Abstract

Quantum electrodynamics theory of the resonance Raman scattering is developed for an atom in a close proximity to a carbon nanotube. The theory predicts a dramatic enhancement of the Raman intensity in the strong atomic coupling regime to nanotube plasmon near-fields. This resonance scattering is a manifestation of the general electromagnetic surface enhanced Raman scattering effect, and can be used in designing efficient nanotube based optical sensing substrates for single atom detection, precision spontaneous emission control, and manipulation.

© 2015 Optical Society of America

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2014 (4)

M. Peng, H. Xu, and M. Shao, “Ultrasensitive surface-enhanced Raman scattering based gold deposited silicon nanowires,” Appl. Phys. Lett. 104, 193103 (2014).
[Crossref]

I. V. Bondarev and A. V. Meliksetyan, “Possibility for exciton Bose-Einstein condensation in carbon nanotubes,” Phys. Rev. B 89, 045414 (2014).
[Crossref]

M. F. Gelin, I. V. Bondarev, and A. V. Meliksetyan, “Optically promoted bipartite atomic entanglement in hybrid metallic carbon nanotube systems,” J. Chem. Phys. 140, 064301 (2014).
[Crossref] [PubMed]

R. Miura, S. Imamura, R. Ohta, A. Ishii, X. Liu, T. Shimada, S. Iwamoto, Y. Arakawa, and Y. K. Kato, “Ultralow mode-volume photonic crystal nanobeam cavities for high-efficiency coupling to individual carbon nanotube emitters,” Nature Commun. 5, 5580 (2014).
[Crossref]

2013 (6)

M. F. Gelin, I. V. Bondarev, and A. V. Meliksetyan, “Monitoring bipartite entanglement in hybrid carbon nanotube systems via optical 2D photon-echo spectroscopy,” Chem. Phys. 413, 123–131 (2013).
[Crossref]

L. M. Woods, A. Popescu, D. Drosdoff, and I. V. Bondarev, “Dispersive interactions in graphitic nanostructures,” Chem. Phys. 413, 116–122 (2013).
[Crossref]

T. Hertel and I. V. Bondarev, eds., Photophysics of Carbon Nanotubes and Nanotube Composites (Special Issue), Chem. Phys. 413, 1–131 (2013).
[Crossref]

Q. Hao, S. M. Morton, B. Wang, Y. Zhao, L. Jensen, and T. J. Huang, “Tuning surface-enhanced Raman scattering from graphene substrates using the electric field effect and chemical doping,” Appl. Phys. Lett. 102, 011102 (2013).
[Crossref]

R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Luo, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
[Crossref] [PubMed]

D. M. Andrada, H. S. Vieira, M. M. Oliveira, A. P. Santos, L. Yin, R. Saito, M. A. Pimenta, C. Fantini, and C. A. Furtado, “Dramatic increase in the Raman signal of functional groups on carbon nanotube surfaces,” Carbon 56, 235–242 (2013).
[Crossref]

2012 (3)

R. Lv, Q. Li, A. R. Botello-Mendez, T. Hayashi, B. Wang, A. Berkdemir, Q. Hao, A. L. Elias, R. Cruz-Silva, H. R. Gutierrez, Y. A. Kim, H. Muramatsu, J. Zhu, M. Endo, H. Terrones, J.-C. Charlie, M. Pan, and M. Terrones, “Nitrogen-doped graphene: beyond single substitution and enhanced molecular sensing,” Sci. Rep. 2, 586 (2012).
[Crossref] [PubMed]

I. V. Bondarev, “Single-wall carbon nanotubes as coherent plasmon generators,” Phys. Rev. B 85, 035448 (2012).
[Crossref]

I. V. Bondarev and T. Antonijevic, “Surface plasmon amplification under controlled exciton-plasmon coupling in individual carbon nanotubes,” Phys. Stat. Sol. C 9, 1259–1264 (2012).
[Crossref]

2011 (3)

A. Popescu, L. M. Woods, and I. V. Bondarev, “Chirality dependent carbon nanotube interactions,” Phys. Rev. B 83, 081406 (2011).
[Crossref]

D. Z. Lin, Y. P. Chen, P. J. Jhuang, J. Y. Chu, J. T. Yeh, and J.-K. Wang, “Optimizing electromagnetic enhancement of flexible nano-imprinted hexagonally patterned surface-enhanced Raman scattering substrates,” Opt. Express 19, 4337–4345 (2011).
[Crossref] [PubMed]

Y.-C. Chen, R. J. Young, J. V. Macpherson, and N. R. Wilson, “Silver-decorated carbon nanotube networks as SERS substrates,” J. Raman Spectrosc. 42, 1255–1256 (2011).
[Crossref]

2010 (4)

Y. Sun, K. Liu, J. Miao, Z. Wang, B. Tian, and L. Zhang, “Highly sensitive surface-enhanced Raman scattering substrate made from superaligned carbon nanotubes,” NanoLett. 10, 1747–1753 (2010).
[Crossref]

I. V. Bondarev, “Surface electromagnetic phenomena in pristine and atomically doped carbon nanotubes,” J. Comp. Theor. Nanoscience 7, 1673–1687 (2010).
[Crossref]

M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single-quantum-dot-nanocavity system,” Nature Phys. 6, 279–283 (2010).
[Crossref]

E. Gaufrès, N. Izard, X. Le Roux, S. Kazaoui, D. Marris-Morini, E. Cassan, and L. Vivien, “Optical microcavity with semiconducting single-wall carbon nanotubes,” Opt. Express 18, 5740–5745 (2010).
[Crossref] [PubMed]

2009 (2)

2008 (2)

N. Noginova, G. Zhu, M. Mavy, and M. A. Noginov, “Magnetic dipole based systems for probing optical magnetism,” J. Appl. Phys. 103, 07E901 (2008).
[Crossref]

F. Xia, M. Steiner, Y.-M. Lin, and Ph. Avouris, “A microcavity-controlled, current-driven, on-chip nanotube emitter at infrared wavelengths,” Nature Nanotechn. 3, 609–613 (2008).
[Crossref]

2007 (1)

I. V. Bondarev and B. Vlahovic, “Entanglement of a pair of atomic qubits near a carbon nanotube,” Phys. Rev. B 75, 033402 (2007).
[Crossref]

2006 (1)

I. V. Bondarev and B. Vlahovic, “Optical absorption by atomically doped carbon nanotubes,” Phys. Rev. B 74, 073401 (2006).
[Crossref]

2005 (4)

I. V. Bondarev and Ph. Lambin, “van der Waals coupling in atomically doped carbon nanotubes,” Phys. Rev. B 72, 035451 (2005).
[Crossref]

T. Ando, “Theory of electronic states and transport in carbon nanotubes,” J. Phys. Soc. Jpn. 74, 777–817 (2005).
[Crossref]

A. Otto, “The chemical (electronic) contribution to surface enhanced Raman scattering,” J. Raman Spectrosc. 36, 497–509 (2005).
[Crossref]

L. Jensen, L. L. Zhao, J. Autschbach, and G. C. Schatz, “Theory and method for calculating resonance Raman scattering from resonance polarizability derivatives,” J. Chem. Phys. 123, 174110 (2005).
[Crossref] [PubMed]

2004 (1)

I. V. Bondarev and Ph. Lambin, “Spontaneous-decay dynamics in atomically doped carbon nanotubes,” Phys. Rev. B 70, 035407 (2004).
[Crossref]

2003 (3)

A. G. Marinopoulos, L. Reining, A. Rubio, and N. Vast, “Optical and loss spectra of carbon nanotubes: Depolarization effects and intertube interactions,” Phys. Rev. Lett. 91, 046402 (2003).
[Crossref] [PubMed]

V. Delgado and J. M. Gomez Llorente, “Weak-coupling-like time evolution of driven four-level systems in the strong-coupling regime,” Phys. Rev. A 68, 022503 (2003).
[Crossref]

J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, and H. J. Kimble, “Experimental realization of a one-atom laser in the regime of strong coupling,” Nature 425, 268–271 (2003).
[Crossref] [PubMed]

2002 (2)

H. Schniepp and V. Sandoghdar, “Spontaneous emission of europium ions embedded in dielectric nanospheres,” Phys. Rev. Lett. 89, 257403 (2002).
[Crossref] [PubMed]

I. V. Bondarev, G.Ya. Slepyan, and S. A. Maksimenko, “Spontaneous decay of excited atomic states near a carbon nanotube,” Phys. Rev. Lett. 89, 115504 (2002).
[Crossref] [PubMed]

2001 (1)

Z. M. Li, Z. K. Tang, H. J. Liu, N. Wang, C. T. Chan, R. Saito, S. Okada, G. D. Li, J. S. Chen, N. Nagasawa, and S. Tsuda, “Polarized absorption spectra of single-walled 4 Å carbon nanotubes aligned in channels of an AlPO4–5 single crystal,” Phys. Rev. Lett. 87, 127401 (2001).
[Crossref]

1999 (1)

1998 (2)

T. Pichler, M. Knupfer, M. S. Golden, J. Fink, A. Rinzler, and R. E. Smalley, “Localized and delocalized electronic states in single-wall carbon nanotubes,” Phys. Rev. Lett. 80, 4729–4732 (1998).
[Crossref]

S. Tasaki, K. Maekawa, and T. Yamabe, “π-band contribution to the optical properties of carbon nanotubes: Effects of chirality,” Phys. Rev. B 57, 9301–9318 (1998).
[Crossref]

Aizpurua, J.

R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Luo, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
[Crossref] [PubMed]

Ando, T.

T. Ando, “Theory of electronic states and transport in carbon nanotubes,” J. Phys. Soc. Jpn. 74, 777–817 (2005).
[Crossref]

Andrada, D. M.

D. M. Andrada, H. S. Vieira, M. M. Oliveira, A. P. Santos, L. Yin, R. Saito, M. A. Pimenta, C. Fantini, and C. A. Furtado, “Dramatic increase in the Raman signal of functional groups on carbon nanotube surfaces,” Carbon 56, 235–242 (2013).
[Crossref]

Antonijevic, T.

I. V. Bondarev and T. Antonijevic, “Surface plasmon amplification under controlled exciton-plasmon coupling in individual carbon nanotubes,” Phys. Stat. Sol. C 9, 1259–1264 (2012).
[Crossref]

Arakawa, Y.

R. Miura, S. Imamura, R. Ohta, A. Ishii, X. Liu, T. Shimada, S. Iwamoto, Y. Arakawa, and Y. K. Kato, “Ultralow mode-volume photonic crystal nanobeam cavities for high-efficiency coupling to individual carbon nanotube emitters,” Nature Commun. 5, 5580 (2014).
[Crossref]

M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single-quantum-dot-nanocavity system,” Nature Phys. 6, 279–283 (2010).
[Crossref]

Autschbach, J.

L. Jensen, L. L. Zhao, J. Autschbach, and G. C. Schatz, “Theory and method for calculating resonance Raman scattering from resonance polarizability derivatives,” J. Chem. Phys. 123, 174110 (2005).
[Crossref] [PubMed]

Avouris, Ph.

F. Xia, M. Steiner, Y.-M. Lin, and Ph. Avouris, “A microcavity-controlled, current-driven, on-chip nanotube emitter at infrared wavelengths,” Nature Nanotechn. 3, 609–613 (2008).
[Crossref]

Berestetskii, V. B.

V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics (Pergamon, 1982).

Berkdemir, A.

R. Lv, Q. Li, A. R. Botello-Mendez, T. Hayashi, B. Wang, A. Berkdemir, Q. Hao, A. L. Elias, R. Cruz-Silva, H. R. Gutierrez, Y. A. Kim, H. Muramatsu, J. Zhu, M. Endo, H. Terrones, J.-C. Charlie, M. Pan, and M. Terrones, “Nitrogen-doped graphene: beyond single substitution and enhanced molecular sensing,” Sci. Rep. 2, 586 (2012).
[Crossref] [PubMed]

Boca, A.

J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, and H. J. Kimble, “Experimental realization of a one-atom laser in the regime of strong coupling,” Nature 425, 268–271 (2003).
[Crossref] [PubMed]

Bogomolov, V. N.

Bondarev, I. V.

I. V. Bondarev and A. V. Meliksetyan, “Possibility for exciton Bose-Einstein condensation in carbon nanotubes,” Phys. Rev. B 89, 045414 (2014).
[Crossref]

M. F. Gelin, I. V. Bondarev, and A. V. Meliksetyan, “Optically promoted bipartite atomic entanglement in hybrid metallic carbon nanotube systems,” J. Chem. Phys. 140, 064301 (2014).
[Crossref] [PubMed]

L. M. Woods, A. Popescu, D. Drosdoff, and I. V. Bondarev, “Dispersive interactions in graphitic nanostructures,” Chem. Phys. 413, 116–122 (2013).
[Crossref]

M. F. Gelin, I. V. Bondarev, and A. V. Meliksetyan, “Monitoring bipartite entanglement in hybrid carbon nanotube systems via optical 2D photon-echo spectroscopy,” Chem. Phys. 413, 123–131 (2013).
[Crossref]

I. V. Bondarev, “Single-wall carbon nanotubes as coherent plasmon generators,” Phys. Rev. B 85, 035448 (2012).
[Crossref]

I. V. Bondarev and T. Antonijevic, “Surface plasmon amplification under controlled exciton-plasmon coupling in individual carbon nanotubes,” Phys. Stat. Sol. C 9, 1259–1264 (2012).
[Crossref]

A. Popescu, L. M. Woods, and I. V. Bondarev, “Chirality dependent carbon nanotube interactions,” Phys. Rev. B 83, 081406 (2011).
[Crossref]

I. V. Bondarev, “Surface electromagnetic phenomena in pristine and atomically doped carbon nanotubes,” J. Comp. Theor. Nanoscience 7, 1673–1687 (2010).
[Crossref]

I. V. Bondarev, L. M. Woods, and K. Tatur, “Strong exciton-plasmon coupling in semiconducting carbon nanotubes,” Phys. Rev. B 80, 085407 (2009).
[Crossref]

I. V. Bondarev and B. Vlahovic, “Entanglement of a pair of atomic qubits near a carbon nanotube,” Phys. Rev. B 75, 033402 (2007).
[Crossref]

I. V. Bondarev and B. Vlahovic, “Optical absorption by atomically doped carbon nanotubes,” Phys. Rev. B 74, 073401 (2006).
[Crossref]

I. V. Bondarev and Ph. Lambin, “van der Waals coupling in atomically doped carbon nanotubes,” Phys. Rev. B 72, 035451 (2005).
[Crossref]

I. V. Bondarev and Ph. Lambin, “Spontaneous-decay dynamics in atomically doped carbon nanotubes,” Phys. Rev. B 70, 035407 (2004).
[Crossref]

I. V. Bondarev, G.Ya. Slepyan, and S. A. Maksimenko, “Spontaneous decay of excited atomic states near a carbon nanotube,” Phys. Rev. Lett. 89, 115504 (2002).
[Crossref] [PubMed]

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Pan, M.

R. Lv, Q. Li, A. R. Botello-Mendez, T. Hayashi, B. Wang, A. Berkdemir, Q. Hao, A. L. Elias, R. Cruz-Silva, H. R. Gutierrez, Y. A. Kim, H. Muramatsu, J. Zhu, M. Endo, H. Terrones, J.-C. Charlie, M. Pan, and M. Terrones, “Nitrogen-doped graphene: beyond single substitution and enhanced molecular sensing,” Sci. Rep. 2, 586 (2012).
[Crossref] [PubMed]

Peng, M.

M. Peng, H. Xu, and M. Shao, “Ultrasensitive surface-enhanced Raman scattering based gold deposited silicon nanowires,” Appl. Phys. Lett. 104, 193103 (2014).
[Crossref]

Petrov, E. P.

Pichler, T.

T. Pichler, M. Knupfer, M. S. Golden, J. Fink, A. Rinzler, and R. E. Smalley, “Localized and delocalized electronic states in single-wall carbon nanotubes,” Phys. Rev. Lett. 80, 4729–4732 (1998).
[Crossref]

Pimenta, M. A.

D. M. Andrada, H. S. Vieira, M. M. Oliveira, A. P. Santos, L. Yin, R. Saito, M. A. Pimenta, C. Fantini, and C. A. Furtado, “Dramatic increase in the Raman signal of functional groups on carbon nanotube surfaces,” Carbon 56, 235–242 (2013).
[Crossref]

Pitaevskii, L. P.

V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics (Pergamon, 1982).

Popescu, A.

L. M. Woods, A. Popescu, D. Drosdoff, and I. V. Bondarev, “Dispersive interactions in graphitic nanostructures,” Chem. Phys. 413, 116–122 (2013).
[Crossref]

A. Popescu, L. M. Woods, and I. V. Bondarev, “Chirality dependent carbon nanotube interactions,” Phys. Rev. B 83, 081406 (2011).
[Crossref]

I. V. Bondarev, L. M. Woods, and A. Popescu, “Exciton-plasmon interactions in individual carbon nanotubes,” in Plasmons: Theory and Applications, K. N. Helsey, ed. (Nova Science, 2011), Ch. 16, pp. 381–435.

Reining, L.

A. G. Marinopoulos, L. Reining, A. Rubio, and N. Vast, “Optical and loss spectra of carbon nanotubes: Depolarization effects and intertube interactions,” Phys. Rev. Lett. 91, 046402 (2003).
[Crossref] [PubMed]

Reitzenstein, S.

Rinzler, A.

T. Pichler, M. Knupfer, M. S. Golden, J. Fink, A. Rinzler, and R. E. Smalley, “Localized and delocalized electronic states in single-wall carbon nanotubes,” Phys. Rev. Lett. 80, 4729–4732 (1998).
[Crossref]

Rogach, A. L.

Rubio, A.

A. G. Marinopoulos, L. Reining, A. Rubio, and N. Vast, “Optical and loss spectra of carbon nanotubes: Depolarization effects and intertube interactions,” Phys. Rev. Lett. 91, 046402 (2003).
[Crossref] [PubMed]

Saito, R.

D. M. Andrada, H. S. Vieira, M. M. Oliveira, A. P. Santos, L. Yin, R. Saito, M. A. Pimenta, C. Fantini, and C. A. Furtado, “Dramatic increase in the Raman signal of functional groups on carbon nanotube surfaces,” Carbon 56, 235–242 (2013).
[Crossref]

Z. M. Li, Z. K. Tang, H. J. Liu, N. Wang, C. T. Chan, R. Saito, S. Okada, G. D. Li, J. S. Chen, N. Nagasawa, and S. Tsuda, “Polarized absorption spectra of single-walled 4 Å carbon nanotubes aligned in channels of an AlPO4–5 single crystal,” Phys. Rev. Lett. 87, 127401 (2001).
[Crossref]

R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Science of Fullerens and Carbon Nanotubes (Imperial College, 1998).

Sandoghdar, V.

H. Schniepp and V. Sandoghdar, “Spontaneous emission of europium ions embedded in dielectric nanospheres,” Phys. Rev. Lett. 89, 257403 (2002).
[Crossref] [PubMed]

Santos, A. P.

D. M. Andrada, H. S. Vieira, M. M. Oliveira, A. P. Santos, L. Yin, R. Saito, M. A. Pimenta, C. Fantini, and C. A. Furtado, “Dramatic increase in the Raman signal of functional groups on carbon nanotube surfaces,” Carbon 56, 235–242 (2013).
[Crossref]

Schatz, G. C.

L. Jensen, L. L. Zhao, J. Autschbach, and G. C. Schatz, “Theory and method for calculating resonance Raman scattering from resonance polarizability derivatives,” J. Chem. Phys. 123, 174110 (2005).
[Crossref] [PubMed]

Schniepp, H.

H. Schniepp and V. Sandoghdar, “Spontaneous emission of europium ions embedded in dielectric nanospheres,” Phys. Rev. Lett. 89, 257403 (2002).
[Crossref] [PubMed]

Shao, M.

M. Peng, H. Xu, and M. Shao, “Ultrasensitive surface-enhanced Raman scattering based gold deposited silicon nanowires,” Appl. Phys. Lett. 104, 193103 (2014).
[Crossref]

Shimada, T.

R. Miura, S. Imamura, R. Ohta, A. Ishii, X. Liu, T. Shimada, S. Iwamoto, Y. Arakawa, and Y. K. Kato, “Ultralow mode-volume photonic crystal nanobeam cavities for high-efficiency coupling to individual carbon nanotube emitters,” Nature Commun. 5, 5580 (2014).
[Crossref]

Slepyan, G.Ya.

I. V. Bondarev, G.Ya. Slepyan, and S. A. Maksimenko, “Spontaneous decay of excited atomic states near a carbon nanotube,” Phys. Rev. Lett. 89, 115504 (2002).
[Crossref] [PubMed]

Smalley, R. E.

T. Pichler, M. Knupfer, M. S. Golden, J. Fink, A. Rinzler, and R. E. Smalley, “Localized and delocalized electronic states in single-wall carbon nanotubes,” Phys. Rev. Lett. 80, 4729–4732 (1998).
[Crossref]

Steiner, M.

F. Xia, M. Steiner, Y.-M. Lin, and Ph. Avouris, “A microcavity-controlled, current-driven, on-chip nanotube emitter at infrared wavelengths,” Nature Nanotechn. 3, 609–613 (2008).
[Crossref]

Sun, Y.

Y. Sun, K. Liu, J. Miao, Z. Wang, B. Tian, and L. Zhang, “Highly sensitive surface-enhanced Raman scattering substrate made from superaligned carbon nanotubes,” NanoLett. 10, 1747–1753 (2010).
[Crossref]

Tang, Z. K.

Z. M. Li, Z. K. Tang, H. J. Liu, N. Wang, C. T. Chan, R. Saito, S. Okada, G. D. Li, J. S. Chen, N. Nagasawa, and S. Tsuda, “Polarized absorption spectra of single-walled 4 Å carbon nanotubes aligned in channels of an AlPO4–5 single crystal,” Phys. Rev. Lett. 87, 127401 (2001).
[Crossref]

Tasaki, S.

S. Tasaki, K. Maekawa, and T. Yamabe, “π-band contribution to the optical properties of carbon nanotubes: Effects of chirality,” Phys. Rev. B 57, 9301–9318 (1998).
[Crossref]

Tatur, K.

I. V. Bondarev, L. M. Woods, and K. Tatur, “Strong exciton-plasmon coupling in semiconducting carbon nanotubes,” Phys. Rev. B 80, 085407 (2009).
[Crossref]

Terrones, H.

R. Lv, Q. Li, A. R. Botello-Mendez, T. Hayashi, B. Wang, A. Berkdemir, Q. Hao, A. L. Elias, R. Cruz-Silva, H. R. Gutierrez, Y. A. Kim, H. Muramatsu, J. Zhu, M. Endo, H. Terrones, J.-C. Charlie, M. Pan, and M. Terrones, “Nitrogen-doped graphene: beyond single substitution and enhanced molecular sensing,” Sci. Rep. 2, 586 (2012).
[Crossref] [PubMed]

Terrones, M.

R. Lv, Q. Li, A. R. Botello-Mendez, T. Hayashi, B. Wang, A. Berkdemir, Q. Hao, A. L. Elias, R. Cruz-Silva, H. R. Gutierrez, Y. A. Kim, H. Muramatsu, J. Zhu, M. Endo, H. Terrones, J.-C. Charlie, M. Pan, and M. Terrones, “Nitrogen-doped graphene: beyond single substitution and enhanced molecular sensing,” Sci. Rep. 2, 586 (2012).
[Crossref] [PubMed]

Tian, B.

Y. Sun, K. Liu, J. Miao, Z. Wang, B. Tian, and L. Zhang, “Highly sensitive surface-enhanced Raman scattering substrate made from superaligned carbon nanotubes,” NanoLett. 10, 1747–1753 (2010).
[Crossref]

Tsuda, S.

Z. M. Li, Z. K. Tang, H. J. Liu, N. Wang, C. T. Chan, R. Saito, S. Okada, G. D. Li, J. S. Chen, N. Nagasawa, and S. Tsuda, “Polarized absorption spectra of single-walled 4 Å carbon nanotubes aligned in channels of an AlPO4–5 single crystal,” Phys. Rev. Lett. 87, 127401 (2001).
[Crossref]

Vast, N.

A. G. Marinopoulos, L. Reining, A. Rubio, and N. Vast, “Optical and loss spectra of carbon nanotubes: Depolarization effects and intertube interactions,” Phys. Rev. Lett. 91, 046402 (2003).
[Crossref] [PubMed]

Vieira, H. S.

D. M. Andrada, H. S. Vieira, M. M. Oliveira, A. P. Santos, L. Yin, R. Saito, M. A. Pimenta, C. Fantini, and C. A. Furtado, “Dramatic increase in the Raman signal of functional groups on carbon nanotube surfaces,” Carbon 56, 235–242 (2013).
[Crossref]

Vivien, L.

Vlahovic, B.

I. V. Bondarev and B. Vlahovic, “Entanglement of a pair of atomic qubits near a carbon nanotube,” Phys. Rev. B 75, 033402 (2007).
[Crossref]

I. V. Bondarev and B. Vlahovic, “Optical absorption by atomically doped carbon nanotubes,” Phys. Rev. B 74, 073401 (2006).
[Crossref]

Wang, B.

Q. Hao, S. M. Morton, B. Wang, Y. Zhao, L. Jensen, and T. J. Huang, “Tuning surface-enhanced Raman scattering from graphene substrates using the electric field effect and chemical doping,” Appl. Phys. Lett. 102, 011102 (2013).
[Crossref]

R. Lv, Q. Li, A. R. Botello-Mendez, T. Hayashi, B. Wang, A. Berkdemir, Q. Hao, A. L. Elias, R. Cruz-Silva, H. R. Gutierrez, Y. A. Kim, H. Muramatsu, J. Zhu, M. Endo, H. Terrones, J.-C. Charlie, M. Pan, and M. Terrones, “Nitrogen-doped graphene: beyond single substitution and enhanced molecular sensing,” Sci. Rep. 2, 586 (2012).
[Crossref] [PubMed]

Wang, J.-K.

Wang, N.

Z. M. Li, Z. K. Tang, H. J. Liu, N. Wang, C. T. Chan, R. Saito, S. Okada, G. D. Li, J. S. Chen, N. Nagasawa, and S. Tsuda, “Polarized absorption spectra of single-walled 4 Å carbon nanotubes aligned in channels of an AlPO4–5 single crystal,” Phys. Rev. Lett. 87, 127401 (2001).
[Crossref]

Wang, Z.

Y. Sun, K. Liu, J. Miao, Z. Wang, B. Tian, and L. Zhang, “Highly sensitive surface-enhanced Raman scattering substrate made from superaligned carbon nanotubes,” NanoLett. 10, 1747–1753 (2010).
[Crossref]

Wilson, N. R.

Y.-C. Chen, R. J. Young, J. V. Macpherson, and N. R. Wilson, “Silver-decorated carbon nanotube networks as SERS substrates,” J. Raman Spectrosc. 42, 1255–1256 (2011).
[Crossref]

Woggon, U.

Woods, L. M.

L. M. Woods, A. Popescu, D. Drosdoff, and I. V. Bondarev, “Dispersive interactions in graphitic nanostructures,” Chem. Phys. 413, 116–122 (2013).
[Crossref]

A. Popescu, L. M. Woods, and I. V. Bondarev, “Chirality dependent carbon nanotube interactions,” Phys. Rev. B 83, 081406 (2011).
[Crossref]

I. V. Bondarev, L. M. Woods, and K. Tatur, “Strong exciton-plasmon coupling in semiconducting carbon nanotubes,” Phys. Rev. B 80, 085407 (2009).
[Crossref]

I. V. Bondarev, L. M. Woods, and A. Popescu, “Exciton-plasmon interactions in individual carbon nanotubes,” in Plasmons: Theory and Applications, K. N. Helsey, ed. (Nova Science, 2011), Ch. 16, pp. 381–435.

Worschech, L.

Xia, F.

F. Xia, M. Steiner, Y.-M. Lin, and Ph. Avouris, “A microcavity-controlled, current-driven, on-chip nanotube emitter at infrared wavelengths,” Nature Nanotechn. 3, 609–613 (2008).
[Crossref]

Xu, H.

M. Peng, H. Xu, and M. Shao, “Ultrasensitive surface-enhanced Raman scattering based gold deposited silicon nanowires,” Appl. Phys. Lett. 104, 193103 (2014).
[Crossref]

Yamabe, T.

S. Tasaki, K. Maekawa, and T. Yamabe, “π-band contribution to the optical properties of carbon nanotubes: Effects of chirality,” Phys. Rev. B 57, 9301–9318 (1998).
[Crossref]

Yang, J. L.

R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Luo, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
[Crossref] [PubMed]

Yarotsky, D. A.

Yeh, J. T.

Yin, L.

D. M. Andrada, H. S. Vieira, M. M. Oliveira, A. P. Santos, L. Yin, R. Saito, M. A. Pimenta, C. Fantini, and C. A. Furtado, “Dramatic increase in the Raman signal of functional groups on carbon nanotube surfaces,” Carbon 56, 235–242 (2013).
[Crossref]

Young, R. J.

Y.-C. Chen, R. J. Young, J. V. Macpherson, and N. R. Wilson, “Silver-decorated carbon nanotube networks as SERS substrates,” J. Raman Spectrosc. 42, 1255–1256 (2011).
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P. Y. Yu and M. Cardona, Fundamentals of Semiconductors, 4. (Springer-Verlag, 2010).
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Zhang, C.

R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Luo, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
[Crossref] [PubMed]

Zhang, L.

R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Luo, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
[Crossref] [PubMed]

Y. Sun, K. Liu, J. Miao, Z. Wang, B. Tian, and L. Zhang, “Highly sensitive surface-enhanced Raman scattering substrate made from superaligned carbon nanotubes,” NanoLett. 10, 1747–1753 (2010).
[Crossref]

Zhang, R.

R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Luo, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
[Crossref] [PubMed]

Zhang, Y.

R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Luo, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
[Crossref] [PubMed]

Zhao, L. L.

L. Jensen, L. L. Zhao, J. Autschbach, and G. C. Schatz, “Theory and method for calculating resonance Raman scattering from resonance polarizability derivatives,” J. Chem. Phys. 123, 174110 (2005).
[Crossref] [PubMed]

Zhao, Y.

Q. Hao, S. M. Morton, B. Wang, Y. Zhao, L. Jensen, and T. J. Huang, “Tuning surface-enhanced Raman scattering from graphene substrates using the electric field effect and chemical doping,” Appl. Phys. Lett. 102, 011102 (2013).
[Crossref]

Zhu, G.

N. Noginova, G. Zhu, M. Mavy, and M. A. Noginov, “Magnetic dipole based systems for probing optical magnetism,” J. Appl. Phys. 103, 07E901 (2008).
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Zhu, J.

R. Lv, Q. Li, A. R. Botello-Mendez, T. Hayashi, B. Wang, A. Berkdemir, Q. Hao, A. L. Elias, R. Cruz-Silva, H. R. Gutierrez, Y. A. Kim, H. Muramatsu, J. Zhu, M. Endo, H. Terrones, J.-C. Charlie, M. Pan, and M. Terrones, “Nitrogen-doped graphene: beyond single substitution and enhanced molecular sensing,” Sci. Rep. 2, 586 (2012).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

M. Peng, H. Xu, and M. Shao, “Ultrasensitive surface-enhanced Raman scattering based gold deposited silicon nanowires,” Appl. Phys. Lett. 104, 193103 (2014).
[Crossref]

Q. Hao, S. M. Morton, B. Wang, Y. Zhao, L. Jensen, and T. J. Huang, “Tuning surface-enhanced Raman scattering from graphene substrates using the electric field effect and chemical doping,” Appl. Phys. Lett. 102, 011102 (2013).
[Crossref]

Carbon (1)

D. M. Andrada, H. S. Vieira, M. M. Oliveira, A. P. Santos, L. Yin, R. Saito, M. A. Pimenta, C. Fantini, and C. A. Furtado, “Dramatic increase in the Raman signal of functional groups on carbon nanotube surfaces,” Carbon 56, 235–242 (2013).
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Chem. Phys. (3)

T. Hertel and I. V. Bondarev, eds., Photophysics of Carbon Nanotubes and Nanotube Composites (Special Issue), Chem. Phys. 413, 1–131 (2013).
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L. M. Woods, A. Popescu, D. Drosdoff, and I. V. Bondarev, “Dispersive interactions in graphitic nanostructures,” Chem. Phys. 413, 116–122 (2013).
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M. F. Gelin, I. V. Bondarev, and A. V. Meliksetyan, “Monitoring bipartite entanglement in hybrid carbon nanotube systems via optical 2D photon-echo spectroscopy,” Chem. Phys. 413, 123–131 (2013).
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J. Appl. Phys. (1)

N. Noginova, G. Zhu, M. Mavy, and M. A. Noginov, “Magnetic dipole based systems for probing optical magnetism,” J. Appl. Phys. 103, 07E901 (2008).
[Crossref]

J. Chem. Phys. (2)

L. Jensen, L. L. Zhao, J. Autschbach, and G. C. Schatz, “Theory and method for calculating resonance Raman scattering from resonance polarizability derivatives,” J. Chem. Phys. 123, 174110 (2005).
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M. F. Gelin, I. V. Bondarev, and A. V. Meliksetyan, “Optically promoted bipartite atomic entanglement in hybrid metallic carbon nanotube systems,” J. Chem. Phys. 140, 064301 (2014).
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J. Comp. Theor. Nanoscience (1)

I. V. Bondarev, “Surface electromagnetic phenomena in pristine and atomically doped carbon nanotubes,” J. Comp. Theor. Nanoscience 7, 1673–1687 (2010).
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J. Lightwave Technol. (1)

J. Phys. Soc. Jpn. (1)

T. Ando, “Theory of electronic states and transport in carbon nanotubes,” J. Phys. Soc. Jpn. 74, 777–817 (2005).
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J. Raman Spectrosc. (2)

A. Otto, “The chemical (electronic) contribution to surface enhanced Raman scattering,” J. Raman Spectrosc. 36, 497–509 (2005).
[Crossref]

Y.-C. Chen, R. J. Young, J. V. Macpherson, and N. R. Wilson, “Silver-decorated carbon nanotube networks as SERS substrates,” J. Raman Spectrosc. 42, 1255–1256 (2011).
[Crossref]

NanoLett. (1)

Y. Sun, K. Liu, J. Miao, Z. Wang, B. Tian, and L. Zhang, “Highly sensitive surface-enhanced Raman scattering substrate made from superaligned carbon nanotubes,” NanoLett. 10, 1747–1753 (2010).
[Crossref]

Nature (2)

R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Luo, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
[Crossref] [PubMed]

J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, and H. J. Kimble, “Experimental realization of a one-atom laser in the regime of strong coupling,” Nature 425, 268–271 (2003).
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Nature Commun. (1)

R. Miura, S. Imamura, R. Ohta, A. Ishii, X. Liu, T. Shimada, S. Iwamoto, Y. Arakawa, and Y. K. Kato, “Ultralow mode-volume photonic crystal nanobeam cavities for high-efficiency coupling to individual carbon nanotube emitters,” Nature Commun. 5, 5580 (2014).
[Crossref]

Nature Nanotechn. (1)

F. Xia, M. Steiner, Y.-M. Lin, and Ph. Avouris, “A microcavity-controlled, current-driven, on-chip nanotube emitter at infrared wavelengths,” Nature Nanotechn. 3, 609–613 (2008).
[Crossref]

Nature Phys. (1)

M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single-quantum-dot-nanocavity system,” Nature Phys. 6, 279–283 (2010).
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Opt. Express (3)

Phys. Rev. A (1)

V. Delgado and J. M. Gomez Llorente, “Weak-coupling-like time evolution of driven four-level systems in the strong-coupling regime,” Phys. Rev. A 68, 022503 (2003).
[Crossref]

Phys. Rev. B (9)

I. V. Bondarev and Ph. Lambin, “van der Waals coupling in atomically doped carbon nanotubes,” Phys. Rev. B 72, 035451 (2005).
[Crossref]

I. V. Bondarev, L. M. Woods, and K. Tatur, “Strong exciton-plasmon coupling in semiconducting carbon nanotubes,” Phys. Rev. B 80, 085407 (2009).
[Crossref]

I. V. Bondarev, “Single-wall carbon nanotubes as coherent plasmon generators,” Phys. Rev. B 85, 035448 (2012).
[Crossref]

I. V. Bondarev and B. Vlahovic, “Optical absorption by atomically doped carbon nanotubes,” Phys. Rev. B 74, 073401 (2006).
[Crossref]

I. V. Bondarev and B. Vlahovic, “Entanglement of a pair of atomic qubits near a carbon nanotube,” Phys. Rev. B 75, 033402 (2007).
[Crossref]

I. V. Bondarev and A. V. Meliksetyan, “Possibility for exciton Bose-Einstein condensation in carbon nanotubes,” Phys. Rev. B 89, 045414 (2014).
[Crossref]

A. Popescu, L. M. Woods, and I. V. Bondarev, “Chirality dependent carbon nanotube interactions,” Phys. Rev. B 83, 081406 (2011).
[Crossref]

S. Tasaki, K. Maekawa, and T. Yamabe, “π-band contribution to the optical properties of carbon nanotubes: Effects of chirality,” Phys. Rev. B 57, 9301–9318 (1998).
[Crossref]

I. V. Bondarev and Ph. Lambin, “Spontaneous-decay dynamics in atomically doped carbon nanotubes,” Phys. Rev. B 70, 035407 (2004).
[Crossref]

Phys. Rev. Lett. (5)

Z. M. Li, Z. K. Tang, H. J. Liu, N. Wang, C. T. Chan, R. Saito, S. Okada, G. D. Li, J. S. Chen, N. Nagasawa, and S. Tsuda, “Polarized absorption spectra of single-walled 4 Å carbon nanotubes aligned in channels of an AlPO4–5 single crystal,” Phys. Rev. Lett. 87, 127401 (2001).
[Crossref]

I. V. Bondarev, G.Ya. Slepyan, and S. A. Maksimenko, “Spontaneous decay of excited atomic states near a carbon nanotube,” Phys. Rev. Lett. 89, 115504 (2002).
[Crossref] [PubMed]

A. G. Marinopoulos, L. Reining, A. Rubio, and N. Vast, “Optical and loss spectra of carbon nanotubes: Depolarization effects and intertube interactions,” Phys. Rev. Lett. 91, 046402 (2003).
[Crossref] [PubMed]

T. Pichler, M. Knupfer, M. S. Golden, J. Fink, A. Rinzler, and R. E. Smalley, “Localized and delocalized electronic states in single-wall carbon nanotubes,” Phys. Rev. Lett. 80, 4729–4732 (1998).
[Crossref]

H. Schniepp and V. Sandoghdar, “Spontaneous emission of europium ions embedded in dielectric nanospheres,” Phys. Rev. Lett. 89, 257403 (2002).
[Crossref] [PubMed]

Phys. Stat. Sol. C (1)

I. V. Bondarev and T. Antonijevic, “Surface plasmon amplification under controlled exciton-plasmon coupling in individual carbon nanotubes,” Phys. Stat. Sol. C 9, 1259–1264 (2012).
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Sci. Rep. (1)

R. Lv, Q. Li, A. R. Botello-Mendez, T. Hayashi, B. Wang, A. Berkdemir, Q. Hao, A. L. Elias, R. Cruz-Silva, H. R. Gutierrez, Y. A. Kim, H. Muramatsu, J. Zhu, M. Endo, H. Terrones, J.-C. Charlie, M. Pan, and M. Terrones, “Nitrogen-doped graphene: beyond single substitution and enhanced molecular sensing,” Sci. Rep. 2, 586 (2012).
[Crossref] [PubMed]

Other (8)

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[Crossref]

I. V. Bondarev, L. M. Woods, and A. Popescu, “Exciton-plasmon interactions in individual carbon nanotubes,” in Plasmons: Theory and Applications, K. N. Helsey, ed. (Nova Science, 2011), Ch. 16, pp. 381–435.

I. V. Bondarev and Ph. Lambin, “Near-field electrodynamics of atomically doped carbon nanotubes,” in Trends in Nanotubes Research, D. A. Martin, ed. (Nova Science, 2006), Ch. 6, pp. 139–183.

R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Science of Fullerens and Carbon Nanotubes (Imperial College, 1998).

Actual DOS resonance frequencies are slightly red shifted relative to their respective plasmon resonance frequencies [cf. Figs. 1(a) and 1(b)]. The shifts are within plasmon resonance widths though, and so are neglected, thereby reducing the number of relevant theory parameters here.

V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics (Pergamon, 1982).

P. Y. Yu and M. Cardona, Fundamentals of Semiconductors, 4. (Springer-Verlag, 2010).
[Crossref]

I. V. Bondarev, M. F. Gelin, and A. V. Meliksetyan, “Tunable plasmon nanooptics with carbon nanotubes,” in Dekker Encyclopedia of Nanoscience and Nanotechnology, S. E. Lyshevski, ed. (3, CRC, 2014), pp. 4989–5001.

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Figures (5)

Fig. 1
Fig. 1 (a) Fragment of the energy dependence of the dimensionless (normalized by e 2 / 2 π h ¯ ) axial surface conductivities σzz for the semiconducting (6,4), (10,0) and (11,0) nanotubes of increasing diameter. Peaks of Reσzz represent excitons (E 11, E 22, …); peaks of Re(1zz ) are inter-band plasmons (P 11, …). (b) Photonic DOS functions for the CNs in (a) with the TLS placed at the distance rA =Rcn +2b (see inset). Dimensionless energy is [Energy]/2γ 0. Conductivities are obtained using the (k · p)-scheme developed by Ando [12]. DOS functions are calculated as described by Bondarev and Lambin in [14, 15]. See text for notations.
Fig. 2
Fig. 2 Schematic of the energy level structure as given by Eq. (8) for the coupled four-level CN–TLS system. Thick red lines show the eigen energy levels as functions of xp . Thin red dashed lines indicate their broadening due to finite Δxp . Horizontal dotted lines are to show Rabi-splitting and the in-resonance strong-coupling solutions given by Eq. (8) with δ = 0.
Fig. 3
Fig. 3 Schematic of the Raman scattering process (top) in terms of the inter-level transitions (levels sketched in Fig. 2) of the coupled CN–TLS system given by Eqs. (3), (8) and (9), and the Feynman diagrams (bottom) for the scattering cross-section calculations.
Fig. 4
Fig. 4 Enhancement factor A ( δ , X , Δ x p ) / Δ x p 4 as given by Eq. (17) for Δxp = 0.005 (left) and for X =0.05 (right) to show the influence of the detuning δ ( = x ˜ A x p ) on the maximum intensity of the plasmon enhanced Raman scattering effect.
Fig. 5
Fig. 5 Raman scattering probability as a function of the incident xi and scattered xs photon energies as given by Eq. (16) with δ = X / 3 (maximum of A / Δ x p 4 in Fig. 4, see text) for x p = 0.35 [ P 11 ( 6 , 4 ) plasmon in Fig. 1(a)], with X and Δxp being varied independently [rows (a) and (b)]. TLS–plasmon coupling strength is represented by the ratio X/Δxp being greater or less than unity for strong and weak coupling, respectively. Raman scattering is seen to be manifestly indicative of the strong TLS–plasmon coupling, dramatically increasing as X/Δxp goes much greater than unity and disappearing when it is comparable with unity.

Equations (27)

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H ^ = H ^ F + H ^ A + H ^ A F = 0 d ω h ¯ ω d R f ^ ( R , ω ) f ^ ( R , ω ) + h ¯ ω ˜ A 2 σ ^ Z + 0 d ω d R [ g ( + ) ( r A , R , ω ) σ ^ g ( ) ( r A , R , ω ) σ ^ ] f ^ ( R , ω ) + h . c . ,
ω ˜ A = ω A [ 1 2 ( h ¯ ω A ) 2 0 d ω d R | g ( r A , R , ω ) | 2 ] .
g ( ± ) ( r A , R , ω ) = g ( r A , R , ω ) ± ω ω A g ( r A , R , ω ) , g ( ) ( r A , R , ω ) = i 4 ω A c 2 d z π h ¯ ω Re σ z z ( ω ) ( ) G z z ( r A , R , ω ) ,
( ) G z z ( r A , R , ω ) = d r δ z z ( ) ( r A r ) G z z ( r , R , ω )
δ α β ( r ) = α β 1 4 π | r | , δ α β ( r ) = δ α β δ ( r ) δ α β ( r )
α = r , φ , z ( × × ω 2 c 2 ) z α G α z ( r , R , ω ) = δ ( r R ) ,
d R | g ( ± ) ( r A , R , ω ) | 2 = h ¯ 2 2 π Γ 0 ( ω ) [ ξ ( r A , ω ) + ω A 2 ω 2 ξ ( r A , ω ) ] ,
ξ ( ) ( r A , ω ) = Im ( ) G z z ( ) ( r A , r A , ω ) Im G z z 0 ( ω )
( ) G z z ( ) ( r A , r A , ω ) = d r d r δ z z ( ) ( r A r ) G z z ( r , r , ω ) δ z z ( ) ( r r A )
| 0 = | l | { 0 } , | 1 , 2 = C u ( 1 , 2 ) | u | { 0 } + 0 d ω d R C l ( 1 , 2 ) ( R , ω ) | l | { 1 ( R , ω ) } , | 3 = | u | { 1 ( R , ω ) } .
E 0 = h ¯ ω ˜ A 2 , E 3 = h ¯ ω ˜ A 2 + h ¯ ω
{ ( h ¯ ω ˜ A 2 E ) C u ( 1 , 2 ) + 0 d ω d R g ( + ) ( r A , R , ω ) C l ( 1 , 2 ) ( R , ω ) = 0 , [ g ( + ) ( r A , R , ω ) ] * C u ( 1 , 2 ) + ( h ¯ ω ˜ A 2 + h ¯ ω E ) C l ( 1 , 2 ) ( R , ω ) = 0.
C l ( 1 , 2 ) ( R , ω ) = [ g ( + ) ( r A , R , ω ) ] * h ¯ ω ˜ A / 2 h ¯ ω + E C u ( 1 , 2 )
E = h ¯ ω ˜ A 2 + 0 d ω d R | g ( + ) ( r A , R , ω ) | 2 h ¯ ω ˜ A / 2 h ¯ ω + E = h ¯ ω ˜ A 2 + h ¯ 2 2 π 0 d ω Γ 0 ( ω ) ( 1 + ω A 2 / ω 2 ) ξ ( r A , ω ) h ¯ ω ˜ A / 2 h ¯ ω + E
0 d ω Γ 0 ( ω ) ( 1 + ω A 2 / ω 2 ) ξ ( r A , ω ) h ¯ ω ˜ A / 2 h ¯ ω + E Γ 0 ( ω p ) ( 1 + ω A 2 / ω p 2 ) ξ ( r A , ω p ) Δ ω 0 2 h ¯ ω ˜ A / 2 h ¯ ω p + E 0 d ω ( ω ω p ) 2 + Δ ω 0 2 ,
ε 0 = x ˜ A 2 , ε 1 , 2 = 1 2 ( x p δ 2 + X 2 i Δ x p ) , ε 3 = x ˜ A 2 + x p i Δ x p .
| C u ( 1 , 2 ) | 2 + 0 d ω d R | C l ( 1 , 2 ) ( R , ω ) | 2 = 1 ,
C u ( 1 , 2 ) = [ 1 2 ( 1 + 1 1 + X 2 / δ 2 1 + X 2 / δ 2 1 + X 2 / δ 2 ) ] 1 / 2 , 0 d ω d R | C l ( 1 , 2 ) ( R , ω ) | 2 = 1 | C u ( 1 , 2 ) | 2 = ( X 2 / 2 δ 2 ) | C u ( 1 , 2 ) | 2 1 + X 2 / 2 δ 2 1 + X 2 / δ 2 .
n | H ^ R ( ω i ) | 0 = i c 2 π h ¯ ω i d z cos ϑ i C u ( n ) * , cos ϑ i = e i · e z , n = 1 , 2
1 | H ^ A F ( e ) | 2 = 0 d ω d R [ C l ( 1 ) ( R , ω ) g ( + ) ( r A , R , ω ) ] * C u ( 2 )
( 2 π h ¯ ) | 0 | H ^ R ( ω s ) | 1 1 | H ^ A F ( e ) | 2 2 | H ^ R ( ω i ) | 0 [ h ¯ ω i h ¯ ω p ( E 1 E 0 ) ] [ h ¯ ω i ( E 2 E 0 ) ] + 0 | H ^ R ( ω i ) | 1 1 | H ^ A F ( e ) | 2 2 | H ^ R ( ω s ) | 0 [ h ¯ ω s h ¯ ω p ( E 1 E 0 ) ] [ h ¯ ω s ( E 2 E 0 ) ] | 2 δ ( h ¯ ω i h ¯ ω p h ¯ ω s ) + | 0 | H ^ R ( ω s ) | 2 2 | H ^ A F ( a ) | 1 1 | H ^ R ( ω i ) | 0 [ h ¯ ω i + h ¯ ω p ( E 2 E 0 ) ] [ h ¯ ω i ( E 1 E 0 ) ] + 0 | H ^ R ( ω i ) | 2 2 | H ^ A F ( a ) | 1 1 | H ^ R ( ω s ) | 0 [ h ¯ ω s + h ¯ ω p ( E 2 E 0 ) ] [ h ¯ ω s ( E 1 E 0 ) ] | 2 δ ( h ¯ ω i + h ¯ ω p h ¯ ω s ) .
1 | H ^ A F ( e ) | 2 = 2 γ 0 C u ( 1 ) * C u ( 2 ) ( X 2 / 4 ) x ˜ A / 2 x p + ε 1 = 2 | H ^ A F ( a ) | 1 ,
0 | H ^ R | 1 1 | H ^ A F ( e ) | 2 2 | H ^ R | 0 = ( 0 | H ^ R | 2 2 | H ^ A F ( a ) | 1 1 | H ^ R | 0 ) = 2 π h ¯ ω i ω s c 2 d z 2 cos ϑ i cos ϑ s 2 γ 0 | C u ( 1 ) C u ( 2 ) | 2 ( X 2 / 4 ) x ˜ A / 2 x p + ε 1
| C u ( 1 ) C u ( 2 ) | 2 = X 2 / 4 δ 2 + X 2 ,
d σ d Ω s = ( 2 γ 0 ) 2 | d z | 4 h ¯ 4 c 4 cos 2 ϑ i cos 2 ϑ s P ( x i , x s ) ,
P ( x i , x s ) = x i x s 3 A ( δ , X , Δ x p ) { 1 [ ( x i x p δ + / 2 ) 2 + Δ x p 2 ] [ ( x s x p δ / 2 ) 2 + Δ x p 2 ] + 1 [ ( x i x p δ / 2 ) 2 + Δ x p 2 ] [ ( x s x p δ + / 2 ) 2 + Δ x p 2 ] } ,
A ( δ , X , Δ x p ) = X 8 2 6 ( δ 2 + X 2 ) 2 ( δ 2 + Δ x p 2 ) ,

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