Abstract

In the literature one finds several conflicting accounts of the phase difference of stimulated and spontaneous emission, as well as absorption, with respect to an existing (triggering) electromagnetic field. One of these approaches proposes that stimulated emission and absorption occur in phase and out of phase with their driving field, respectively, whereas spontaneous emission occurs under an arbitrary phase difference with respect to an existing field. It has served as a basis for explaining quantum-mechanically the laser linewidth, its narrowing by a factor of 2 around the laser threshold, as well as its broadening due to amplitude–phase coupling, resulting in Henry’s α-factor. Assuming the validity of Maxwell’s equations, all three processes would, thus, violate the law of energy conservation. In semi-classical approaches, we investigate stimulated emission in a Fabry–Perot resonator, analyze the Lorentz oscillator model, apply the Kramers–Kronig relations to the complex susceptibility, understand the summation of quantized electric fields, and quantitatively interpret emission and absorption in the amplitude–phase diagram. In all cases, we derive that the phase of stimulated emission is 90° in lead of the driving field, and the phase of absorption lags 90° behind the transmitted field. Also spontaneous emission must obey energy conservation, hence it occurs with 90° phase in lead of an existing field. These semi-classical findings agree with recent experimental investigations regarding the interaction of attosecond pulses with an atom, thereby questioning the physical explanation of the laser linewidth and its narrowing or broadening.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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2016 (3)

N. Ismail, C. C. Kores, D. Geskus, and M. Pollnau, “Fabry–Pérot resonator: spectral line shapes, generic and related Airy distributions, linewidths, finesses, and performance at low or frequency-dependent reflectivity,” Opt. Express 24, 16366–16389 (2016).
[Crossref]

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref]

T. V. Raziman and O. J. F. Martin, “Does the real part contain all the physical information?” J. Opt. 18, 095002 (2016).
[Crossref]

2015 (1)

M. Eichhorn and M. Pollnau, “Spectroscopic foundations of lasers: spontaneous emission into a resonator mode,” IEEE J. Sel. Top. Quantum Electron. 21, 9000216 (2015).
[Crossref]

2010 (1)

M. Motsch, M. Zeppenfeld, P. W. H. Pinkse, and G. Rempe, “Cavity-enhanced Rayleigh scattering,” New J. Phys. 12, 063022 (2010).
[Crossref]

2007 (1)

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nat. Phys. 3, 106–110 (2007).
[Crossref]

2002 (1)

1997 (1)

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357–4360 (1997).
[Crossref]

1991 (1)

G. Björk and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. 27, 2386–2396 (1991).
[Crossref]

1989 (3)

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
[Crossref]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[Crossref]

W. A. Hamel and J. P. Woerdman, “Nonorthogonality of the longitudinal eigenmodes of a laser,” Phys. Rev. A 40, 2785–2787 (1989).
[Crossref]

1986 (1)

A. Kastler, “Review: masers and lasers,” Eur. J. Phys. 7, 69–76 (1986).
[Crossref]

1982 (4)

M. Gross and S. Haroche, “Superradiance: an essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982).
[Crossref]

H. S. Sommers, “Threshold and oscillation of injection lasers: a critical review of laser theory,” Solid State Electron. 25, 25–44 (1982).
[Crossref]

M. Cray, M. L. Shih, and P. W. Milonni, “Stimulated emission, absorption, and interference,” Am. J. Phys. 50, 1016–1021 (1982).
[Crossref]

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[Crossref]

1976 (1)

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

1974 (1)

H. S. Sommers, “Spontaneous power and the coherent state of injection lasers,” J. Appl. Phys. 45, 1787–1793 (1974).
[Crossref]

1972 (2)

H. Gerhardt, H. Welling, and A. Güttner, “Measurements of the laser linewidth due to quantum phase and quantum amplitude noise above and below threshold. I,” Z. Phys. 253, 113–126 (1972).
[Crossref]

H. Gerhardt, H. Welling, and A. Güttner, “Observation of quantum-phase and quantum-amplitude noise for a laser below and above threshold,” Phys. Lett. A 40, 191–193 (1972).
[Crossref]

1968 (1)

V. F. Weisskopf, “How light interacts with matter,” Sci. Am. 219, 60–71 (1968).
[Crossref]

1967 (2)

R. D. Hempstead and M. Lax, “Classical noise. VI. Noise in self-sustained oscillators near threshold,” Phys. Rev. 161, 350–366 (1967).
[Crossref]

M. Lax, “Classical noise. V. Noise in self-sustained oscillators,” Phys. Rev. 160, 290–307 (1967).
[Crossref]

1966 (1)

H. Risken, “Correlation function of the amplitude and of the intensity fluctuations for a laser model near threshold,” Z. Phys. 191, 302–312 (1966).
[Crossref]

1963 (1)

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

1958 (1)

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[Crossref]

1954 (1)

R. H. Dieke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[Crossref]

1929 (1)

H. P. Robertson, “The uncertainty principle,” Phys. Rev. 34, 163–164 (1929).
[Crossref]

1928 (1)

H. Kopfermann and R. Ladenburg, “Untersuchungen über die anomale Dispersion angeregter Gase. II. Teil. Anomale Dispersion in angeregtem Neon (Einfluß von Strom und Druck, Bildung und Vernichtung angeregter Atome),” Z. Phys. 48, 26–50 (1928).
[Crossref]

1927 (2)

W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Phys. 43, 172–198 (1927).
[Crossref]

P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proc. R. Soc. A 114, 243–265 (1927).
[Crossref]

1926 (1)

1917 (1)

A. Einstein, “Zur Quantentheorie der Strahlung,” Phys. Z. 18, 121–128 (1917).

1901 (1)

M. Planck, “Ueber das Gesetz der Energieverteilung im Normalspectrum,” Ann. Phys. 309, 553–563 (1901).
[Crossref]

1880 (1)

H. A. Lorentz, “Über die Beziehung zwischen der Fortpflanzungsgeschwindigkeit des Lichtes und der Körperdichte,” Ann. Phys. 245, 641–665 (1880).
[Crossref]

1865 (1)

J. C. Maxwell, “VIII. A dynamical theory of the electromagnetic field,” Philos. Trans. R. Soc. London 155, 459–512 (1865).
[Crossref]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Bacher, G.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nat. Phys. 3, 106–110 (2007).
[Crossref]

Björk, G.

G. Björk and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. 27, 2386–2396 (1991).
[Crossref]

Blaquière, A.

P. Grivet and A. Blaquière, “Masers and classical oscillators,” in Symposium on Optical Masers, J. Fox, ed. (Polytechnique Institute, 1963), pp. 69–93.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Bothschafter, E. M.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref]

Bruhat, G.

G. Bruhat and A. Kastler, “Interférences de la lumière,” in Optique, 6th ed. (Masson, 1965), pp. 60–61, 95–96, and 175.

Cray, M.

M. Cray, M. L. Shih, and P. W. Milonni, “Stimulated emission, absorption, and interference,” Am. J. Phys. 50, 1016–1021 (1982).
[Crossref]

de L. Kronig, R.

Dieke, R. H.

R. H. Dieke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[Crossref]

Dirac, P. A. M.

P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proc. R. Soc. A 114, 243–265 (1927).
[Crossref]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Eberly, J. H.

P. W. Milonni and J. H. Eberly, “The electron oscillator model,” in Lasers (Wiley, 1988), Ch. 2.2, pp. 27–33.

Eichhorn, M.

M. Eichhorn and M. Pollnau, “Spectroscopic foundations of lasers: spontaneous emission into a resonator mode,” IEEE J. Sel. Top. Quantum Electron. 21, 9000216 (2015).
[Crossref]

M. Eichhorn and M. Pollnau are preparing a manuscript to be called “Laser linewidth, spectral coherence, and the laser eigenvalue.”

Einstein, A.

A. Einstein, “Zur Quantentheorie der Strahlung,” Phys. Z. 18, 121–128 (1917).

Fabre, C.

Fattahi, H.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref]

Fermi, E.

E. Fermi, Nuclear Physics (The University of Chicago, 1950), formula VIII.2.

Feynman, R. P.

R. P. Feynman, R. B. Leighton, and M. Sands, “The origin of the refractive index,” in The Feynman Lectures on Physics (Addison-Wesley, 1963), Vol. I, Ch. 31, pp. 31–34.

Forchel, A.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nat. Phys. 3, 106–110 (2007).
[Crossref]

Gerhardt, H.

H. Gerhardt, H. Welling, and A. Güttner, “Measurements of the laser linewidth due to quantum phase and quantum amplitude noise above and below threshold. I,” Z. Phys. 253, 113–126 (1972).
[Crossref]

H. Gerhardt, H. Welling, and A. Güttner, “Observation of quantum-phase and quantum-amplitude noise for a laser below and above threshold,” Phys. Lett. A 40, 191–193 (1972).
[Crossref]

Geskus, D.

Glauber, R. J.

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

Grivet, P.

P. Grivet and A. Blaquière, “Masers and classical oscillators,” in Symposium on Optical Masers, J. Fox, ed. (Polytechnique Institute, 1963), pp. 69–93.

Gross, M.

M. Gross and S. Haroche, “Superradiance: an essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982).
[Crossref]

Güttner, A.

H. Gerhardt, H. Welling, and A. Güttner, “Measurements of the laser linewidth due to quantum phase and quantum amplitude noise above and below threshold. I,” Z. Phys. 253, 113–126 (1972).
[Crossref]

H. Gerhardt, H. Welling, and A. Güttner, “Observation of quantum-phase and quantum-amplitude noise for a laser below and above threshold,” Phys. Lett. A 40, 191–193 (1972).
[Crossref]

Haken, H.

H. Haken, Laser Theory, Vol. XXV/2c of Encyclopedia of Physics (Springer, 1970).

Hamel, W. A.

W. A. Hamel and J. P. Woerdman, “Nonorthogonality of the longitudinal eigenmodes of a laser,” Phys. Rev. A 40, 2785–2787 (1989).
[Crossref]

Haroche, S.

M. Gross and S. Haroche, “Superradiance: an essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982).
[Crossref]

Heisenberg, W.

W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Phys. 43, 172–198 (1927).
[Crossref]

Hempstead, R. D.

R. D. Hempstead and M. Lax, “Classical noise. VI. Noise in self-sustained oscillators near threshold,” Phys. Rev. 161, 350–366 (1967).
[Crossref]

Henry, C. H.

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[Crossref]

Hommel, D.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nat. Phys. 3, 106–110 (2007).
[Crossref]

Ismail, N.

Jakubeit, C.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref]

Jobst, M.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref]

Karpowicz, N.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref]

Kastler, A.

A. Kastler, “Review: masers and lasers,” Eur. J. Phys. 7, 69–76 (1986).
[Crossref]

G. Bruhat and A. Kastler, “Interférences de la lumière,” in Optique, 6th ed. (Masson, 1965), pp. 60–61, 95–96, and 175.

Kienberger, R.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
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Kopfermann, H.

H. Kopfermann and R. Ladenburg, “Untersuchungen über die anomale Dispersion angeregter Gase. II. Teil. Anomale Dispersion in angeregtem Neon (Einfluß von Strom und Druck, Bildung und Vernichtung angeregter Atome),” Z. Phys. 48, 26–50 (1928).
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Kores, C. C.

Kramers, H. A.

H. A. Kramers, “La diffusion de la lumière par les atoms,” in Atti del Congresso Internazionale del Fisica (Transactions of Volta Centenary Congress) Como (1927), Vol. 2, pp. 545–557.

Krausz, F.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
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Ladenburg, R.

H. Kopfermann and R. Ladenburg, “Untersuchungen über die anomale Dispersion angeregter Gase. II. Teil. Anomale Dispersion in angeregtem Neon (Einfluß von Strom und Druck, Bildung und Vernichtung angeregter Atome),” Z. Phys. 48, 26–50 (1928).
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Lamb, W. E.

M. Sargent, M. O. Scully, and W. E. Lamb, “Dipole oscillators,” in Laser Physics, 6th ed. (Westview, 1993), Ch. 3.2, pp. 34–42.

M. Sargent, M. O. Scully, and W. E. Lamb, “Coherent states,” in Laser Physics, 6th ed. (Westview, 1993), Ch. XV, pp. 242–255.

Latka, T.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
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Lax, M.

R. D. Hempstead and M. Lax, “Classical noise. VI. Noise in self-sustained oscillators near threshold,” Phys. Rev. 161, 350–366 (1967).
[Crossref]

M. Lax, “Classical noise. V. Noise in self-sustained oscillators,” Phys. Rev. 160, 290–307 (1967).
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Leighton, R. B.

R. P. Feynman, R. B. Leighton, and M. Sands, “The origin of the refractive index,” in The Feynman Lectures on Physics (Addison-Wesley, 1963), Vol. I, Ch. 31, pp. 31–34.

Lindberg, Å. M.

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357–4360 (1997).
[Crossref]

Lorentz, H. A.

H. A. Lorentz, “Über die Beziehung zwischen der Fortpflanzungsgeschwindigkeit des Lichtes und der Körperdichte,” Ann. Phys. 245, 641–665 (1880).
[Crossref]

Martin, O. J. F.

T. V. Raziman and O. J. F. Martin, “Does the real part contain all the physical information?” J. Opt. 18, 095002 (2016).
[Crossref]

Maxwell, J. C.

J. C. Maxwell, “VIII. A dynamical theory of the electromagnetic field,” Philos. Trans. R. Soc. London 155, 459–512 (1865).
[Crossref]

Mieremet, A. L.

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357–4360 (1997).
[Crossref]

Milonni, P. W.

M. Cray, M. L. Shih, and P. W. Milonni, “Stimulated emission, absorption, and interference,” Am. J. Phys. 50, 1016–1021 (1982).
[Crossref]

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

P. W. Milonni and J. H. Eberly, “The electron oscillator model,” in Lasers (Wiley, 1988), Ch. 2.2, pp. 27–33.

Motsch, M.

M. Motsch, M. Zeppenfeld, P. W. H. Pinkse, and G. Rempe, “Cavity-enhanced Rayleigh scattering,” New J. Phys. 12, 063022 (2010).
[Crossref]

Passow, T.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nat. Phys. 3, 106–110 (2007).
[Crossref]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Pinkse, P. W. H.

M. Motsch, M. Zeppenfeld, P. W. H. Pinkse, and G. Rempe, “Cavity-enhanced Rayleigh scattering,” New J. Phys. 12, 063022 (2010).
[Crossref]

Planck, M.

M. Planck, “Ueber das Gesetz der Energieverteilung im Normalspectrum,” Ann. Phys. 309, 553–563 (1901).
[Crossref]

Pollnau, M.

N. Ismail, C. C. Kores, D. Geskus, and M. Pollnau, “Fabry–Pérot resonator: spectral line shapes, generic and related Airy distributions, linewidths, finesses, and performance at low or frequency-dependent reflectivity,” Opt. Express 24, 16366–16389 (2016).
[Crossref]

M. Eichhorn and M. Pollnau, “Spectroscopic foundations of lasers: spontaneous emission into a resonator mode,” IEEE J. Sel. Top. Quantum Electron. 21, 9000216 (2015).
[Crossref]

M. Eichhorn and M. Pollnau are preparing a manuscript to be called “Laser linewidth, spectral coherence, and the laser eigenvalue.”

M. Pollnau is preparing a manuscript to be called “Reconciling Einstein, Planck, and Heisenberg: the factor of two in vacuum energy and laser linewidth.”

Raziman, T. V.

T. V. Raziman and O. J. F. Martin, “Does the real part contain all the physical information?” J. Opt. 18, 095002 (2016).
[Crossref]

Razskazovskaya, O.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref]

Rempe, G.

M. Motsch, M. Zeppenfeld, P. W. H. Pinkse, and G. Rempe, “Cavity-enhanced Rayleigh scattering,” New J. Phys. 12, 063022 (2010).
[Crossref]

Risken, H.

H. Risken, “Correlation function of the amplitude and of the intensity fluctuations for a laser model near threshold,” Z. Phys. 191, 302–312 (1966).
[Crossref]

Robertson, H. P.

H. P. Robertson, “The uncertainty principle,” Phys. Rev. 34, 163–164 (1929).
[Crossref]

Rosencher, E.

Sands, M.

R. P. Feynman, R. B. Leighton, and M. Sands, “The origin of the refractive index,” in The Feynman Lectures on Physics (Addison-Wesley, 1963), Vol. I, Ch. 31, pp. 31–34.

Sargent, M.

M. Sargent, M. O. Scully, and W. E. Lamb, “Coherent states,” in Laser Physics, 6th ed. (Westview, 1993), Ch. XV, pp. 242–255.

M. Sargent, M. O. Scully, and W. E. Lamb, “Dipole oscillators,” in Laser Physics, 6th ed. (Westview, 1993), Ch. 3.2, pp. 34–42.

Sato, S. A.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref]

Schawlow, A. L.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[Crossref]

Scheibner, M.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nat. Phys. 3, 106–110 (2007).
[Crossref]

Schiff, L. I.

L. I. Schiff, “Interaction between charged particles and the electromagnetic field,” in Quantum Mechanics, 3rd ed. (McGraw-Hill, 1968), Sec. 57, pp. 531–533.

L. I. Schiff, “Absorption and induced emission,” in Quantum Mechanics, 3rd ed. (McGraw-Hill, 1968), Sec. 44, p. 398–406.

Schmidt, T.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nat. Phys. 3, 106–110 (2007).
[Crossref]

Schultze, M.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref]

Schweinberger, W.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref]

Scully, M. O.

M. Sargent, M. O. Scully, and W. E. Lamb, “Coherent states,” in Laser Physics, 6th ed. (Westview, 1993), Ch. XV, pp. 242–255.

M. Sargent, M. O. Scully, and W. E. Lamb, “Dipole oscillators,” in Laser Physics, 6th ed. (Westview, 1993), Ch. 3.2, pp. 34–42.

Shih, M. L.

M. Cray, M. L. Shih, and P. W. Milonni, “Stimulated emission, absorption, and interference,” Am. J. Phys. 50, 1016–1021 (1982).
[Crossref]

Shirvanyan, V.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref]

Siegman, A. E.

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
[Crossref]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[Crossref]

A. E. Siegman, “Oscillation dynamics and oscillation threshold,” in Lasers (University Science Books, 1986), Ch. 13, pp. 510–524.

Sommer, A.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref]

Sommers, H. S.

H. S. Sommers, “Threshold and oscillation of injection lasers: a critical review of laser theory,” Solid State Electron. 25, 25–44 (1982).
[Crossref]

H. S. Sommers, “Spontaneous power and the coherent state of injection lasers,” J. Appl. Phys. 45, 1787–1793 (1974).
[Crossref]

Townes, C. H.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[Crossref]

van der Lee, A. M.

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357–4360 (1997).
[Crossref]

van Druten, N. J.

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357–4360 (1997).
[Crossref]

van Eijkelenborg, M. A.

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357–4360 (1997).
[Crossref]

van Exter, M. P.

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357–4360 (1997).
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Weisskopf, V. F.

V. F. Weisskopf, “How light interacts with matter,” Sci. Am. 219, 60–71 (1968).
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Welling, H.

H. Gerhardt, H. Welling, and A. Güttner, “Observation of quantum-phase and quantum-amplitude noise for a laser below and above threshold,” Phys. Lett. A 40, 191–193 (1972).
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H. Gerhardt, H. Welling, and A. Güttner, “Measurements of the laser linewidth due to quantum phase and quantum amplitude noise above and below threshold. I,” Z. Phys. 253, 113–126 (1972).
[Crossref]

Woerdman, J. P.

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357–4360 (1997).
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W. A. Hamel and J. P. Woerdman, “Nonorthogonality of the longitudinal eigenmodes of a laser,” Phys. Rev. A 40, 2785–2787 (1989).
[Crossref]

Worschech, L.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nat. Phys. 3, 106–110 (2007).
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Yabana, K.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
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Yakovlev, V. S.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
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G. Björk and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. 27, 2386–2396 (1991).
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M. Motsch, M. Zeppenfeld, P. W. H. Pinkse, and G. Rempe, “Cavity-enhanced Rayleigh scattering,” New J. Phys. 12, 063022 (2010).
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Am. J. Phys. (1)

M. Cray, M. L. Shih, and P. W. Milonni, “Stimulated emission, absorption, and interference,” Am. J. Phys. 50, 1016–1021 (1982).
[Crossref]

Ann. Phys. (2)

M. Planck, “Ueber das Gesetz der Energieverteilung im Normalspectrum,” Ann. Phys. 309, 553–563 (1901).
[Crossref]

H. A. Lorentz, “Über die Beziehung zwischen der Fortpflanzungsgeschwindigkeit des Lichtes und der Körperdichte,” Ann. Phys. 245, 641–665 (1880).
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G. Björk and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. 27, 2386–2396 (1991).
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IEEE J. Sel. Top. Quantum Electron. (1)

M. Eichhorn and M. Pollnau, “Spectroscopic foundations of lasers: spontaneous emission into a resonator mode,” IEEE J. Sel. Top. Quantum Electron. 21, 9000216 (2015).
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J. Appl. Phys. (1)

H. S. Sommers, “Spontaneous power and the coherent state of injection lasers,” J. Appl. Phys. 45, 1787–1793 (1974).
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J. Opt. (1)

T. V. Raziman and O. J. F. Martin, “Does the real part contain all the physical information?” J. Opt. 18, 095002 (2016).
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J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

Nat. Phys. (1)

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nat. Phys. 3, 106–110 (2007).
[Crossref]

Nature (1)

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light-matter energy transfer in solids,” Nature 534, 86–90 (2016).
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New J. Phys. (1)

M. Motsch, M. Zeppenfeld, P. W. H. Pinkse, and G. Rempe, “Cavity-enhanced Rayleigh scattering,” New J. Phys. 12, 063022 (2010).
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M. Gross and S. Haroche, “Superradiance: an essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982).
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Phys. Rev. (7)

R. D. Hempstead and M. Lax, “Classical noise. VI. Noise in self-sustained oscillators near threshold,” Phys. Rev. 161, 350–366 (1967).
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R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[Crossref]

M. Lax, “Classical noise. V. Noise in self-sustained oscillators,” Phys. Rev. 160, 290–307 (1967).
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H. P. Robertson, “The uncertainty principle,” Phys. Rev. 34, 163–164 (1929).
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R. H. Dieke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
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J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
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Phys. Rev. A (3)

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
[Crossref]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[Crossref]

W. A. Hamel and J. P. Woerdman, “Nonorthogonality of the longitudinal eigenmodes of a laser,” Phys. Rev. A 40, 2785–2787 (1989).
[Crossref]

Phys. Rev. Lett. (1)

A. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg, M. P. van Exter, and J. P. Woerdman, “Excess quantum noise due to nonorthogonal polarization modes,” Phys. Rev. Lett. 79, 4357–4360 (1997).
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H. Gerhardt, H. Welling, and A. Güttner, “Measurements of the laser linewidth due to quantum phase and quantum amplitude noise above and below threshold. I,” Z. Phys. 253, 113–126 (1972).
[Crossref]

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M. Eichhorn and M. Pollnau are preparing a manuscript to be called “Laser linewidth, spectral coherence, and the laser eigenvalue.”

M. Pollnau is preparing a manuscript to be called “Reconciling Einstein, Planck, and Heisenberg: the factor of two in vacuum energy and laser linewidth.”

G. Bruhat and A. Kastler, “Interférences de la lumière,” in Optique, 6th ed. (Masson, 1965), pp. 60–61, 95–96, and 175.

M. Sargent, M. O. Scully, and W. E. Lamb, “Dipole oscillators,” in Laser Physics, 6th ed. (Westview, 1993), Ch. 3.2, pp. 34–42.

P. W. Milonni and J. H. Eberly, “The electron oscillator model,” in Lasers (Wiley, 1988), Ch. 2.2, pp. 27–33.

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H. A. Kramers, “La diffusion de la lumière par les atoms,” in Atti del Congresso Internazionale del Fisica (Transactions of Volta Centenary Congress) Como (1927), Vol. 2, pp. 545–557.

M. Sargent, M. O. Scully, and W. E. Lamb, “Coherent states,” in Laser Physics, 6th ed. (Westview, 1993), Ch. XV, pp. 242–255.

A. E. Siegman, “Oscillation dynamics and oscillation threshold,” in Lasers (University Science Books, 1986), Ch. 13, pp. 510–524.

L. I. Schiff, “Interaction between charged particles and the electromagnetic field,” in Quantum Mechanics, 3rd ed. (McGraw-Hill, 1968), Sec. 57, pp. 531–533.

L. I. Schiff, “Absorption and induced emission,” in Quantum Mechanics, 3rd ed. (McGraw-Hill, 1968), Sec. 44, p. 398–406.

E. Fermi, Nuclear Physics (The University of Chicago, 1950), formula VIII.2.

R. P. Feynman, R. B. Leighton, and M. Sands, “The origin of the refractive index,” in The Feynman Lectures on Physics (Addison-Wesley, 1963), Vol. I, Ch. 31, pp. 31–34.

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Figures (8)

Fig. 1.
Fig. 1. (a) Amplitude–phase diagram visualizing the interpretation of quantum noise and laser linewidth by Lax [9], Haken [10], and Henry [11]. Quantum noise in a laser is said to be induced by adding with an arbitrary phase difference θ a spontaneously emitted photon (red dashed–dotted arrow) of intensity 1 to the intra-cavity laser field (red solid arrows) of intensity I and phase ϕ, resulting in an intra-cavity laser field of intensity I+ΔI (orange solid arrow) and inducing a phase shift Δϕ. Below laser threshold, all phase differences θ are proposed to generate noise, whereas, above laser threshold, amplitude fluctuations a (θ=0 or π, i.e., the projection of noise onto the direction of the green dashed arrow) are rapidly damped out by relaxation oscillations, and only phase fluctuations p (θ=±π/2, i.e., the projection of noise onto the direction of the blue dotted arrow) contribute to noise, thereby reducing the laser linewidth compared to the Schawlow–Townes linewidth [12] by a factor of 2 [911,1315]. (b) Number ϕ of photons resulting from the interference according to Eq. (1) between one photon and 100 photons (red solid curve) versus phase difference θ and medium of 101 photons averaged over all θ (green dashed line).
Fig. 2.
Fig. 2. Schematic of a Fabry–Perot resonator and the relevant electric fields E for (a) light launched from outside [33] and (b) light generated inside the resonator.
Fig. 3.
Fig. 3. (a) Requirement of energy conservation, Aemit=1 (black line), and violation of energy conservation for different potential phase shifts Δϕem=0,π/50,π/30,π/18,π/10,π/6,π/4, and π/2 (see legend) from the field ERT to the field Ecirc induced by interference of ERT with the field Egen generated by stimulated emission, (b) the phase shift Δϕem that is required to obtain energy conservation, and (c) the ratio ϕRT/ϕgen of triggering photon number ϕRT over generated photon number ϕgen as a function of (R1R2)1/2. (d) Phase shift Δϕem induced by stimulated emission as a function of the ratio ϕRT/ϕgen. For ϕRT/ϕgen=1 one obtains Δϕem=π/4 (dashed lines).
Fig. 4.
Fig. 4. Real part χe (solid lines) and imaginary part χe (dashed lines) of the susceptibility, calibrated to χ0, for γe=±0.00333ω0 (blue curves) and γe=±0.01ω0 (red curves) in (a) stimulated emission and (b) absorption. (c) Phase difference θ between the amplitudes of a driven atomic oscillator and its driving electric field as a function of driving frequency. (d) Complex susceptibility χe, calibrated to χ0, as a function of ωext, for the four examples displayed in (a) and (b). For the examples of γe=±0.00333ω0, the arrows indicate the situations of (ωextω0)/ω0=1.38×103, 0 (resonance), and 3.34×103, resulting in θ=3/8π (dotted arrow), π/2 (resonance, dashed arrow), and 3/4π (dashed–dotted arrow), respectively. The phase difference in resonance of θ=π/2 is indicated by the black curved arrow, which points in the direction of increasing ωext.
Fig. 5.
Fig. 5. (a) Quadrant of the amplitude–phase diagram illustrating the process of stimulated emission (with the dark-red arrows of the emitted field pointing toward the upper left): a field of ϕext photons triggers an atom in its excited state to emit ϕem=1 photon, in the two situations of (i) ϕext=1 and (ii) ϕext=7. In both situations, the indicated right angle is 90°=180°θ, hence θ=90°. The color code denotes the amplitude in units of ϕ1/2, from ϕ=1 photon (dark red) to ϕ=9 photons (violet). The same diagram holds true for absorption (with the dark-red arrows of the absorbed field pointing toward the lower right). (b) Build-up of a light beam by the consecutive addition of single photons in the amplitude–phase diagram.
Fig. 6.
Fig. 6. Quadrant of the amplitude–phase diagram comparing the simultaneous, independent addition of several photons (dashed arrows; here, a field of ϕext=5 photons triggers atoms in their excited state to simultaneously emit ϕem=2 photons), resulting in a phase shift Δϕem, with the consecutive addition of two single photons (solid arrows), resulting in an accumulated phase shift Δϕn>Δϕem.
Fig. 7.
Fig. 7. (a) Example of the summation according to Eq. (1) of two electric fields of amplitudes equivalent to ϕ1/2=4 and 1, with a phase difference of θ=π/2. The phase shift Δϕem=0.1476π is equal to Eq. (10), and the energy is conserved, as 16+1=17 photons emerge. (b) Consecutive addition of single photons to an existing electromagnetic field. Intensity of the light beam in units of ϕ. The black dashed line calculated from Eq. (11) indicates the phase shift Δϕn=Σ(Δϕem) accumulated with increasing number ϕ of photons.
Fig. 8.
Fig. 8. Relation between vacuum fluctuations, violating the conservation of energy, and spontaneous emission, obeying the conservation of energy. (a) Quadrant of the amplitude-phase diagram illustrating the process of (stimulated and spontaneous) emission in the presence of vacuum fluctuations. In the example, an existing field representing 4 photons (solid yellow line and arrow) plus the vacuum fluctuation, i.e., in average 4.5 photons (dashed yellow line and arrow), is increased by the emission of a photon to a field representing 5 photons (solid light-green line and arrow) plus the vacuum fluctuation, i.e., in average 5.5 photons (dashed light-green line and arrow). The dashed black arrows indicate the addition of half a vacuum photon to the field of real photons, which can occur under any phase difference θ with the existing field, resulting in a state on the (yellow or green) circle, but averaging out over many such extremely short-lived events to a state on the (yellow or green) dashed line. Although occurring under all phase differences θ, the dashed black arrows are shown only for the two specific cases of ±90° phase difference, where energy conservation happens not to be violated, equivalently to the intensity resulting from averaging over all phase angles. The dark-red arrow represents the field of the one emitted photon. The phase shift Δϕem is calculated from Eq. (13). (b) Number ϕ of photons resulting from the interference according to Eq. (1) between ½ vacuum photon and 4 photons (yellow solid curve) or 5 photons (green solid curve) versus phase difference θ. Medium of 4.5 photons (yellow dashed line) or 5.5 photons (green dashed line) averaged over all θ. The deviations in amplitude from the average are consistent with the deviations predicted by the yellow and green rings in part (a).

Equations (13)

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ϕI1+2=cϵ02|E1+E2|2=cϵ02[|E1|2+2|E1E2|cos(θ)+|E2|2].
Acirc=Icirc/Ilaun=1(1R1R2)2+4R1R2sin2(ΔϕRT/2),Atrans=Itrans/Iinc=(1R1)(1R2)Acirc,Aback=Iback/Iinc=(1R1)2R2Acirc,Arefl=Irefl/Iinc=[(R1R2)2+4R1R2sin2(ΔϕRT/2)]Acirc,Atrans+Arefl=(Itrans+Irefl)/Iinc=1,
Acirc=Icirc/Igen=1(1R1R2)2+4R1R2sin2[(ΔϕRT+Δϕem)/2],Atrans=Itrans/Igen=(1R2)Acirc,Aback=Iback/Igen=(1R1)R2Acirc,Aemit=Atrans+Aback=(Itrans+Iback)/Igen=(1R1R2)Acirc,Aemit>1for  ΔϕRT=Δϕem=0.
(1R1R2)Acirc=12sin2(Δϕem/2)=1R1R2cos(Δϕem)=R1R2tan(Δϕem)=1cos2(Δϕem)cos2(Δϕem)=1R1R2R1R2
ϕRT/ϕgen=IRT/Igen=ART=R1R2Acirc=R1R2(1R1R2)2+4R1R2sin2[(ΔϕRT+Δϕem)/2]=R1R21R1R2tan(Δϕem)=ϕgen/ϕRT,
θ=πarctanϕgen/ϕRTarctanϕRT/ϕgen=π/2.
mex¨(t)+2γemex˙(t)+ω02mex(t)=eEextexp(iωextt),Pe=Neex(t)=Nee2/meω02ωext2i2γeωextEextexp(iωextt+θ),tan(θ)=2γeωextω02ωext2.
Pe=ϵ0χeEextexp(iωextt+θ),χe=χ0ω02ω02ωext2i2γeωext,χ0=Nee2ϵ0meω02,χe=χ0(ω02ωext2)ω02(ω02ωext2)2+(2γeωext)2,χe=χ02γeωextω02(ω02ωext2)2+(2γeωext)2,tan(θ)=χeχe=2γeωextω02ωext2.
ddtϕ=RstRdecay=cgϕ1τcϕ.
tan(Δϕem)=1/ϕext.
Δϕn=i=1n1Δϕem,i=i=1n1arctan(1/i)for  n1.
Δϕem(ϕem>1)<Δϕn(n=ϕem>1).
tan(Δϕem)=1/(ϕext+1/2).

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