Abstract

We provide a simple semi-classical formalism to describe the coupling between one or several quantum emitters and a structured environment. Describing the emitter by an electric polarizability, and the surrounding medium by a Green function, we show that an intuitive scattering picture allows one to derive a coupling equation from which the eigenfrequencies of the coupled system can be extracted. The model covers a variety of regimes observed in light–matter interaction, including weak and strong coupling, coherent collective interactions, and incoherent energy transfer. It provides a unified description of many processes, showing that different interaction regimes are actually rooted on the same ground. It can also serve as a basis for the development of more refined models in a full quantum electrodynamics framework.

© 2019 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2018 (1)

B. Barnes, F. G. Vidal, and J. Aizpurua, “Special issue on strong coupling of molecules to cavities,” ACS Photon. 5, 1 (2018).
[Crossref]

2016 (3)

D. Bouchet, D. Cao, R. Carminati, Y. De Wilde, and V. Krachmalnicoff, “Long-range plasmon-assisted energy transfer between fluorescent emitters,” Phys. Rev. Lett. 116, 037401 (2016).
[Crossref]

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

2015 (2)

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

A. F. Koenderink, A. Alù, and A. Polman, “Nanophotonics: shrinking light-based technology,” Science 348, 516–521 (2015).
[Crossref]

2011 (3)

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[Crossref]

G. D. Scholes, G. R. Fleming, A. Olaya-Castro, and R. van Grondelle, “Lessons from nature about solar light harvesting,” Nat. Chem. 3, 763–774 (2011).
[Crossref]

R. Vincent and R. Carminati, “Magneto-optical control of Förster energy transfer,” Phys. Rev. B 83, 165426 (2011).
[Crossref]

2008 (1)

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[Crossref]

2004 (2)

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and superradiance in inhomogeneous dielectrics,” Phys. Rev. A 70, 053823 (2004).
[Crossref]

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

2002 (2)

H. T. Dung, L. Knöll, and D.-G. Welsch, “Resonant dipole-dipole interaction in the presence of dispersing and absorbing surroundings,” Phys. Rev. A 66, 063810 (2002).
[Crossref]

H. T. Dung, L. Knöll, and D.-G. Welsch, “Intermolecular energy transfer in the presence of dispersing and absorbing media,” Phys. Rev. A 65, 043813 (2002).
[Crossref]

2000 (1)

P. R. Selvin, “The renaissance of fluorescence resonance energy transfer,” Nat. Struct. Mol. Biol. 7, 730–734 (2000).
[Crossref]

1998 (1)

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998).
[Crossref]

1992 (1)

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314–3317 (1992).
[Crossref]

1984 (1)

G. S. Agarwal, “Vacuum-field Rabi splittings in microwave absorption by Rydberg atoms in a cavity,” Phys. Rev. Lett. 53, 1732–1734 (1984).
[Crossref]

1982 (1)

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

1970 (1)

K. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 1, 693–701 (1970).
[Crossref]

1951 (1)

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
[Crossref]

1948 (1)

T. Förster, “Zwischenmolekulare Energiewanderung und Fluoreszenz,” Ann. Phys. 437, 55–75 (1948).
[Crossref]

1946 (1)

E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

1945 (1)

L. L. Foldy, “The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945).
[Crossref]

Agarwal, G. S.

G. S. Agarwal, “Vacuum-field Rabi splittings in microwave absorption by Rydberg atoms in a cavity,” Phys. Rev. Lett. 53, 1732–1734 (1984).
[Crossref]

G. S. Agarwal, Quantum Optics: Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches (Springer, 1974).

Aizpurua, J.

B. Barnes, F. G. Vidal, and J. Aizpurua, “Special issue on strong coupling of molecules to cavities,” ACS Photon. 5, 1 (2018).
[Crossref]

Allen, L.

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

Alù, A.

A. F. Koenderink, A. Alù, and A. Polman, “Nanophotonics: shrinking light-based technology,” Science 348, 516–521 (2015).
[Crossref]

Arakawa, Y.

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314–3317 (1992).
[Crossref]

Aspect, A.

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics (Cambridge University, 2010).

Atikian, H. A.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Barnes, B.

B. Barnes, F. G. Vidal, and J. Aizpurua, “Special issue on strong coupling of molecules to cavities,” ACS Photon. 5, 1 (2018).
[Crossref]

Barrow, S. J.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Baumberg, J. J.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Benz, F.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Berman, P.

P. Berman, Cavity Quantum Electrodynamics (Academic, 1994).

Bhaskar, M. K.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Bielejec, E.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Bohren, C. F.

C. F. Bohren and R. D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

Borregaard, J.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Bouchet, D.

D. Bouchet, D. Cao, R. Carminati, Y. De Wilde, and V. Krachmalnicoff, “Long-range plasmon-assisted energy transfer between fluorescent emitters,” Phys. Rev. Lett. 116, 037401 (2016).
[Crossref]

Burek, M. J.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Camacho, R. M.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Cao, D.

D. Bouchet, D. Cao, R. Carminati, Y. De Wilde, and V. Krachmalnicoff, “Long-range plasmon-assisted energy transfer between fluorescent emitters,” Phys. Rev. Lett. 116, 037401 (2016).
[Crossref]

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

Carmichael, H. J.

H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations, Theoretical and Mathematical Physics (Springer, 1999).

Carminati, R.

D. Bouchet, D. Cao, R. Carminati, Y. De Wilde, and V. Krachmalnicoff, “Long-range plasmon-assisted energy transfer between fluorescent emitters,” Phys. Rev. Lett. 116, 037401 (2016).
[Crossref]

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

R. Vincent and R. Carminati, “Magneto-optical control of Förster energy transfer,” Phys. Rev. B 83, 165426 (2011).
[Crossref]

Cazé, A.

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

Chikkaraddy, R.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, J. Dunpont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications (Wiley, 2010).

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1991), Vol. 1.

de Nijs, B.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

de Vries, P.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998).
[Crossref]

De Wilde, Y.

D. Bouchet, D. Cao, R. Carminati, Y. De Wilde, and V. Krachmalnicoff, “Long-range plasmon-assisted energy transfer between fluorescent emitters,” Phys. Rev. Lett. 116, 037401 (2016).
[Crossref]

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

Demetriadou, A.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Deppe, D. G.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

Dignam, M. M.

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[Crossref]

Diu, B.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1991), Vol. 1.

Drexhage, K.

K. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 1, 693–701 (1970).
[Crossref]

Dung, H. T.

H. T. Dung, L. Knöll, and D.-G. Welsch, “Resonant dipole-dipole interaction in the presence of dispersing and absorbing surroundings,” Phys. Rev. A 66, 063810 (2002).
[Crossref]

H. T. Dung, L. Knöll, and D.-G. Welsch, “Intermolecular energy transfer in the presence of dispersing and absorbing media,” Phys. Rev. A 65, 043813 (2002).
[Crossref]

Dunpont-Roc, J.

C. Cohen-Tannoudji, J. Dunpont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications (Wiley, 2010).

Eberly, J. H.

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

Ell, C.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

Evans, R. E.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Fabre, C.

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics (Cambridge University, 2010).

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, 1st ed. (McGraw-Hill, 1953).

Fleming, G. R.

G. D. Scholes, G. R. Fleming, A. Olaya-Castro, and R. van Grondelle, “Lessons from nature about solar light harvesting,” Nat. Chem. 3, 763–774 (2011).
[Crossref]

Foldy, L. L.

L. L. Foldy, “The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945).
[Crossref]

Förster, T.

T. Förster, “Zwischenmolekulare Energiewanderung und Fluoreszenz,” Ann. Phys. 437, 55–75 (1948).
[Crossref]

Fox, P.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Fussell, D. P.

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[Crossref]

Gibbs, H. M.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

Gross, M.

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

Grynberg, G.

C. Cohen-Tannoudji, J. Dunpont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications (Wiley, 2010).

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics (Cambridge University, 2010).

Haroche, S.

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

S. Haroche, “Cavity quantum electrodynamics,” in Fundamental Systems in Quantum Optics, J. Dalibard, J. Raimond, and J. Zinn-Justin, eds. (Elsevier, 1992), pp. 767–940.

S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford University, 2013).

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

L. Novotny and B. Hecht, “Dipole-dipole interactions and energy transfer,” in Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012), pp. 256–264. See Eqs. (8.159) and (8.160) for the expression of the energy transfer rate.

Hendrickson, J.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

Hess, O.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Hildebrandt, N.

I. Medintz and N. Hildebrandt, FRET—Förster Resonance Energy Transfer: From Theory to Applications (Wiley, 2013).

Huffman, R. D.

C. F. Bohren and R. D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

Hughes, S.

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[Crossref]

Ishikawa, A.

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314–3317 (1992).
[Crossref]

Jelezko, F.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Kavokin, A.

A. Kavokin and G. Malpuech, Cavity Polaritons (Elsevier, 2003).

Khitrova, G.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

Knöll, L.

H. T. Dung, L. Knöll, and D.-G. Welsch, “Intermolecular energy transfer in the presence of dispersing and absorbing media,” Phys. Rev. A 65, 043813 (2002).
[Crossref]

H. T. Dung, L. Knöll, and D.-G. Welsch, “Resonant dipole-dipole interaction in the presence of dispersing and absorbing surroundings,” Phys. Rev. A 66, 063810 (2002).
[Crossref]

Koenderink, A. F.

A. F. Koenderink, A. Alù, and A. Polman, “Nanophotonics: shrinking light-based technology,” Science 348, 516–521 (2015).
[Crossref]

Krachmalnicoff, V.

D. Bouchet, D. Cao, R. Carminati, Y. De Wilde, and V. Krachmalnicoff, “Long-range plasmon-assisted energy transfer between fluorescent emitters,” Phys. Rev. Lett. 116, 037401 (2016).
[Crossref]

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

Lagendijk, A.

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and superradiance in inhomogeneous dielectrics,” Phys. Rev. A 70, 053823 (2004).
[Crossref]

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998).
[Crossref]

Laloe, F.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1991), Vol. 1.

Lax, M.

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
[Crossref]

Loncar, M.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Lukin, M. D.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Malpuech, G.

A. Kavokin and G. Malpuech, Cavity Polaritons (Elsevier, 2003).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Medintz, I.

I. Medintz and N. Hildebrandt, FRET—Förster Resonance Energy Transfer: From Theory to Applications (Wiley, 2013).

Meuwly, C.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, 1st ed. (McGraw-Hill, 1953).

Nguyen, C. T.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Nishioka, M.

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314–3317 (1992).
[Crossref]

Novotny, L.

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[Crossref]

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

L. Novotny and B. Hecht, “Dipole-dipole interactions and energy transfer,” in Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012), pp. 256–264. See Eqs. (8.159) and (8.160) for the expression of the energy transfer rate.

Olaya-Castro, A.

G. D. Scholes, G. R. Fleming, A. Olaya-Castro, and R. van Grondelle, “Lessons from nature about solar light harvesting,” Nat. Chem. 3, 763–774 (2011).
[Crossref]

Pacheco, J. L.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Park, H.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Peragut, F.

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

Pierrat, R.

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

Polman, A.

A. F. Koenderink, A. Alù, and A. Polman, “Nanophotonics: shrinking light-based technology,” Science 348, 516–521 (2015).
[Crossref]

Purcell, E.

E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Raimond, J.-M.

S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford University, 2013).

Rosta, E.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Rupper, G.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

Scherer, A.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

Scherman, O. A.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Scholes, G. D.

G. D. Scholes, G. R. Fleming, A. Olaya-Castro, and R. van Grondelle, “Lessons from nature about solar light harvesting,” Nat. Chem. 3, 763–774 (2011).
[Crossref]

Selvin, P. R.

P. R. Selvin, “The renaissance of fluorescence resonance energy transfer,” Nat. Struct. Mol. Biol. 7, 730–734 (2000).
[Crossref]

Shchekin, O. B.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

Sipahigil, A.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Sukachev, D. D.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Suttorp, L. G.

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and superradiance in inhomogeneous dielectrics,” Phys. Rev. A 70, 053823 (2004).
[Crossref]

van Coevorden, D. V.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998).
[Crossref]

van Grondelle, R.

G. D. Scholes, G. R. Fleming, A. Olaya-Castro, and R. van Grondelle, “Lessons from nature about solar light harvesting,” Nat. Chem. 3, 763–774 (2011).
[Crossref]

van Hulst, N.

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[Crossref]

Vidal, F. G.

B. Barnes, F. G. Vidal, and J. Aizpurua, “Special issue on strong coupling of molecules to cavities,” ACS Photon. 5, 1 (2018).
[Crossref]

Vincent, R.

R. Vincent and R. Carminati, “Magneto-optical control of Förster energy transfer,” Phys. Rev. B 83, 165426 (2011).
[Crossref]

Weisbuch, C.

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314–3317 (1992).
[Crossref]

Welsch, D.-G.

H. T. Dung, L. Knöll, and D.-G. Welsch, “Resonant dipole-dipole interaction in the presence of dispersing and absorbing surroundings,” Phys. Rev. A 66, 063810 (2002).
[Crossref]

H. T. Dung, L. Knöll, and D.-G. Welsch, “Intermolecular energy transfer in the presence of dispersing and absorbing media,” Phys. Rev. A 65, 043813 (2002).
[Crossref]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Wubs, M.

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and superradiance in inhomogeneous dielectrics,” Phys. Rev. A 70, 053823 (2004).
[Crossref]

Yoshie, T.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

ACS Photon. (1)

B. Barnes, F. G. Vidal, and J. Aizpurua, “Special issue on strong coupling of molecules to cavities,” ACS Photon. 5, 1 (2018).
[Crossref]

Ann. Phys. (1)

T. Förster, “Zwischenmolekulare Energiewanderung und Fluoreszenz,” Ann. Phys. 437, 55–75 (1948).
[Crossref]

J. Lumin. (1)

K. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 1, 693–701 (1970).
[Crossref]

Nat. Chem. (1)

G. D. Scholes, G. R. Fleming, A. Olaya-Castro, and R. van Grondelle, “Lessons from nature about solar light harvesting,” Nat. Chem. 3, 763–774 (2011).
[Crossref]

Nat. Photonics (1)

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[Crossref]

Nat. Struct. Mol. Biol. (1)

P. R. Selvin, “The renaissance of fluorescence resonance energy transfer,” Nat. Struct. Mol. Biol. 7, 730–734 (2000).
[Crossref]

Nature (2)

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Phys. Rep. (1)

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

Phys. Rev. (2)

E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

L. L. Foldy, “The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945).
[Crossref]

Phys. Rev. A (3)

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and superradiance in inhomogeneous dielectrics,” Phys. Rev. A 70, 053823 (2004).
[Crossref]

H. T. Dung, L. Knöll, and D.-G. Welsch, “Resonant dipole-dipole interaction in the presence of dispersing and absorbing surroundings,” Phys. Rev. A 66, 063810 (2002).
[Crossref]

H. T. Dung, L. Knöll, and D.-G. Welsch, “Intermolecular energy transfer in the presence of dispersing and absorbing media,” Phys. Rev. A 65, 043813 (2002).
[Crossref]

Phys. Rev. B (2)

R. Vincent and R. Carminati, “Magneto-optical control of Förster energy transfer,” Phys. Rev. B 83, 165426 (2011).
[Crossref]

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[Crossref]

Phys. Rev. Lett. (3)

D. Bouchet, D. Cao, R. Carminati, Y. De Wilde, and V. Krachmalnicoff, “Long-range plasmon-assisted energy transfer between fluorescent emitters,” Phys. Rev. Lett. 116, 037401 (2016).
[Crossref]

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314–3317 (1992).
[Crossref]

G. S. Agarwal, “Vacuum-field Rabi splittings in microwave absorption by Rydberg atoms in a cavity,” Phys. Rev. Lett. 53, 1732–1734 (1984).
[Crossref]

Rev. Mod. Phys. (2)

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
[Crossref]

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998).
[Crossref]

Science (2)

A. F. Koenderink, A. Alù, and A. Polman, “Nanophotonics: shrinking light-based technology,” Science 348, 516–521 (2015).
[Crossref]

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Surf. Sci. Rep. (1)

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

Other (16)

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

I. Medintz and N. Hildebrandt, FRET—Förster Resonance Energy Transfer: From Theory to Applications (Wiley, 2013).

G. S. Agarwal, Quantum Optics: Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches (Springer, 1974).

L. Novotny and B. Hecht, “Dipole-dipole interactions and energy transfer,” in Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012), pp. 256–264. See Eqs. (8.159) and (8.160) for the expression of the energy transfer rate.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1991), Vol. 1.

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics (Cambridge University, 2010).

H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations, Theoretical and Mathematical Physics (Springer, 1999).

C. F. Bohren and R. D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, 1st ed. (McGraw-Hill, 1953).

A. Kavokin and G. Malpuech, Cavity Polaritons (Elsevier, 2003).

C. Cohen-Tannoudji, J. Dunpont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications (Wiley, 2010).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

S. Haroche, “Cavity quantum electrodynamics,” in Fundamental Systems in Quantum Optics, J. Dalibard, J. Raimond, and J. Zinn-Justin, eds. (Elsevier, 1992), pp. 767–940.

S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford University, 2013).

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

P. Berman, Cavity Quantum Electrodynamics (Academic, 1994).

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Figures (6)

Fig. 1.
Fig. 1. Jablonski diagram of a three-level system. For Γbc=0, the system reduced to the model of a two-level atom. For ΓbcΓba, the three-level system is the simplest relevant model of a fluorescent molecule. In this case, Γbc corresponds to a fast non-radiative decay towards state |c, and Γca corresponds to the radiative transition.
Fig. 2.
Fig. 2. Representation of the two scattering processes involved in the electrodynamic interaction between a dipole emitter and a structured environment.
Fig. 3.
Fig. 3. (a) Evolution of the eigenfrequencies of the coupled system in the complex plane when increasing the Purcell factor Fm of the cavity. (b) Normalized frequency shift of the two eigenmodes versus the Purcell factor Fm. Error bars represent intervals bounded by ωp±γp/2.
Fig. 4.
Fig. 4. (a) Evolution of the eigenfrequencies in the complex plane when decreasing the distance d between the emitters in free space. (b) Normalized frequency shift of the two eigenmodes versus the distance d. Error bars represent intervals bounded by ωp±γp/2.
Fig. 5.
Fig. 5. Solid line: normalized modification in the donor linewidth due to the presence of the acceptor Γinter/Γ1 versus the distance d between donor and acceptor. Dashed line: normalized energy transfer rate Γet/Γ1 calculated using expression (55).
Fig. 6.
Fig. 6. (a) Evolution of the eigenfrequencies in the complex plane when increasing the Purcell factor F2 of emitter 2. (b) Normalized frequency shift of the three eigenmodes versus the Purcell factor F2. Error bars represent intervals bounded by ωp±γp/2.

Equations (65)

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

dρ^dt=1i[H^,ρ^].
H^0=ωabσ^ab+σ^ab+ωacσ^ac+σ^ac,
H^1=dab·E(t)(σ^ab++σ^ab),
dρ^dt=1i[H^0+H^1,ρ^]+{dρ^dt}relax
dρbbdt=(Γba+Γbc)ρbb+2Im[ρabΩ(+)(t)],
dρccdt=Γcaρcc+Γbcρbb,
dρbadt=γab2ρbaiωabρba+i(ρbbρaa)Ω(+)(t),
Ω(+)(t)=0+Ω(ω)eiωtdω.
ρba(ω)=Ω(+)(ω)ωabωiγab/2(11+s),
s=2(2Γca+Γbc)Γca(Γba+Γbc)+Im[|Ω(+)(ω)|2ωabωiγab/2]dω.
d(+)(ω)=1(1ωabωiγab/2)[dab·E(+)(ω)]dab.
d(+)(ω)=αab(ω)ε0E(+)(ω).
αab(ω)=3πc3ωab3Γbaspωabωiγab/2uu.
Γbasp=ωab3dab23πε0c3.
σe(ω)=3πc22ωab2γabΓbasp(ωabω)2+γab2/4,
σs(ω)=3πc22ωab2(Γbasp)2(ωabω)2+γab2/4.
××G(r,r,ω)ω2c2ε(r,ω)G(r,r,ω)=δ(rr)I
E(r,ω)=μ0ω2G(r,r,ω)d(ω).
G(r,r,ω)=G0(r,r,ω)+S(r,r,ω),
α0(ω)=3πc3ω03Γ0ω0ωiγ0/2.
d(+)(ω)=α0(ω)ε0Eexc(+)(rs,ω)+α0(ω)k02S(rs,rs,ω)d(+)(ω),
d(+)(ω)=α(ω)ε0Eexc(+)(rs,ω).
α(ω)1=α0(ω)1k02S(rs,rs,ω),
ω2c2S(rs,rs,ω)α0(ω)=I.
ω2c2[u·S(rs,rs,ω)u]α0(ω)=1,
S(ϖp)=ω0ϖpiγ0/2,
S(ω)=3πcΓ0ω0[u·S(rs,rs,ω)u].
ϖp=ω0i2γ0S(ω0).
ωp=ω0Re[S(ω0)],
γp=γ0+2Im[S(ω0)].
γpΓ0=1+2Im[S(ω0)]Γ0.
γpΓ0=ρu(rs,ω)ρu,0,
G(r,r,ω)=c22ωmem(r)em*(r)ωmωiγm/2,
S(ω)=FmΓ0γm/4ωmωiγm/2iΓ0/2,
Fm=6πc3ωm2γm|em(rs)·u|2.
1=FmΓ0γm/4(ω0ϖpiγ0/2)(ωmϖpiγm/2)iΓ0/2ω0ϖpiγ0/2.
ϖp±=ϖ0+ϖm2±(ϖmϖ02)2+g2,
ωp±=ω0+ωm2±FmΓ0γm4,
γp±=(γ0Γ0)+γm2.
αi(ω)=3πc3ω03Γiωiωiγi/2for  i=1,2.
Sii(ω)=3πcΓiω0[ui·S(ri,ri,ω)ui],
Gij(ω)=3πcΓiΓjω0[ui·G(ri,rj,ω)uj].
X=(d1(+)(ω)d2(+)(ω)),
Y=(α01(ω)ε0u1·Eexc(+)(r1,ω)α02(ω)ε0u2·Eexc(+)(r2,ω)),
M=(1S11(ω)ω1ωiγ1/2G12(ω)ω1ωiγ1/2G21(ω)ω2ωiγ2/21S22(ω)ω2ωiγ2/2).
0=1S11(ϖp)ω1ϖpiγ1/2S22(ϖp)ω2ϖpiγ2/2+S11(ϖp)S22(ϖp)G122(ϖp)(ω1ϖpiγ1/2)(ω2ϖpiγ2/2).
ϖp±=ϖ1+ϖ22±(ϖ2ϖ12)2+G12(ω0)2,
ωp±=ω1+ω22Re[S11(ω0)+S22(ω0)2]±Re[G12(ω0)],
γp±=γ1+γ22+2Im[S11(ω0)+S22(ω0)2]2Im[G12(ω0)].
ϖp±=ϖ1+ϖ2±(ϖ2ϖ1)2±G12(ω0)2ϖ2ϖ1.
γp+=γ2,
γp=γ1+2Im[S11(ω0)]+4Re[G12(ω0)2]γ2.
σe(ω0)=6πc2ω02Γ2γ2,
σs(ω0)=6πc2ω02(Γ2γ2)2.
Γet=6πΓ1σa(ω0)|u1·G(r1,r2,ω0)u2|2,
Fi=6πc3ωm2γm|em(ri)·ui|2for  i=1,2.
Sii(ω)=FiΓiγm/4ωmωiγm/2iΓi/2,
Gij(ω)2=FiΓiFjΓjγm2/16(ωmωiγm/2)2.
gi=FiΓiγm4for  i=1,2.
(1S(ϖp)G(ϖp)ω0ϖpiγ0/2)N1
×(1S(ϖp)+(N1)G(ϖp)ω0ϖpiγ0/2)=0.
ϖp=ω0i2γ0S(ω0)(N1)G(ω0),
ϖp+=ω0i2γ0S(ω0)+G(ω0),
ϖp±=ϖ0+ϖm2±(ϖmϖ02)2+Ng2,
ϖp=ϖ0,

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