Abstract

We study the dynamics of two two-level atoms embedded near to the interface of paired meta-material slabs, one of negative permeability and the other of negative permittivity. This combination generates a strong surface plasmon field at the interface between the meta-materials. It is found that the symmetric and antisymmetric modes of the two-atom system couple to the plasmonic field with different Rabi frequencies. Including the Ohmic losses of the materials we find that the Rabi frequencies exhibit threshold behaviour which distinguish between the non-Markovian (memory preserving) and Markovian (memoryless) regimes of the evolution. Moreover, it is found that significantly different dynamics occur for the resonant and an off-resonant couplings of the plasmon field to the atoms. In the case of the resonant coupling, the field does not appear as a dissipative reservoir to the atoms. We adopt the image method and show that the dynamics of the two atoms coupled to the plasmon field are analogous to the dynamics of a four-atom system in a rectangular configuration. A large and long living entanglement mediated by the plasmonic field in both Markovian and non-Markovian regimes of the evolution is predicted. We also show that a simultaneous Markovian and non-Markovian regime of the evolution may occur in which the memory effects exist over a finite evolution time. In the case of an off-resonant coupling of the atoms to the plasmon field, the atoms interact with each other by exchanging virtual photons which results in the dynamics corresponding to those of two atoms coupled to a common reservoir. In addition, the entanglement is significantly enhanced.

© 2017 Optical Society of America

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2016 (1)

W. Fang, Q. L. Wu, S. P. Wu, and G.-x. Li, “Enhancement of squeezing in resonance fluorescence of a driven quantum dot close to a graphene sheet,” Phys. Rev. A 93, 053831 (2016).
[Crossref]

2015 (5)

R. C. Ge and S. Hughes, “Quantum dynamics of two quantum dots coupled through localized plasmons: An intuitive and accurate quantum optics approach using quasinormal modes”, Phys. Rev. B 92, 205420 (2015).
[Crossref]

M. Otten, R. A. Shah, N. F. Scherer, M. Min, M. Pelton, and S. K. Gray, “Entanglement of two, three, or four plasmonically coupled quantum dots,” Phys. Rev. B 92, 125432 (2015).
[Crossref]

S. Scheel, S. Y. Buhmann, C. Clausen, and P. Schneeweiss, “Directional spontaneous emission and lateral Casimir-Polder force on an atom close to a nanofiber,” Phys. Rev. A 92, 043819 (2015).
[Crossref]

Z. X. Man, Y. J. Xia, and R. Lo Franco, “Harnessing non-Markovian quantum memory by environmental coupling,” Phys. Rev. A 92, 012315 (2015).
[Crossref]

Z. X. Man, N. B. An, and Y. J. Xia, “Non-Markovian dynamics of a two-level system in the presence of hierarchical environments,” Opt. Express 23, 5763 (2015).
[Crossref] [PubMed]

2014 (8)

T. J. G. Apollaro, S. Lorenzo, C. Di Franco, F. Plastina, and M. Paternostro, “Competition between memory-keeping and memory-erasing decoherence channels,” Phys. Rev. A 90, 012310 (2014).
[Crossref]

A. G. Tudela, P. A. Huidobro, L. M. Moreno, C. Tejedor, and F. J. García-Vidal, “Reversible dynamics of single quantum emitters near metal-dielectric interfaces,” Phys. Rev. B 89, 041402 (2014).
[Crossref]

V. Karanikolas, C. A. Marocico, and A. L. Bradley, “Spontaneous emission and energy transfer rates near a coated metallic cylinder,” Phys. Rev. A 89, 063817 (2014).
[Crossref]

A. Delga, J. Feist, J. Bravo-Abad, and F. J. Garcia-Vidal, “Quantum emitters near a metal nanoparticle: strong coupling and quenching,” Phys. Rev. Lett. 112, 253601 (2014).
[Crossref] [PubMed]

Y. Li and N. Engheta, “Supercoupling of surface waves with ęÅ-near-zero metastructures,” Phys. Rev. B 90, 201107 (2014).
[Crossref]

G. Song, J. P. Xu, and Y. P. Yang, “Quantum interference between Zeeman levels near structures made of left-handed materials and matched zero-index metamaterials,” Phys. Rev. A 89, 053830 (2014).
[Crossref]

C. Gonzalez-Ballestero, E. Moren, and F. J. Garcia-Vidal, “Generation, manipulation, and detection of two-qubit entanglement in waveguide QED,” Phys. Rev. A 89, 042328 (2014).
[Crossref]

W. Tan, Y. Sun, H. Chen, and S. Q. Shen, “Photonic simulation of topological excitations in metamaterials,” Scientific Reports 4, 3842 (2014).
[Crossref] [PubMed]

2013 (8)

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7, 791–795 (2013).
[Crossref]

R. C. Ge, C. V. Vlack, P. Yao, J. F. Young, and S. Hughes, “Accessing quantum nanoplasmonics in a hybrid quantum dot–metal nanosystem: Mollow triplet of a quantum dot near a metal nanoparticle,” Phys. Rev. B 87, 205425 (2013).
[Crossref]

H. X. Zheng and H. U. Baranger, “Persistent quantum beats and long-distance entanglement from waveguide-mediated interactions,” Phys. Rev. Lett. 110, 113601 (2013).
[Crossref] [PubMed]

J. Barthes, A. Bouhelier, A. Dereux, and G. C. des Francs, “Coupling of a dipolar emitter into one-dimensional surface plasmon,” Sci. Rep. 3, 2734 (2013).
[Crossref] [PubMed]

A. G. Tudela, P. A. Huidobro, L. M. Moreno, C. Tejedor, and F. J. García-Vidal, “Theory of strong coupling between quantum emitters and propagating surface plasmons,” Phys. Rev. Lett. 110, 126801 (2013).
[Crossref]

Y. Luo, A. I. Fernandez-Dominguez, A. Wiener, S. A. Maier, and J. B. Pendry, “Surface plasmons and nonlocality: A simple model,” Phys. Rev. Lett. 111, 093901 (2013).
[Crossref] [PubMed]

Z. H. Han and S. I. Bozhevolnyi, “Radiation guiding with surface plasmon polaritons,” Rep. Prog. Phys. 76, 016402 (2013).
[Crossref]

M. S. Tame, K. R. McEnery, S. K. Özdemir, J. Lee, S. A. Maier, and M. S. Kim, “Quantum plasmonics,” Nat. Phys. 9, 329 (2013).
[Crossref]

2012 (4)

K. J. Russell, T. L. Liu, S. Y. Cui, and E. L. Hu, “Large spontaneous emission enhancement in plasmonic nanocavities,” Nat. Photonics 6, 459 (2012).
[Crossref]

Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108, 213903 (2012).
[Crossref] [PubMed]

O. Hess, J. B. Pendry, S. A. Maier, R. F. Oulton, J. M. Hamm, and K. L. Tsakmakidis, “Active nanoplasmonic metamaterials,” Nat. Mater. 11, 573 (2012).
[Crossref] [PubMed]

J. X. Zhang, L. Zhang, and W. Xu, “Surface plasmon polaritons: physics and applications,” J. Phys. D: Appl. Phys. 45, 113001 (2012).
[Crossref]

2011 (8)

M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express 19, 22029–22106 (2011).
[Crossref] [PubMed]

J. P. Xu, M. Al-Amri, Y. P. Yang, S. Y. Zhu, and M. S. Zubairy, “Entanglement generation between two atoms via surface modes,” Phys. Rev. A 84, 032334 (2011).
[Crossref]

G. Y. Chen, N. Lambert, C. H. Chou, Y. N. Chen, and F. Nori, “Surface plasmons in a metal nanowire coupled to colloidal quantum dots: Scattering properties and quantum entanglement,” Phys. Rev. B 84, 045310 (2011).
[Crossref]

D. Martõn-Cano, A. Gonzalez-Tudela, L. Martõn-Moreno, F. J. Garcia-Vidal, C. Tejedor, and E. Moreno, “Dissipation-driven generation of two-qubit entanglement mediated by plasmonic waveguides,” Phys. Rev. B 84, 235306 (2011).
[Crossref]

X. D. Zeng, J. P. Xu, and Y. P. Yang, “Spontaneous emission interference enhancement with a mu-negative meta-material slab,” Phys. Rev. A 84, 033834 (2011).
[Crossref]

I. Bergmair, B. Dastmalchi, M. Bergmair, A. Saeed, W. Hilber, G. Hesser, C. Helgert, E. Pshenay-Severin, T. Pertsch, E. B. Kley, U. Hübner, N. H. Shen, R. Penciu, M. Kafesaki, C. M. Soukoulis, K. Hingerl, M. Muehlberger, and R. Schoeftner, “Single and multilayer metamaterials fabricated by nanoimprint lithography,” Nano Technol. 22, 325301 (2011).

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582 (2011).
[Crossref] [PubMed]

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[Crossref] [PubMed]

2010 (4)

D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Green’s function approach,” Phys. Rev. B 82, 075427 (2010).
[Crossref]

B. Stein, J. Y. Laluet, E. Devaux, C. Genet, and T. W. Ebbesen, “Surface plasmon mode steering and negative refraction,” Phys. Rev. Lett. 105, 266804 (2010).
[Crossref]

E. R-Gómez, N. Raigoza, S. B. Cavalcanti, C. A. A. de Carvalho, and L. E. Oliveira, “Plasmon polaritons in photonic metamaterial Fibonacci superlattices,” Phys. Rev. B 81, 153101 (2010).
[Crossref]

L. H. Sun, G. X. Li, and Z. Ficek, “Continuous variables approach to entanglement creation and processing,” Appl. Math. Inf. Sci. 4, 315 (2010).

2009 (6)

Z. Ficek, “Quantum entanglement processing with atoms,” Appl. Math. Inf. Sci. 3, 375 (2009).

A. Archambault, T. V. Teperik, F. Marquier, and J. J. Greffet, “Surface plasmon Fourier optics,” Phys. Rev. B 79, 195414 (2009).
[Crossref]

M. Cuevas and R. A. Depine, “Radiation characteristics of electromagnetic eigenmodes at the corrugated interface of a left-handed material,” Phys. Rev. Lett. 103, 097401 (2009).
[Crossref] [PubMed]

G. X. Li, J. Evers, and C. H. Keitel, “Spontaneous emission interference in negative-refractive-index waveguides,” Phys. Rev. B 80, 045102 (2009).
[Crossref]

A. Huck, S. Smolka, P. Loudahl, A. S. Sørenson, A. Boltasseva, J. Janousek, and U. L. Andersen, “Demonstration of quadrature-squeezed surface plasmons in a gold waveguide,” Phys. Rev. Lett. 102, 246802 (2009).
[Crossref] [PubMed]

N. T. Tung, V. D. Lam, J. W. Park, M. H. Cho, J. Y. Rhee, W. H. Jang, and Y. P. Lee, “Single- and double-negative refractive indices of combined metamaterial structure,” J. App. Phys. 106, 053109 (2009).
[Crossref]

2007 (5)

S. Y. Buhmann and D.-G. Welsch, “Dispersion forces in macroscopic quantum electrodynamics,” Prog. Quant. Electr. 31, 51 (2007).
[Crossref]

H. T. Jiang, H. Chen, and S. Y. Zhu, “Rabi splitting with excitons in effective (near) zero-index media,” Opt. Lett. 32, 1980 (2007).
[Crossref] [PubMed]

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450, 402 (2007).
[Crossref] [PubMed]

H. K. Yuan, U. K. Chettiar, W. S. Cai, A. V. Kildishev, A. Boltasseva, V. P. Drachev, and V. M. Shalaev, “A negative permeability material at red light,” Opt. Express 15, 1076 (2007).
[Crossref] [PubMed]

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75, 241402 (2007).
[Crossref]

2006 (5)

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97, 053002 (2006).
[Crossref] [PubMed]

L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E 74, 056615 (2006).
[Crossref]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892 (2006).
[Crossref] [PubMed]

L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E 74, 056615 (2006).
[Crossref]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800 (2006).
[Crossref] [PubMed]

2005 (1)

2004 (2)

S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 Terahertz,” Science 306, 1351 (2004).
[Crossref] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788 (2004).
[Crossref] [PubMed]

2003 (2)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003).
[Crossref] [PubMed]

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68, 043816 (2003).
[Crossref]

2001 (2)

R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489 (2001).
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R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys. Condens. Matter 13, 1811 (2001).
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2000 (3)

G. S. Agarwal, “Anisotropic vacuum-induced Interference in decay channels,” Phys. Rev. Lett. 84, 5500 (2000).
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R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000).
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L.-W. Li, M.-S. Leong, T.-S. Yeo, and P.-S. Kooi, “Electromagnetic dyadic Green’s functions in spectral domain for multilayered cylinders,” J. Electro. Waves Appl. 14, 961 (2000).
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1999 (2)

S. Scheel, L. Knöll, and D. G. Welsch, “Spontaneous decay of an excited atom in an absorbing dielectric,” Phys. Rev. A 60, 4094 (1999).
[Crossref]

H. P. Breuer, D. Faller, B. Kappler, and F. Petruccione, “Non-Markovian dynamics in pulsed- and continuous-wave atom lasers,” Phys. Rev. A 60, 3188 (1999).
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1996 (1)

J. B. Pendry, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773 (1996).
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1995 (3)

M. S. Tomaš, “Green function for multilayers: Light scattering in planar cavities,” Phys. Rev. A 51, 2545 (1995).
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C. Szymanowski, C. H. Keitel, B. J. Dalton, and P. L. Knight, “Switching between Rayleigh-like and Lorentzian lineshapes of the dispersion in driven two-level atoms,” J. Mod. Opt. 42, 985 (1995).
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C. H. Bennett, “Quantum information and computation,” Phys. Today 48(10), 24 (1995).
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1985 (1)

K. C. Liu and T. F. George, “Spontaneous emission by two atoms with different resonance frequencies near a metal surface,” Phys. Rev. B 32, 3622 (1985).
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1984 (1)

J. M. Wylie and J. E. Sipe, “Quantum electrodynamics near an interface,” Phys. Rev. A 30, 1185 (1984).
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1980 (1)

W. Lukosz, “Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin layers,” Phys. Rev. B 22, 3030 (1980).
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1970 (1)

R. H. Lehmberg, “Radiation from an N-atom system. II. Spontaneous emission from a pair of atoms,” Phys. Rev. A 2, 889 (1970).
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1954 (1)

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99 (1954).
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Agarwal, G. S.

G. S. Agarwal, “Anisotropic vacuum-induced Interference in decay channels,” Phys. Rev. Lett. 84, 5500 (2000).
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Akimov, A. V.

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450, 402 (2007).
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Al-Amri, M.

J. P. Xu, M. Al-Amri, Y. P. Yang, S. Y. Zhu, and M. S. Zubairy, “Entanglement generation between two atoms via surface modes,” Phys. Rev. A 84, 032334 (2011).
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An, N. B.

Andersen, U. L.

A. Huck, S. Smolka, P. Loudahl, A. S. Sørenson, A. Boltasseva, J. Janousek, and U. L. Andersen, “Demonstration of quadrature-squeezed surface plasmons in a gold waveguide,” Phys. Rev. Lett. 102, 246802 (2009).
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Anderson, Z.

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7, 791–795 (2013).
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Apollaro, T. J. G.

T. J. G. Apollaro, S. Lorenzo, C. Di Franco, F. Plastina, and M. Paternostro, “Competition between memory-keeping and memory-erasing decoherence channels,” Phys. Rev. A 90, 012310 (2014).
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Archambault, A.

A. Archambault, T. V. Teperik, F. Marquier, and J. J. Greffet, “Surface plasmon Fourier optics,” Phys. Rev. B 79, 195414 (2009).
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Asenjo-Garcia, A.

J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. Yu, D. E. Chang, and H. J. Kimble, “Atom-atom interaction around the band edge of a photonic crystal waveguide,” arXiv: 1603.02771 (2016).

Avrutsky, I.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75, 241402 (2007).
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Baranger, H. U.

H. X. Zheng and H. U. Baranger, “Persistent quantum beats and long-distance entanglement from waveguide-mediated interactions,” Phys. Rev. Lett. 110, 113601 (2013).
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Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003).
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Barthes, J.

J. Barthes, A. Bouhelier, A. Dereux, and G. C. des Francs, “Coupling of a dipolar emitter into one-dimensional surface plasmon,” Sci. Rep. 3, 2734 (2013).
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Bennett, C. H.

C. H. Bennett, “Quantum information and computation,” Phys. Today 48(10), 24 (1995).
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Bergmair, I.

I. Bergmair, B. Dastmalchi, M. Bergmair, A. Saeed, W. Hilber, G. Hesser, C. Helgert, E. Pshenay-Severin, T. Pertsch, E. B. Kley, U. Hübner, N. H. Shen, R. Penciu, M. Kafesaki, C. M. Soukoulis, K. Hingerl, M. Muehlberger, and R. Schoeftner, “Single and multilayer metamaterials fabricated by nanoimprint lithography,” Nano Technol. 22, 325301 (2011).

Bergmair, M.

I. Bergmair, B. Dastmalchi, M. Bergmair, A. Saeed, W. Hilber, G. Hesser, C. Helgert, E. Pshenay-Severin, T. Pertsch, E. B. Kley, U. Hübner, N. H. Shen, R. Penciu, M. Kafesaki, C. M. Soukoulis, K. Hingerl, M. Muehlberger, and R. Schoeftner, “Single and multilayer metamaterials fabricated by nanoimprint lithography,” Nano Technol. 22, 325301 (2011).

Boltasseva, A.

A. Huck, S. Smolka, P. Loudahl, A. S. Sørenson, A. Boltasseva, J. Janousek, and U. L. Andersen, “Demonstration of quadrature-squeezed surface plasmons in a gold waveguide,” Phys. Rev. Lett. 102, 246802 (2009).
[Crossref] [PubMed]

H. K. Yuan, U. K. Chettiar, W. S. Cai, A. V. Kildishev, A. Boltasseva, V. P. Drachev, and V. M. Shalaev, “A negative permeability material at red light,” Opt. Express 15, 1076 (2007).
[Crossref] [PubMed]

Bouhelier, A.

J. Barthes, A. Bouhelier, A. Dereux, and G. C. des Francs, “Coupling of a dipolar emitter into one-dimensional surface plasmon,” Sci. Rep. 3, 2734 (2013).
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Bozhevolnyi, S. I.

Z. H. Han and S. I. Bozhevolnyi, “Radiation guiding with surface plasmon polaritons,” Rep. Prog. Phys. 76, 016402 (2013).
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Bradley, A. L.

V. Karanikolas, C. A. Marocico, and A. L. Bradley, “Spontaneous emission and energy transfer rates near a coated metallic cylinder,” Phys. Rev. A 89, 063817 (2014).
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Bravo-Abad, J.

A. Delga, J. Feist, J. Bravo-Abad, and F. J. Garcia-Vidal, “Quantum emitters near a metal nanoparticle: strong coupling and quenching,” Phys. Rev. Lett. 112, 253601 (2014).
[Crossref] [PubMed]

Breuer, H. P.

H. P. Breuer, D. Faller, B. Kappler, and F. Petruccione, “Non-Markovian dynamics in pulsed- and continuous-wave atom lasers,” Phys. Rev. A 60, 3188 (1999).
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H. P. Breuer and F. Petruccione, Theory of Open Quantum Systems (Oxford University Press, 2002).

Briggs, D. P.

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7, 791–795 (2013).
[Crossref]

Buhmann, S. Y.

S. Scheel, S. Y. Buhmann, C. Clausen, and P. Schneeweiss, “Directional spontaneous emission and lateral Casimir-Polder force on an atom close to a nanofiber,” Phys. Rev. A 92, 043819 (2015).
[Crossref]

S. Y. Buhmann and D.-G. Welsch, “Dispersion forces in macroscopic quantum electrodynamics,” Prog. Quant. Electr. 31, 51 (2007).
[Crossref]

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68, 043816 (2003).
[Crossref]

Cai, W. S.

Cavalcanti, S. B.

E. R-Gómez, N. Raigoza, S. B. Cavalcanti, C. A. A. de Carvalho, and L. E. Oliveira, “Plasmon polaritons in photonic metamaterial Fibonacci superlattices,” Phys. Rev. B 81, 153101 (2010).
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Chan, C. T.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582 (2011).
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Chang, D. E.

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450, 402 (2007).
[Crossref] [PubMed]

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97, 053002 (2006).
[Crossref] [PubMed]

J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. Yu, D. E. Chang, and H. J. Kimble, “Atom-atom interaction around the band edge of a photonic crystal waveguide,” arXiv: 1603.02771 (2016).

Chen, G. Y.

G. Y. Chen, N. Lambert, C. H. Chou, Y. N. Chen, and F. Nori, “Surface plasmons in a metal nanowire coupled to colloidal quantum dots: Scattering properties and quantum entanglement,” Phys. Rev. B 84, 045310 (2011).
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Chen, H.

W. Tan, Y. Sun, H. Chen, and S. Q. Shen, “Photonic simulation of topological excitations in metamaterials,” Scientific Reports 4, 3842 (2014).
[Crossref] [PubMed]

H. T. Jiang, H. Chen, and S. Y. Zhu, “Rabi splitting with excitons in effective (near) zero-index media,” Opt. Lett. 32, 1980 (2007).
[Crossref] [PubMed]

L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E 74, 056615 (2006).
[Crossref]

L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E 74, 056615 (2006).
[Crossref]

Chen, Y. N.

G. Y. Chen, N. Lambert, C. H. Chou, Y. N. Chen, and F. Nori, “Surface plasmons in a metal nanowire coupled to colloidal quantum dots: Scattering properties and quantum entanglement,” Phys. Rev. B 84, 045310 (2011).
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Cheng, Q.

Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108, 213903 (2012).
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Chettiar, U. K.

Cho, M. H.

N. T. Tung, V. D. Lam, J. W. Park, M. H. Cho, J. Y. Rhee, W. H. Jang, and Y. P. Lee, “Single- and double-negative refractive indices of combined metamaterial structure,” J. App. Phys. 106, 053109 (2009).
[Crossref]

Chou, C. H.

G. Y. Chen, N. Lambert, C. H. Chou, Y. N. Chen, and F. Nori, “Surface plasmons in a metal nanowire coupled to colloidal quantum dots: Scattering properties and quantum entanglement,” Phys. Rev. B 84, 045310 (2011).
[Crossref]

Clausen, C.

S. Scheel, S. Y. Buhmann, C. Clausen, and P. Schneeweiss, “Directional spontaneous emission and lateral Casimir-Polder force on an atom close to a nanofiber,” Phys. Rev. A 92, 043819 (2015).
[Crossref]

Cuevas, M.

M. Cuevas and R. A. Depine, “Radiation characteristics of electromagnetic eigenmodes at the corrugated interface of a left-handed material,” Phys. Rev. Lett. 103, 097401 (2009).
[Crossref] [PubMed]

Cui, S. Y.

K. J. Russell, T. L. Liu, S. Y. Cui, and E. L. Hu, “Large spontaneous emission enhancement in plasmonic nanocavities,” Nat. Photonics 6, 459 (2012).
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Cui, T. J.

Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108, 213903 (2012).
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Dalton, B. J.

C. Szymanowski, C. H. Keitel, B. J. Dalton, and P. L. Knight, “Switching between Rayleigh-like and Lorentzian lineshapes of the dispersion in driven two-level atoms,” J. Mod. Opt. 42, 985 (1995).
[Crossref]

Dastmalchi, B.

I. Bergmair, B. Dastmalchi, M. Bergmair, A. Saeed, W. Hilber, G. Hesser, C. Helgert, E. Pshenay-Severin, T. Pertsch, E. B. Kley, U. Hübner, N. H. Shen, R. Penciu, M. Kafesaki, C. M. Soukoulis, K. Hingerl, M. Muehlberger, and R. Schoeftner, “Single and multilayer metamaterials fabricated by nanoimprint lithography,” Nano Technol. 22, 325301 (2011).

de Carvalho, C. A. A.

E. R-Gómez, N. Raigoza, S. B. Cavalcanti, C. A. A. de Carvalho, and L. E. Oliveira, “Plasmon polaritons in photonic metamaterial Fibonacci superlattices,” Phys. Rev. B 81, 153101 (2010).
[Crossref]

Delga, A.

A. Delga, J. Feist, J. Bravo-Abad, and F. J. Garcia-Vidal, “Quantum emitters near a metal nanoparticle: strong coupling and quenching,” Phys. Rev. Lett. 112, 253601 (2014).
[Crossref] [PubMed]

Depine, R. A.

M. Cuevas and R. A. Depine, “Radiation characteristics of electromagnetic eigenmodes at the corrugated interface of a left-handed material,” Phys. Rev. Lett. 103, 097401 (2009).
[Crossref] [PubMed]

Dereux, A.

J. Barthes, A. Bouhelier, A. Dereux, and G. C. des Francs, “Coupling of a dipolar emitter into one-dimensional surface plasmon,” Sci. Rep. 3, 2734 (2013).
[Crossref] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003).
[Crossref] [PubMed]

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J. Barthes, A. Bouhelier, A. Dereux, and G. C. des Francs, “Coupling of a dipolar emitter into one-dimensional surface plasmon,” Sci. Rep. 3, 2734 (2013).
[Crossref] [PubMed]

Devaux, E.

B. Stein, J. Y. Laluet, E. Devaux, C. Genet, and T. W. Ebbesen, “Surface plasmon mode steering and negative refraction,” Phys. Rev. Lett. 105, 266804 (2010).
[Crossref]

Di Franco, C.

T. J. G. Apollaro, S. Lorenzo, C. Di Franco, F. Plastina, and M. Paternostro, “Competition between memory-keeping and memory-erasing decoherence channels,” Phys. Rev. A 90, 012310 (2014).
[Crossref]

Dicke, R. H.

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

Dolling, G.

Drachev, V. P.

Dung, H. T.

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68, 043816 (2003).
[Crossref]

Dzsotjan, D.

D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Green’s function approach,” Phys. Rev. B 82, 075427 (2010).
[Crossref]

Ebbesen, T. W.

B. Stein, J. Y. Laluet, E. Devaux, C. Genet, and T. W. Ebbesen, “Surface plasmon mode steering and negative refraction,” Phys. Rev. Lett. 105, 266804 (2010).
[Crossref]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003).
[Crossref] [PubMed]

Elser, J.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75, 241402 (2007).
[Crossref]

Engheta, N.

Y. Li and N. Engheta, “Supercoupling of surface waves with ęÅ-near-zero metastructures,” Phys. Rev. B 90, 201107 (2014).
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Enkrich, C.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892 (2006).
[Crossref] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800 (2006).
[Crossref] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. 30, 3198 (2005).
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S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 Terahertz,” Science 306, 1351 (2004).
[Crossref] [PubMed]

Evers, J.

G. X. Li, J. Evers, and C. H. Keitel, “Spontaneous emission interference in negative-refractive-index waveguides,” Phys. Rev. B 80, 045102 (2009).
[Crossref]

Faller, D.

H. P. Breuer, D. Faller, B. Kappler, and F. Petruccione, “Non-Markovian dynamics in pulsed- and continuous-wave atom lasers,” Phys. Rev. A 60, 3188 (1999).
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Fang, W.

W. Fang, Q. L. Wu, S. P. Wu, and G.-x. Li, “Enhancement of squeezing in resonance fluorescence of a driven quantum dot close to a graphene sheet,” Phys. Rev. A 93, 053831 (2016).
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Feist, J.

A. Delga, J. Feist, J. Bravo-Abad, and F. J. Garcia-Vidal, “Quantum emitters near a metal nanoparticle: strong coupling and quenching,” Phys. Rev. Lett. 112, 253601 (2014).
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Y. Luo, A. I. Fernandez-Dominguez, A. Wiener, S. A. Maier, and J. B. Pendry, “Surface plasmons and nonlocality: A simple model,” Phys. Rev. Lett. 111, 093901 (2013).
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Ficek, Z.

L. H. Sun, G. X. Li, and Z. Ficek, “Continuous variables approach to entanglement creation and processing,” Appl. Math. Inf. Sci. 4, 315 (2010).

Z. Ficek, “Quantum entanglement processing with atoms,” Appl. Math. Inf. Sci. 3, 375 (2009).

Fleischhauer, M.

D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Green’s function approach,” Phys. Rev. B 82, 075427 (2010).
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Figures (13)

Fig. 1
Fig. 1 (Color online) Schematic illustration of the system showing the electric field (yellow arrows) of an electromagnetic wave and dislocated surface charges at the interface between MN and EN slabs. The z axis is taken normal to the interface with its origin at the interface. The slabs have thickness d1 and d2, respectively, and are assumed to have infinite extents in the xy plane. Two atoms are embedded in the MN slab at fixed positions (x1, 0, z0) and (x2, 0, z0), where z0 is the distance of the atoms from the interface between the materials. The atomic transition dipole moments p1 and p2 are parallel to each other and oriented in the xz plane.
Fig. 2
Fig. 2 Dependence of U(x21, z0) on separation between the atoms x21s for several different distances of the atoms from the interface: z0 = 0.05λs (solid black line), z0 = 0.1λs (dashed red line), z0 = 0.25λs (dashed-dotted blue line), and z0 = 0.5λs (solid green line).
Fig. 3
Fig. 3 (Color online) Two atoms located at a distance z0 from the interface between two materials and their images located at a distance z0 behind the interface. The atoms are not directly coupled to each other, but can be coupled by the radiation reflected from the interface. A photon emitted by atom 1 and reflected from the interface towards atom 2 can be viewed as being emitted from the image of the atom 1.
Fig. 4
Fig. 4 Time evolution of the populations P1(t) = |C1(t)|2 (black solid line) and P2(t) = |C2(t)|2 (red dashed line) for δ = 0, Ω0 = 0.15γ, and U(x21, z0) = 0.95, corresponding to both Ωs and Ωa below the threshold of 0.25γ, Ωs = 0.21γ and Ωa = 0.033γ. The atoms were initially in the state |Ψ(0)〉 = |e1〉 |g2〉.
Fig. 5
Fig. 5 Time evolution of the populations P1(t) = |C1(t)|2 (black solid line) and P2(t) = |C2(t)|2 (red dashed line) for δ = 0, Ω0 = 0.5γ, and U(x21, z0) = 0.99 corresponding to Ωs above and Ωa below the threshold of 0.25γ, Ωs = 0.705γ and Ωa = 0.05γ. The atoms were initially in the state |Ψ(0)〉 = |e1〉 |g2〉.
Fig. 6
Fig. 6 Time evolution of the populations (a) P1(t) = |C1(t)|2 and (b) P2(t) = |C2(t)|2 for δ = 0, Ω0 = 25γ, and U(x21, z0) = 0.1. The atoms were initially in the state |Ψ(0)〉 = |e1〉 |g2〉.
Fig. 7
Fig. 7 Time evolution of the populations (a) P1(t) = |C1(t)|2 and (b) P2(t) = |C2(t)|2 for δ = 0, Ω0 = 25γ, and U(x21, z0) = 0.8. The atoms were initially in the state |Ψ(0)〉 = |e1〉 |g2〉.
Fig. 8
Fig. 8 Concurrence versus time for the case of above threshold with Ω0 = 25γ, δ = 0, (a) U(x21, z0) = 0.1 and (b) U(x21, z0) = 0.8. The atoms were initially in the state |Ψ(0)〉 = |e1〉 |g2〉.
Fig. 9
Fig. 9 Concurrence versus time for Ω0 = γ, δ = 0 and U(x21, z0) ≈ 1 corresponding to the case of Ωs above threshold but Ωa below the threshold. Frame (a) shows the concurrence for U(x21, z0) = 0.95 (solid black line) and U(x21, z0) = 0.99 (dashed red line). The atoms were initially in a separable state |Ψ(0)〉 = |e1〉 |g2〉. Frame (b) shows the concurrence for U(x21, z0) = 0.95 and two different initial states, the maximally entangled symmetric state |Ψ(0)〉 = |s〉 (solid black line) and the maximally entangled antisymmetric state |Ψ(0)〉 = |a〉 (dashed red line).
Fig. 10
Fig. 10 The time evolution of the populations of the atoms for the situation presented in Fig. 9. Frame (a) shows the populations P1(t)(solid black line) and P2(t) (dashed red line) for Ω0 = γ, δ = 0 and U(x21, z0) = 0.95. The atoms were initially in the state |Ψ(0)〉 = |e1〉 |g2〉. Frame (b) shows the time evolution of the population P1(t) for two different initial states, |Ψ(0)〉 = |s〉 (solid black line) and |Ψ(0)〉 = |a〉 (dashed green line). Not shown is P2(t) since in this case P2(t) = P1(t).
Fig. 11
Fig. 11 Concurrence versus time for the case of below threshold and two different initial states (a) |Ψ(0)〉 = |e1〉 |g2〉 and (b) |Ψ(0)〉 = |s〉 with Ω0 = 0.15γ, δ = 0 and different U(x21, z0): U(x21, z0) = 0.99 (solid black line), U(x21, z0) = 0.5 (dashed red line), U(x21, z0) = 0.25 (dashed-dotted blue line).
Fig. 12
Fig. 12 Time evolution of the populations (a) P1(t) = |C1(t)|2 and (b) P2(t) = |C2(t)|2 for δ = 50γ, Ω0 = 25γ, and U(x21, z0) = 0.1. The atoms were initially in the state |Ψ(0)〉 = |e1〉 |g2〉.
Fig. 13
Fig. 13 Concurrence versus time for the case of above threshold with Ω0 = 25γ, δ = 50γ and different U(x21, z0): (a) U(x21, z0) = 0.1 and (b) U(x21, z0) = 0.95. The atoms were initially in the state |Ψ(0)〉 = |e1〉 |g2〉.

Equations (45)

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ε r = ( d 1 ε 1 + d 2 ε 2 ) / ( d 1 + d 2 ) , μ r = ( d 1 μ 1 + d 2 μ 2 ) / ( d 1 + d 2 )
H ^ = H ^ 0 + H ^ I ,
H ^ 0 = 1 2 ω a ( σ ^ z 1 + σ ^ z 2 ) + λ = e , m d r 0 d ω ω f ^ λ ( r , ω ) f ^ λ ( r , ω )
H ^ I = l = 1 , 2 [ p l 0 d ω E ^ ( + ) ( r l , ω ) σ ^ l + H . c . ]
E ^ ( + ) ( r l , ω ) = i π ε 0 ω c d r { ω c [ ε ( r , ω ) ] G ( r l , r , ω ) f ^ e ( r , ω ) + [ κ ( r , ω ) ] × G ( r l , r , ω ) f ^ m ( r , ω ) } ,
ε 2 ( ω ) ε 0 = 1 + ω e p 2 ω e o   2 ω 2 i ω γ e , μ 1 ( ω ) μ 0 = 1 + ω m p 2 ω m o   2 ω 2 i ω γ m ,
| Ψ ( t ) = C 1 ( t ) e i ω a t | { 0 } | e 1 , g 2 + C 2 ( t ) e i ω a t | { 0 } | g 1 , e 2 + λ = e , m 0 d ω e i ω t d r C λ ( r , ω , t ) | 1 λ ( r , ω ) | g 1 , g 2 ,
C ˙ s ( t ) = 0 t d t K s ( t , t ) C s ( t ) , C ˙ a ( t ) = 0 t d t K a ( t , t ) C a ( t ) ,
K j ( t , t ) = Ω j 2 e ( 1 2 γ + i δ ) ( t t ) , j = s , a .
Ω 0 = { [ 3 + 4 π 2 | [ μ 1 ( ω s ) ] | ( 2 z 0 / λ s ) 2 ] ω s Γ A 64 π 3 ( 2 z 0 / λ s ) 3 } 1 / 2
U ( x 21 , z 0 ) = F [ 1 2 , 1 , 2 ; x 21 2 ( 2 z 0 ) 2 ] + 1 3 + 4 π 2 | [ μ 1 ( ω s ) ] | ( 2 z 0 / λ s ) 2 { F [ 3 2 , 2 , 2 ; x 21 2 ( 2 z 0 ) 2 ] + 2 F [ 3 2 , 2 , 1 ; x 21 2 ( 2 z 0 ) 2 ] 3 F [ 1 2 , 1 , 2 ; x 21 2 ( 2 z 0 ) 2 ] 3 x 21 2 ( 2 z 0 ) 2 F [ 5 2 , 3 , 3 ; x 21 2 ( 2 z 0 ) 2 ] }
C j ( t ) = 1 2 C j ( 0 ) e 1 2 i δ t [ ( 1 + u e i θ Ω ˜ j ) e ( 1 4 γ Ω ˜ j ) t + ( 1 u e i θ Ω ˜ j ) e ( 1 4 γ + Ω ˜ j ) t ] , j = s , a ,
C ˜ s ( t ) = ( γ + 2 i δ ) 8 Ω ˜ s [ D ˜ a ( t ) cos ϕ s + i D ˜ s ( t ) sin ϕ s ] , C ˜ a ( t ) = ( γ + 2 i δ ) 8 Ω ˜ a [ G ˜ a ( t ) cos ϕ a + i G ˜ s ( t ) sin ϕ a ] ,
D ˜ s ( t ) = i C ˜ s ( t ) sin ϕ s + C ˜ s I ( t ) cos ϕ s = i C s ( 0 ) e [ 1 4 ( γ + 2 i δ ) Ω ˜ s ] t sin ϕ s , D ˜ a ( t ) = C ˜ s ( t ) cos ϕ s i C ˜ s I ( t ) sin ϕ s = C s ( 0 ) e [ 1 4 ( γ + 2 i δ ) + Ω ˜ s ] t cos ϕ s , G ˜ s ( t ) = i C a ( 0 ) e [ 1 4 ( γ + 2 i δ ) Ω ˜ a ] t sin ϕ a , G ˜ a ( t ) = C a ( 0 ) e [ 1 4 ( γ + 2 i δ ) + Ω ˜ a ] t cos ϕ a ,
cos 2 ϕ j = 1 2 + 2 Ω ˜ j ( γ + 2 i δ ) , j = s , a .
C s ( t ) = C s ( 0 ) e 1 4 ( γ + 2 i δ ) t [ cos Ω ¯ s t + u e i θ Ω ¯ s sin Ω ¯ s t ] ,
C j ( t ) = C j ( 0 ) e 1 4 ( γ + 2 i δ ) t ( cos Ω ¯ j t + u e i θ Ω ¯ j sin Ω ¯ j t ) ,
C ( t ) = | [ C s ( t ) C a ( t ) ] [ C s * ( t ) + C a * ( t ) ] | .
Ω ˜ s 1 4 γ ( 1 2 Ω s 2 δ 2 ) + 1 2 i δ ( 1 + 2 Ω s 2 δ 2 ) .
C s ( t ) C s ( 0 ) { e ( γ 2 i δ ) Ω s 2 2 δ 2 t + Ω s 2 4 δ 2 e 1 2 ( γ + 2 i δ ) t } .
C ˙ 1 ( t ) = 1 π ε 0 0 d ω ω 2 c 2 e i ( ω ω a ) t d r { [ ε ( r , ω ) ] p 1 G ( r 1 , r , ω ) C e ( r , ω , t ) + [ κ ( r , ω ) ] p 1 [ G ( r 1 , r , ω ) × ] C m ( r , ω , t ) } ,
C ˙ 2 ( t ) = 1 π ε 0 0 d ω ω 2 c 2 e i ( ω ω a ) t d r { [ ε ( r , ω ) ] p 2 G ( r 2 , r , ω ) C e ( r , ω , t ) + [ κ ( r , ω ) ] p 2 [ G ( r 2 , r , ω ) × ] C m ( r , ω , t ) } ,
C ˙ e ( r , ω , t ) = e i ( ω ω a ) t π ε 0 ω 2 c 2 [ ε ( r , ω ) ] [ G * ( r 1 , r , ω ) p 1 * C 1 ( t ) + G * ( r 2 , r , ω ) p 2 * C 2 ( t ) ] ,
C ˙ m ( r , ω , t ) = e i ( ω ω a ) t π ε 0 ω c [ κ ( r , ω ) ] × [ G * ( r 1 , r , ω ) p 1 * C 1 ( t ) + G * ( r 2 , r , ω ) p 2 * C 2 ( t ) ] .
C ˙ 1 ( t ) = 0 t d t K 11 ( t , t ) C 1 ( t ) + 0 t d t K 12 ( t , t ) C 2 ( t ) ,
C ˙ 2 ( t ) = 0 t d t K 22 ( t , t ) C 2 ( t ) + 0 t d t K 21 ( t , t ) C 1 ( t ) ,
K i j ( t , t ) = 1 π ε 0 c 2 0 d ω ω 2 e i ( ω ω a ) ( t t ) p i [ G ( r i , r j , ω ) ] p j * , i , j = 1 , 2 .
p i [ G ( r i , r j , ω ) ] p j * = | p i | | p j | ( x ¯ + z ¯ ) [ G ( r i , r j , ω ) ] ( x ¯ + z ¯ ) = | p i | | p j | { [ G ( r i , r j , ω ) ] x x + [ G ( r i , r j , ω ) ] z z + [ G ( r i , r j , ω ) ] x z + [ G ( r i , r j , ω ) ] z x } .
G ( r , r i , ω ) = c 2 ω 2 ε 1 ( + k 1 2 I ) e i k 1 R R ,
G ( k , ω , z , z 0 ) = i μ 1 2 ( 2 π ) 2 d 2 k ξ q e i β 1 d 1 β 1 D q [ U q + ( k , ω , z ) U q ( k , ω , z 0 ) Θ ( z z 0 ) + U q ( k , ω , z ) U q + ( k , ω , z 0 ) Θ ( z 0 z ) ] e i k ( ρ ρ 0 ) ,
U q + ( k , ω , z ) = e q + ( k ) e i β 1 ( z d 1 ) + r + q e q ( k ) e i β 1 ( z d 1 ) , U q ( k , ω , z ) = e q ( k ) e i β 1 z + r q e q + ( k ) e i β 1 z ,
G n m ( r , r i , ω ) = n ( G ( r , r i , ω ) ) m , n , m = x , y , z , n , m = x ¯ , y ¯ , z ¯ .
G x x ( r , r i , ω ) = i μ 1 4 π d k k { β 1 k 1 2 [ J 1 ( α i ) α i J 2 ( α i ) ] R ( p ) ( z ) + J 1 ( α i ) β 1 α i R + ( s ) ( z ) } , G y y ( r , r i , ω ) = i μ 1 4 π d k k { β 1 J 1 ( α i ) k 1 2 α i R ( p ) ( z ) + 1 β 1 [ J 1 ( α i ) α i J 2 ( α i ) ] R + ( s ) ( z ) } , G z z ( r , r i , ω ) = i μ 1 4 π d k k 3 β 1 k 1 2 J 0 ( α i ) R + ( p ) ( z ) ,
G x y ( r , r i , ω ) = G y x ( r , r i , ω ) = 0 , G y z ( r , r i , ω ) = G z y ( r , r i , ω ) = 0 , G x z ( r , r i , ω ) = G z x ( r , r i , ω ) = μ 1 4 π d k k 2 J 1 ( α i ) k 1 2 D p [ r + p e i β 1 ( z + z 0 2 d 1 ) r p e i β 1 ( z + z 0 ) ] ,
R ± ( q ) ( z ) = 1 D q [ e i β 1 ( z z 0 ) ± r q e i β 1 ( z + z 0 ) ± r + q e i β 1 ( z + z 0 2 d 1 ) + r + q r q e i β 1 ( z z 0 2 d 1 ) ] .
r i j p = ( β i ε j β j ε i ) / ( β i ε j + β j ε i ) , r i j s = ( β i μ j β j μ i ) / ( β i μ j + β j μ i ) , i j = ± ,
r i j k q = ( r i j q + r j k q e 2 i β j d j ) / ( 1 r j i q r j k q e 2 i β j d j ) .
r + p r 1 0 p = ( β 1 ε 0 β 0 ε 1 ) / ( β 1 ε 0 + β 0 ε 1 ) ( ε 0 ε 1 ) / ( ε 0 + ε 1 ) ,
r p = ( r 1 2 p + r 2 0 p e 2 i β 2 d 2 ) / ( 1 r 2 0 p + r 2 1 p e 2 i β 2 d 2 ) ( ε 2 ε 1 ) / ( ε 1 + ε 2 ) .
r p = 1 ε 1 ω s ( Δ ω 1 2 i γ ) ( ε 1 + 1 ) ( Δ ω 2 + 1 4 γ 2 ) ,
r s = 1 + μ 1 ω s ( Δ ω 1 2 i γ ) ( μ 1 + 1 ) ( Δ ω 2 + 1 4 γ 2 ) ,
[ G z z ( r 1 , r 1 , ω ) ] = γ ω s 12 π k 2 ( Δ ω 2 + 1 4 γ 2 ) ( 2 z 0 ) 3 ,
[ G z z ( r 2 , r 1 , ω ) ] = γ ω s 12 π k 2 ( Δ ω 2 + 1 4 γ 2 ) ( 2 z 0 ) 3 F [ 3 2 , 2 , 1 ; x 21 2 ( 2 z 0 ) 2 ] .
[ G x x ( r 1 , r 1 , ω ) ] = [ G x x ( r 2 , r 2 , ω ) ] = γ ω s { 1 k s 2 [ μ 1 ( ω s ) ] ( 2 z 0 ) 2 } 24 π k 2 ( Δ ω 2 + 1 4 γ 2 ) ( 2 z 0 ) 3 ,
[ G x x ( r 2 , r 1 , ω ) ] = [ G x x ( r 1 , r 2 , ω ) ] = γ ω s 24 π k 2 ( Δ ω 2 + 1 4 γ 2 ) ( 2 z 0 ) 3 × { F [ 3 2 , 2 , 2 ; x 21 2 ( 2 z 0 ) 2 ] 3 x 21 2 ( 2 z 0 ) 2 F [ 5 2 , 3 , 3 ; x 21 2 ( 2 z 0 ) 2 ] [ μ 1 ( ω s ) ] ( 2 z 0 k s ) 2 F [ 1 2 , 1 , 2 ; x 21 2 ( 2 z 0 ) 2 ] } .

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