88–93 (1997).
    [CrossRef]
  8. J. R. Ackerhalt, P. L. Knight, and J. H. Eberly, “Radiation reaction and radiative frequency shifts,” Phys. Rev. Lett. 30, 456–460 (1973); I. R. Senitzky, “Radiation-reaction and vacuum-field effects in Heisenberg-picture quantum electrodynamics,” Phys. Rev. Lett. 31, 955–958 (1973); P. W. Milonni, J. R. Ackerhalt, and W. A. Smith, “Interpretation of radiative corrections in spontaneous emission,” Phys. Rev. Lett. PRLTAO 31, 958–960 (1973); J. Dalibard, J. Dupont-Roc, and C. Cohen-Tannoudji, “Vacuum fluctuations and radiative reaction: identification of their respective contributions,” J. Phys. (Paris) JOPQAG 43, 1617–1638 (1982).
    [CrossRef]
  9. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962), Chap. 21; J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 17.
  10. P. L. Knight, “Dynamics of spontaneous emission,” Phys. Scr. T70, 94–100 (1997); M. B. Plenio and P. L. Knight, “The quantum jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998).
    [CrossRef]
  11. A. G. Kofman, G. Kurizki, and B. Sherman, “Spontaneous and induced atomic decay in photonic band structures,” J. Mod. Opt. 41, 353–384 (1994).
    [CrossRef]
  12. P. W. Milonni, The Quantum Vacuum (Academic, Boston, Mass., 1994).
  13. An important exception to this statement occurs in the theory of QED based on self-energy. See A. O. Barut and J. F. Van Huele, “Quantum electrodynamics based on self-energy: Lamb shift and spontaneous emission without field quantization,” Phys. Rev. A 32, 3187–3195 (1985).
    [CrossRef] [PubMed]
  14. It is worth mentioning here that (as was pointed out by Eberly) when a classical Lagrangian for the field–atom system is supplanted by the assumption of stochastic, homogenous solutions, a description of the field–atom interaction is obtained that “...doesn’t differ from QED at all in its second-order predictions of decay rates and level shifts.” See J. H. Eberly, “Unified view of spontaneous emission in several theories of radiation,” in Foundations of Radiation Theory and Quantum Electrodynamics, A. O. Barut, ed. (Plenum, New York, 1980).
  15. T. H. Boyer, “A brief survey of stochastic electrodynamics,” in Foundations of Radiation Theory and Quantum Electrodynamics, A. O. Barut, ed. (Plenum, New York, 1980); L. de la Peña, “Stochastic electrodynamics: its development, present situation and perspectives,” in Stochastic Processes Applied to Physics and Other Related Fields, B. Gomez, S. M. Moore, A. M. Rodriguez-Vargas, and A. Reuda, eds. (World Scientific, Singapore, 1983).
  16. D. K. Ross and W. Moreau, “Stochastic gravity,” Gen. Relativ. Gravit. 27, 845–858 (1995).
    [CrossRef]
  17. L. de la Peña and A. M. Cetto, The Quantum Dice: An Introduction to Stochastic Electrodynamics (Kluwer, Dordrecht, The Netherlands, 1996).
  18. H. M. Franca and T. W. Marshall, “Excited states in stochastic electrodynamics,” Phys. Rev. A 38, 3258–3263 (1988); L. de la Peña and A. M. Cetto, “Quantum phenomena and the zeropoint radiation field,” Found. Phys. 25, 573–604 (1995).
    [CrossRef] [PubMed]
  19. For a discussion of these issues, see G. N. Plass, “Classical electrodynamic equations of motion with radiative reaction,” Rev. Mod. Phys. 33, 37–62 (1961); F. Rohrlich, “The dynamics of a charged sphere and the electron,” Am. J. Phys. 65, 1051–1056 (1997); G. W. Ford and R. F. O’Connell, “Radiative reaction in electrodynamics and the elimination of runaway solutions,” Phys. Lett. A PYLAAG 157, 217–220 (1991).
    [CrossRef]
  20. J. A. Wheeler and R. P. Feynman, “Interaction with the absorber as the mechanism of radiation,” Rev. Mod. Phys. 17, 157–181 (1945).
    [CrossRef]
  21. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, New York, 1977), Vol. 1, pp. 315–328; R. R. Schlicher, W. Becker, J. Bergou, and M. O. Scully, “Interaction Hamiltonian in quantum optics or: p⋅A vs. E⋅r revisited,” in Quantum Electrodynamics and Quantum Optics, A. O. Barut, ed. (Plenum, New York, 1984), pp. 405–441.
  22. Physically, in SRED the missing factor of 2 in Eq. (5b) is to be attributed to an explicit role for the vacuum field in describing an oscillating dipole’s dynamics.
  23. T. H. Boyer, “Random electrodynamics: the theory of classical electrodynamics with classical electromagnetic zero-point radiation,” Phys. Rev. D 11, 790–808 (1975).
    [CrossRef]
  24. T. H. Boyer, “Classical statistical thermodynamics and electromagnetic zero-point radiation,” Phys. Rev. 186, 1304–1318 (1969).
    [CrossRef]
  25. M. Ibison and B. Haisch, “Quantum and classical statistics of the electromagnetic zero-point field,” Phys. Rev. A 54, 2737–2744 (1996).
    [CrossRef] [PubMed]
  26. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynbert, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley, New York, 1989), Chap. I.
  27. Heitler, The Quantum Theory of Radiation (Dover, New York, 1984), Chap. 2.
  28. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), p. 446.
  29. H. A. Bethe, “The electromagnetic shift of energy levels,” Phys. Rev. 72, 339 (1947).
    [CrossRef]
  30. It should at least be mentioned in passing that the connection between the Lamb shift and the ac Stark shift goes back to some of the first ac Stark shift investigations. See, for example, S. Pancharatnam, “Lights shifts in semiclassical dispersion theory,” J. Opt. Soc. Am. 56, 1636 (1966), and references therein.
    [CrossRef]
  31. H. M. Franca, T. W. Marshall, and E. Santos, “Spontaneous emission in confined space according to stochastic electrodynamics,” Phys. Rev. A 45, 6436–6442 (1992).
    [CrossRef] [PubMed]
  32. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), Chap. 15; H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev. 83, 34–40 (1951).
    [CrossRef]
  33. E. A. Power, Introductory Quantum Electrodynamics (Elsevier, New York, 1964), Chap. 2.
  34. O. S. Heavens, “Radiative transition probabilities of the lower excited states of the alkali metals,” J. Opt. Soc. Am. 51, 1058–1061 (1961).
    [CrossRef]
  35. R. C. Hilborn, “Einstein coefficients, cross sections, f values, dipole moments, and all that,” Am. J. Phys. 50, 982–986 (1982).
    [CrossRef]
  36. The computation first simulated each mode of the ZPF over the ~100-ns simulation time (Δtstep~1 ns) and then summed over modes. For the case of Δν=10 kHz the entire simulation of P2(t) required only 4 min. on a 133-MHz Pentium computer.

1996 (1)

M. Ibison and B. Haisch, “Quantum and classical statistics of the electromagnetic zero-point field,” Phys. Rev. A 54, 2737–2744 (1996).
[CrossRef] [PubMed]

1995 (1)

D. K. Ross and W. Moreau, “Stochastic gravity,” Gen. Relativ. Gravit. 27, 845–858 (1995).
[CrossRef]

1994 (1)

A. G. Kofman, G. Kurizki, and B. Sherman, “Spontaneous and induced atomic decay in photonic band structures,” J. Mod. Opt. 41, 353–384 (1994).
[CrossRef]

1992 (1)

H. M. Franca, T. W. Marshall, and E. Santos, “Spontaneous emission in confined space according to stochastic electrodynamics,” Phys. Rev. A 45, 6436–6442 (1992).
[CrossRef] [PubMed]

1985 (1)

An important exception to this statement occurs in the theory of QED based on self-energy. See A. O. Barut and J. F. Van Huele, “Quantum electrodynamics based on self-energy: Lamb shift and spontaneous emission without field quantization,” Phys. Rev. A 32, 3187–3195 (1985).
[CrossRef] [PubMed]

1982 (1)

R. C. Hilborn, “Einstein coefficients, cross sections, f values, dipole moments, and all that,” Am. J. Phys. 50, 982–986 (1982).
[CrossRef]

1976 (1)

P. W. Milonni, “Semiclassical and quantum-electrodynamical approaches in nonrelativistic radiation theory,” Phys. Rep. 25, 1–81 (1976).
[CrossRef]

1975 (2)

We do not consider here the absorber theory of radiation, which posits the existence of advanced radiation fields from distant absorbers as the seat of radiation reaction. For a review of absorber theory see D. T. Pegg, “Absorber theory of radiation,” Rep. Prog. Phys. 38, 1339–1383 (1975).
[CrossRef]

T. H. Boyer, “Random electrodynamics: the theory of classical electrodynamics with classical electromagnetic zero-point radiation,” Phys. Rev. D 11, 790–808 (1975).
[CrossRef]

1969 (1)

T. H. Boyer, “Classical statistical thermodynamics and electromagnetic zero-point radiation,” Phys. Rev. 186, 1304–1318 (1969).
[CrossRef]

1966 (1)

1961 (1)

1947 (1)

H. A. Bethe, “The electromagnetic shift of energy levels,” Phys. Rev. 72, 339 (1947).
[CrossRef]

1945 (1)

J. A. Wheeler and R. P. Feynman, “Interaction with the absorber as the mechanism of radiation,” Rev. Mod. Phys. 17, 157–181 (1945).
[CrossRef]

Barut, A. O.

An important exception to this statement occurs in the theory of QED based on self-energy. See A. O. Barut and J. F. Van Huele, “Quantum electrodynamics based on self-energy: Lamb shift and spontaneous emission without field quantization,” Phys. Rev. A 32, 3187–3195 (1985).
[CrossRef] [PubMed]

Bethe, H. A.

H. A. Bethe, “The electromagnetic shift of energy levels,” Phys. Rev. 72, 339 (1947).
[CrossRef]

Boyer, T. H.

T. H. Boyer, “Random electrodynamics: the theory of classical electrodynamics with classical electromagnetic zero-point radiation,” Phys. Rev. D 11, 790–808 (1975).
[CrossRef]

T. H. Boyer, “Classical statistical thermodynamics and electromagnetic zero-point radiation,” Phys. Rev. 186, 1304–1318 (1969).
[CrossRef]

Feynman, R. P.

J. A. Wheeler and R. P. Feynman, “Interaction with the absorber as the mechanism of radiation,” Rev. Mod. Phys. 17, 157–181 (1945).
[CrossRef]

Franca, H. M.

H. M. Franca, T. W. Marshall, and E. Santos, “Spontaneous emission in confined space according to stochastic electrodynamics,” Phys. Rev. A 45, 6436–6442 (1992).
[CrossRef] [PubMed]

Haisch, B.

M. Ibison and B. Haisch, “Quantum and classical statistics of the electromagnetic zero-point field,” Phys. Rev. A 54, 2737–2744 (1996).
[CrossRef] [PubMed]

Heavens, O. S.

Hilborn, R. C.

R. C. Hilborn, “Einstein coefficients, cross sections, f values, dipole moments, and all that,” Am. J. Phys. 50, 982–986 (1982).
[CrossRef]

Ibison, M.

M. Ibison and B. Haisch, “Quantum and classical statistics of the electromagnetic zero-point field,” Phys. Rev. A 54, 2737–2744 (1996).
[CrossRef] [PubMed]

Kofman, A. G.

A. G. Kofman, G. Kurizki, and B. Sherman, “Spontaneous and induced atomic decay in photonic band structures,” J. Mod. Opt. 41, 353–384 (1994).
[CrossRef]

Kurizki, G.

A. G. Kofman, G. Kurizki, and B. Sherman, “Spontaneous and induced atomic decay in photonic band structures,” J. Mod. Opt. 41, 353–384 (1994).
[CrossRef]

Marshall, T. W.

H. M. Franca, T. W. Marshall, and E. Santos, “Spontaneous emission in confined space according to stochastic electrodynamics,” Phys. Rev. A 45, 6436–6442 (1992).
[CrossRef] [PubMed]

Milonni, P. W.

P. W. Milonni, “Semiclassical and quantum-electrodynamical approaches in nonrelativistic radiation theory,” Phys. Rep. 25, 1–81 (1976).
[CrossRef]

Moreau, W.

D. K. Ross and W. Moreau, “Stochastic gravity,” Gen. Relativ. Gravit. 27, 845–858 (1995).
[CrossRef]

Pancharatnam, S.

Pegg, D. T.

We do not consider here the absorber theory of radiation, which posits the existence of advanced radiation fields from distant absorbers as the seat of radiation reaction. For a review of absorber theory see D. T. Pegg, “Absorber theory of radiation,” Rep. Prog. Phys. 38, 1339–1383 (1975).
[CrossRef]

Ross, D. K.

D. K. Ross and W. Moreau, “Stochastic gravity,” Gen. Relativ. Gravit. 27, 845–858 (1995).
[CrossRef]

Santos, E.

H. M. Franca, T. W. Marshall, and E. Santos, “Spontaneous emission in confined space according to stochastic electrodynamics,” Phys. Rev. A 45, 6436–6442 (1992).
[CrossRef] [PubMed]

Sherman, B.

A. G. Kofman, G. Kurizki, and B. Sherman, “Spontaneous and induced atomic decay in photonic band structures,” J. Mod. Opt. 41, 353–384 (1994).
[CrossRef]

Van Huele, J. F.

An important exception to this statement occurs in the theory of QED based on self-energy. See A. O. Barut and J. F. Van Huele, “Quantum electrodynamics based on self-energy: Lamb shift and spontaneous emission without field quantization,” Phys. Rev. A 32, 3187–3195 (1985).
[CrossRef] [PubMed]

Wheeler, J. A.

J. A. Wheeler and R. P. Feynman, “Interaction with the absorber as the mechanism of radiation,” Rev. Mod. Phys. 17, 157–181 (1945).
[CrossRef]

Am. J. Phys. (1)

R. C. Hilborn, “Einstein coefficients, cross sections, f values, dipole moments, and all that,” Am. J. Phys. 50, 982–986 (1982).
[CrossRef]

Gen. Relativ. Gravit. (1)

D. K. Ross and W. Moreau, “Stochastic gravity,” Gen. Relativ. Gravit. 27, 845–858 (1995).
[CrossRef]

J. Mod. Opt. (1)

A. G. Kofman, G. Kurizki, and B. Sherman, “Spontaneous and induced atomic decay in photonic band structures,” J. Mod. Opt. 41, 353–384 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

Phys. Rep. (1)

P. W. Milonni, “Semiclassical and quantum-electrodynamical approaches in nonrelativistic radiation theory,” Phys. Rep. 25, 1–81 (1976).
[CrossRef]

Phys. Rev. (2)

T. H. Boyer, “Classical statistical thermodynamics and electromagnetic zero-point radiation,” Phys. Rev. 186, 1304–1318 (1969).
[CrossRef]

H. A. Bethe, “The electromagnetic shift of energy levels,” Phys. Rev. 72, 339 (1947).
[CrossRef]

Phys. Rev. A (3)

H. M. Franca, T. W. Marshall, and E. Santos, “Spontaneous emission in confined space according to stochastic electrodynamics,” Phys. Rev. A 45, 6436–6442 (1992).
[CrossRef] [PubMed]

M. Ibison and B. Haisch, “Quantum and classical statistics of the electromagnetic zero-point field,” Phys. Rev. A 54, 2737–2744 (1996).
[CrossRef] [PubMed]

An important exception to this statement occurs in the theory of QED based on self-energy. See A. O. Barut and J. F. Van Huele, “Quantum electrodynamics based on self-energy: Lamb shift and spontaneous emission without field quantization,” Phys. Rev. A 32, 3187–3195 (1985).
[CrossRef] [PubMed]

Phys. Rev. D (1)

T. H. Boyer, “Random electrodynamics: the theory of classical electrodynamics with classical electromagnetic zero-point radiation,” Phys. Rev. D 11, 790–808 (1975).
[CrossRef]

Rep. Prog. Phys. (1)

We do not consider here the absorber theory of radiation, which posits the existence of advanced radiation fields from distant absorbers as the seat of radiation reaction. For a review of absorber theory see D. T. Pegg, “Absorber theory of radiation,” Rep. Prog. Phys. 38, 1339–1383 (1975).
[CrossRef]

Rev. Mod. Phys. (1)

J. A. Wheeler and R. P. Feynman, “Interaction with the absorber as the mechanism of radiation,” Rev. Mod. Phys. 17, 157–181 (1945).
[CrossRef]

Other (22)

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, New York, 1977), Vol. 1, pp. 315–328; R. R. Schlicher, W. Becker, J. Bergou, and M. O. Scully, “Interaction Hamiltonian in quantum optics or: p⋅A vs. E⋅r revisited,” in Quantum Electrodynamics and Quantum Optics, A. O. Barut, ed. (Plenum, New York, 1984), pp. 405–441.

Physically, in SRED the missing factor of 2 in Eq. (5b) is to be attributed to an explicit role for the vacuum field in describing an oscillating dipole’s dynamics.

The computation first simulated each mode of the ZPF over the ~100-ns simulation time (Δtstep~1 ns) and then summed over modes. For the case of Δν=10 kHz the entire simulation of P2(t) required only 4 min. on a 133-MHz Pentium computer.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynbert, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley, New York, 1989), Chap. I.

Heitler, The Quantum Theory of Radiation (Dover, New York, 1984), Chap. 2.

E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), p. 446.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), Chap. 15; H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev. 83, 34–40 (1951).
[CrossRef]

E. A. Power, Introductory Quantum Electrodynamics (Elsevier, New York, 1964), Chap. 2.

J. N. Dodd, Atoms and Light: Interactions (Plenum, New York, 1991), Chap. 10.

P. Meystre and M. Sargent III, Elements of Quantum Optics (Springer-Verlag, Berlin, 1991).

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).

C. P. Slichter, Principles of Magnetic Resonance (Harper & Row, New York, 1963).

See, for example, G. W. Series, “A long journey with classical fields,” Phys. Scr. T12, 5–13 (1986); J. N. Dodd, “Spontaneous emission and the vacuum field,” Phys. Scr. T70, 88–93 (1997).
[CrossRef]

J. R. Ackerhalt, P. L. Knight, and J. H. Eberly, “Radiation reaction and radiative frequency shifts,” Phys. Rev. Lett. 30, 456–460 (1973); I. R. Senitzky, “Radiation-reaction and vacuum-field effects in Heisenberg-picture quantum electrodynamics,” Phys. Rev. Lett. 31, 955–958 (1973); P. W. Milonni, J. R. Ackerhalt, and W. A. Smith, “Interpretation of radiative corrections in spontaneous emission,” Phys. Rev. Lett. PRLTAO 31, 958–960 (1973); J. Dalibard, J. Dupont-Roc, and C. Cohen-Tannoudji, “Vacuum fluctuations and radiative reaction: identification of their respective contributions,” J. Phys. (Paris) JOPQAG 43, 1617–1638 (1982).
[CrossRef]

H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962), Chap. 21; J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 17.

P. L. Knight, “Dynamics of spontaneous emission,” Phys. Scr. T70, 94–100 (1997); M. B. Plenio and P. L. Knight, “The quantum jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998).
[CrossRef]

It is worth mentioning here that (as was pointed out by Eberly) when a classical Lagrangian for the field–atom system is supplanted by the assumption of stochastic, homogenous solutions, a description of the field–atom interaction is obtained that “...doesn’t differ from QED at all in its second-order predictions of decay rates and level shifts.” See J. H. Eberly, “Unified view of spontaneous emission in several theories of radiation,” in Foundations of Radiation Theory and Quantum Electrodynamics, A. O. Barut, ed. (Plenum, New York, 1980).

T. H. Boyer, “A brief survey of stochastic electrodynamics,” in Foundations of Radiation Theory and Quantum Electrodynamics, A. O. Barut, ed. (Plenum, New York, 1980); L. de la Peña, “Stochastic electrodynamics: its development, present situation and perspectives,” in Stochastic Processes Applied to Physics and Other Related Fields, B. Gomez, S. M. Moore, A. M. Rodriguez-Vargas, and A. Reuda, eds. (World Scientific, Singapore, 1983).

P. W. Milonni, The Quantum Vacuum (Academic, Boston, Mass., 1994).

L. de la Peña and A. M. Cetto, The Quantum Dice: An Introduction to Stochastic Electrodynamics (Kluwer, Dordrecht, The Netherlands, 1996).

H. M. Franca and T. W. Marshall, “Excited states in stochastic electrodynamics,” Phys. Rev. A 38, 3258–3263 (1988); L. de la Peña and A. M. Cetto, “Quantum phenomena and the zeropoint radiation field,” Found. Phys. 25, 573–604 (1995).
[CrossRef] [PubMed]

For a discussion of these issues, see G. N. Plass, “Classical electrodynamic equations of motion with radiative reaction,” Rev. Mod. Phys. 33, 37–62 (1961); F. Rohrlich, “The dynamics of a charged sphere and the electron,” Am. J. Phys. 65, 1051–1056 (1997); G. W. Ford and R. F. O’Connell, “Radiative reaction in electrodynamics and the elimination of runaway solutions,” Phys. Lett. A PYLAAG 157, 217–220 (1991).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Numerical simulation of the sodium 32P1/2 state’s spontaneous decay using SRED: squares, mode spacing of 100 kHz; circles, mode spacing of 10 kHz; diamonds, mode spacing of 1 kHz. For comparison, the solid curve corresponds to exponential decay given the 32P1/2 state’s 17.0-ns lifetime.

Equations (82)

Equations on this page are rendered with MathJax. Learn more.

ERR=2e3c3t3r=23c3t3μ,
it|φ=H0|φ  t -iH0,
-itφ|=φ|H0  t iH0.
ERR-i33c3(H03μ-μH03).
φ|ERR|φ=-i33c3n[φ|H03|nn|μ|φ-φ|μ|nn|H03|φ],
|ω21|3ω2ω1,ω21ω2-ω1
φ|ERR|φ=±|μ21|ω2133c3sin(ω21t+θ),
VRR=-μ·ERR=i33c3(μ·H03μ-μ·μH03).
A0=L-3/2kλˆkλ{ckλ exp[i(k·r-ωst)]+ckλ* exp[-i(k·r-ωst)]}.
k·k=2πL2[nx2+ny2+nz2]=ωsc2,
E0=-1c2L3sλˆsλ{-iωscsλ exp[i(ks·r-ωst)]+iωscsλ* exp[-i(ks·r-ωst)]}.
zsλ12(usλ-ivsλ),
E{zsλzrμ*}=δsrδλμ,
E{zsλzrμ}=E{zsλ*zrμ*}=0,
U0=14πE{|E0|2}d3r=sλ(ωscsλ0)22πc2.
csλ0=[(πc2)/ωs]1/2  U0=sλωs2.
E0=[(2π)/L3]1/2sλˆsλωs[vsλ cos(ks·r-ωst)-usλ sin(ks·r-ωst)].
E0[(2π)/L3]1/2sλˆsλωs[vsλ cos(ωst)+usλ sin(ωst)]
E0i[(π)/L3]1/2sλˆsλωs[zsλ exp(-iωst)-zsλ* exp(iωst)].
V0(t)=-μ·E0=-i[(π)/L3]1/2sλμ·ˆsλ×ωs[zsλ exp(-iωst)-zsλ* exp(iωst)].
E{Vmn0(t)}=0.
E{Vmn0(t)Vpq0(t)}
=-πL3sλrμωsωrm|μ·ˆsλ|np|μ·ˆrμ|q
×E{[zsλ exp(-iωst)-zsλ* exp(iωst)]×[zrμ exp(-iωrt)-zrμ* exp(iωrt)]}.
E{Vmn0(t)Vpq0(t)}
=πL3sλrμωsωrm|μ·ˆsλ|np|μ·ˆrμ|q-
×{δsrδλμ exp[-iωs(t-t)]+δsrδλμ exp[iωs(t-t)]}
E{Vmn0(t)Vpq0(t)}
=πL3sλωsm|μ·ˆsλ|np|μ·ˆsλ|q×{exp[-iωs(t-t)]+exp[iωs(t-t)]}.
i|φt=[H0+VRR+V0(t)]|φ,
|φ=nan(t)|nexp(-iωnt).
a˙n(t)=-ij[n|VRR|j+n|V0|j]aj(t) exp(iωnjt),
an(t)=an0-ij0t[VnjRR+Vnj0(t)]aj(t)×exp(iωnjt)dt.
a˙nRR(t)=134c3jaj(t)×n|(μ·H03μ-μ·μH03)|j exp(iωnjt).
b˙nRR(t)=134c3pqjbj(t)[μnpp|H03|qμqj-μnpμpqq|H03|j]exp(iωnjt)
b˙nRR(t)=13c3pjbj(t)μnpμpj(ωp3-ωj3) exp(iωnjt).
b˙nRR(t)=-bn(t)3c3p|μnp|2ωnp3.
a˙nZPF=-ijVnj0(t)aj(t) exp(iωnjt),
a˙mZPF(t)=-inVmn0(t) exp(iωmnt)×an0-ij0t[VnjRR+Vnj0(t)]aj(t)×exp(iωnjt)dt.
b˙mZPF(t)=-12njexp(iωmnt)×0tE{Vmn0(t)Vnj0(t)}bj(t) exp(iωnjt)dt.
E[Vmn0(t)Vnj0(t)aj(t)]E[Vmn0(t)Vnj0(t)]E[aj(t)]=E[Vmn0(t)Vnj0(t)]bj(t).
b˙mZPF(t)=-πL3njbj(t) exp(iωmnt)×0tsλωsm|μ·ˆsλ|nn|μ·ˆsλ|j×{exp[-iωs(t-t)]+exp[iωs(t-t)]}×exp(iωnjt)dt.
λsλ,zsλ,z=1-kz2|k|2,
b˙mZPF(t)=-πL3njbj(t) exp(iωmnt)μmnμnj×0tsωs sin2(θs){exp[-iωs(t-t)]+exp[iωs(t-t)]} exp(iωnjt)dt,
sL38π3d3k,
b˙mZPF(t)=-18π2c3njbj(t) exp(iωmnt)μmnμnj×0tω3{exp[-iω(t-t)]+exp[iω(t-t)]} sin3(θ)dθdφdω×exp(iωnjt)dt.
b˙mZPF(t)=-13πc3njbj(t) exp(iωmnt)μmnμnj×0t0ω3{exp[-iω(t-t)]+exp[iω(t-t)]}dω exp(iωnjt)dt
b˙mZPF(t)=-13πc3njbj(t) exp(iωmnt)μmnμnj×0ω3exp(-iωt)0t exp[i(ω+ωnj)t]dt+exp(iωt)0t exp[-i(ω-ωnj)t]dtdω,
b˙mZPF(t)=-13πc3njbj(t)×exp(iωmnt)μmnμnj[Int(1)+Int(2)].
Int(1)lima0 0ω3 exp(-iωt)×0t exp[at+i(ω+ωnj)t]dtdω,
Int(1)=lima0exp{[(a+iωnj)t]}0ω3dωa+i(ω+ωnj)-0ω3 exp(-iωt)dωa+i(ω+ωnj).
Int(1)lima0exp{[(a+iωnj)t]}×0ω3[a-i(ω+ωnj)]dωa2+(ω+ωnj)2.
Int(1)=exp(iωnjt)π0ω3δ(ω+ωnj)dω-iP0ω3dω(ω+ωnj).
Int(2)=exp(iωnjt)π0ω3δ(ω-ωnj)dω+iP0ω3dω(ω-ωnj).
b˙mZPF(t)=-13c3njbj(t) exp(iωmjt)μmnμnj×0ω3[δ(ω+ωnj)+δ(ω-ωnj)]dω-iΔ˜,
Δ˜13πc3njbj(t) exp(iωmjt)μmnμnj×P0ω3ω-ωnj-ω3ω+ωnjdω
b˙mZPF(t)=-bm(t)3c3n|μmn|20ω3δ(ω+ωnm)dω+0ω3δ(ω-ωnm)dω-ibm(t)Δ,
Δ23πc3nωnm|μmn|2P0ω3dωω2-ωnm2.
b˙mZPF(t)=-bm(t)3c3n|μmn|2ωmn3ωm>ωnωnm3ωn>ωm-ibm(t)Δ.
b˙m(t)=-bm(t)3c3×n|μmn|2+ωmn3ωm>ωnωmn3-ωmn3ωn>ωm,
b˙m(t)=-2bm(t)3c3n<m|μmn|2ωmn3.
dPmdt=-ΓmPm,
Γm43c3n<m|μmn|2ωmn3.
Δ23πc3nωnm|μmn|20ωdωΔ0.
Δ0=e23πmc30ωdω.
Δ0=e22mc2|A0|2z.
Δ=23πc3nωnm3|μmn|2P0ωdωω2-ωnm2,
Δ=23πc3nωnm3|μmn|2 lnmc2|En-Em|.
a˙2=a˙2RR+a˙2ZPF.
Rea˙2RR=Rea˙2ZPF=-A4a2(t).
a˙2(t)=2iμ21ˆz·E0(t) exp(iω21t)×a10+i0t[μ12ˆz·E0(t)-V12RR]×a2(t) exp(-iω21t)dt.
Y(t)τ1τt-τ/2t+τ/2Y(t)dt,
a˙2(t)=-2|μ21|220texp(iω21t)ˆz·E0(t)ˆz·E0(t)×exp(-iω21t)τa2(t)dt.
a˙2(t)=-2π|μ21|2L3sλ,pηzsλzpη*ωsωp(ˆsλ,zˆpη,z)×0texp(-iδst) exp(iδpt)τa2(t)dt.
limΔω0 zsλpηzpη*=|zsλ|2δspδλη.
a˙2(t)-2π|μ21|2L3sλ|zsλ|2ωs(ˆsλ,zˆsλ,z)×0texp[-iδs(t-t)]τa2(t)dt.
a˙2(t)=-2π|μ21|2L3a2(t)sλ|zsλ|2ωsiδs(ˆsλ,zˆsλ,z)×[1-exp(-iδst)].
a˙2(t)=-4π|μ21|23L3a2(t)s|zs|2ωsiδs[1-exp(-iδst)].
a˙2(t)=-2|μ21|2ω212Δω3πc3a2(t)|z0|2ω21t+s|zs|2ωsiδs[1-exp(-iδst)],
lna2(t)a2(0)=-κ|z0|2ω21t22+s|zs|2ωsδs2[1-cos(δst)]+iωsδs2[sin(δst)-δst],
κ2|μ21|2ω212Δω3πc3.
P2(t)=P2(0)exp-κ|z0|2ω21t2-2κs|zs|2ωsδs2×[1-cos(δst)].

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