Abstract

The controlled coupling of photon emitters with tailored nanophotonic structures offers an exciting platform for studying fundamental quantum electrodynamics (QED) and developing on-chip quantum information processing at telecom-compatible wavelengths. Here we introduce a three-dimensional polariton waveguide structure, capable of achieving strong coupling with a single quantum emitter. This polariton waveguide consists of a nanowire-based photonic crystal (PC) waveguide with a quantum dot (QD) embedded in each unit cell. Using realistic designs and parameters, we derive and calculate the fundamental electromagnetic properties of these polariton waveguides, with an emphasis on the local optical density of states (LDOS) and the photon Green function. We demonstrate dramatic increases, and rich fundamental control, of the LDOS due to strong light–matter interactions in each unit cell through periodic QD interactions and further show that these results are quite robust to structural disorder. As an example application, we consider the coupling of an external target QD with a finite-sized polariton waveguide, and show that the single QD strong coupling regime is easily accessible, even for modest dipole strengths. Our polaronic structures are fundamentally interesting and allow for the exploration of new regimes of waveguide QED. While our calculations are exemplified for a PC nanowire system, the general results apply to a wide range of PC waveguides including planar PC slabs and “alligator” PCs, as well as circuit-QED systems.

© 2016 Optical Society of America

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References

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    [Crossref]

2015 (6)

G. Angelatos and S. Hughes, “Entanglement dynamics and Mollow nonuplets between two coupled quantum dots in a nanowire photonic-crystal system,” Phys. Rev. A 91, 051803(R) (2015).
[Crossref]

J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
[Crossref]

A. Goban, C. L. Hung, J. D. Hood, S. P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” Phys. Rev. Lett. 115, 1–5 (2015).
[Crossref]

B. le Feber, N. Rotenberg, and L. Kuipers, “Nanophotonic control of circular dipole emission,” Nat. Commun. 6, 6695 (2015).
[Crossref]

I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Deterministic photon-emitter coupling in chiral photonic circuits,” Nat. Nanotechnol. 10, 775–778 (2015).
[Crossref]

A. B. Young, A. C. T. Thijssen, D. M. Beggs, P. Androvitsaneas, L. Kuipers, J. G. Rarity, S. Hughes, and R. Oulton, “Polarization engineering in photonic crystal waveguides for spin-photon entanglers,” Phys. Rev. Lett. 115, 1–5 (2015).

2014 (4)

N. Dhindsa, A. Chia, J. Boulanger, I. Khodadad, R. LaPierre, and S. S. Saini, “Highly ordered vertical GaAs nanowire arrays with dry etching and their optical properties,” Nanotechnology 25, 305303 (2014).
[Crossref]

G. Angelatos and S. Hughes, “Theory and design of quantum light sources from quantum dots embedded in semiconductor-nanowire photonic-crystal systems,” Phys. Rev. B 90, 205406 (2014).
[Crossref]

S. P. Yu, J. D. Hood, J. A. Muniz, M. J. Martin, R. Norte, C. L. Hung, S. M. Meenehan, J. D. Cohen, O. Painter, and H. J. Kimble, “Nanowire photonic crystal waveguides for single-atom trapping and strong light-matter interactions,” Appl. Phys. Lett. 104, 2012–2017 (2014).

D. O. Krimer, M. Liertzer, S. Rotter, and H. E. Türeci, “Route from spontaneous decay to complex multimode dynamics in cavity QED,” Phys. Rev. A 89, 1–8 (2014).
[Crossref]

2013 (2)

M. N. Makhonin, A. P. Foster, A. B. Krysa, P. W. Fry, D. G. Davies, T. Grange, T. Walther, M. S. Skolnick, and L. R. Wilson, “Homogeneous array of nanowire-embedded quantum light emitters,” Nano Lett. 13, 861–865 (2013).
[Crossref]

J. Gao, S. Combrie, B. Liang, P. Schmitteckert, G. Lehoucq, S. Xavier, X. Xu, K. Busch, D. L. Huffaker, A. De Rossi, and C. W. Wong, “Strongly coupled slow-light polaritons in one-dimensional disordered localized states,” Sci. Rep. 3, 1994 (2013).

2012 (5)

R. Bose, D. Sridharan, H. Kim, G. S. Solomon, and E. Waks, “Low-photon-number optical switching with a single quantum dot coupled to a photonic crystal cavity,” Phys. Rev. Lett. 108, 227402 (2012).
[Crossref]

T. Ba Hoang, J. Beetz, L. Midolo, M. Skacel, M. Lermer, M. Kamp, S. Höfling, L. Balet, N. Chauvin, and A. Fiore, “Enhanced spontaneous emission from quantum dots in short photonic crystal waveguides,” Appl. Phys. Lett. 100, 061122 (2012).
[Crossref]

S. Weiler, A. Ulhaq, S. M. Ulrich, D. Richter, M. Jetter, P. Michler, C. Roy, and S. Hughes, “Phonon-assisted incoherent excitation of a quantum dot and its emission properties,” Phys. Rev. B 86, 241304 (2012).
[Crossref]

T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6, 607–611 (2012).
[Crossref]

M. Février, P. Gogol, A. Aassime, R. Mégy, C. Delacour, A. Chelnokov, A. Apuzzo, S. Blaize, J. M. Lourtioz, and B. Dagens, “Giant coupling effect between metal nanoparticle chain and optical waveguide,” Nano Lett. 12, 1032–1037 (2012).
[Crossref]

2011 (3)

P. T. Kristensen, J. Mørk, P. Lodahl, and S. Hughes, “Decay dynamics of radiatively coupled quantum dots in photonic crystal slabs,” Phys. Rev. B 83, 075305 (2011).
[Crossref]

H. J. Joyce, Q. Gao, H. Hoe Tan, C. Jagadish, Y. Kim, J. Zou, L. M. Smith, H. E. Jackson, J. M. Yarrison-Rice, P. Parkinson, and M. B. Johnston, “III-V semiconductor nanowires for optoelectronic device applications,” Prog. Quant. Electron. 35, 23–75 (2011).
[Crossref]

S. L. Diedenhofen, O. T. A. Janssen, M. Hocevar, A. Pierret, E. P. A. M. Bakkers, H. P. Urbach, and J. Gómez Rivas, “Controlling the directional emission of light by periodic arrays of heterostructured semiconductor nanowires,” ACS Nano 5, 5830–5837 (2011).
[Crossref]

2010 (3)

P. Yao, V. S. C. Manga Rao, and S. Hughes, “On-chip single photon sources using planar photonic crystals and single quantum dots,” Laser Photon. Rev. 4, 499–516 (2010).
[Crossref]

M. Patterson and S. Hughes, “Interplay between disorder-induced scattering and local field effects in photonic crystal waveguides,” Phys. Rev. B 81, 245321 (2010).
[Crossref]

F. J. Rodríguez-Fortuño, B. Tomás-Navarro, C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Zero-bandwidth mode in a split-ring-resonator-loaded one-dimensional photonic crystal,” Phys. Rev. B 81, 2–5 (2010).

2009 (6)

M. Patterson, S. Hughes, S. Combrié, N.-V.-Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouën, “Disorder-induced coherent scattering in slow-light photonic crystal waveguides,” Phys. Rev. Lett. 102, 253903 (2009).
[Crossref]

J.-C. Harmand, L. Liu, G. Patriarche, M. Tchernycheva, N. Akopian, U. Perinetti, and V. Zwiller, “Potential of semiconductor nanowires for single photon sources,” Proc. SPIE 7222, 722219 (2009).
[Crossref]

V. G. Dubrovskii, G. E. Cirlin, and V. M. Ustinov, “Semiconductor nanowhiskers: synthesis, properties, and applications,” Semiconductors 43, 1539–1584 (2009).
[Crossref]

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, “Modes of shallow photonic crystal waveguides: semi-analytic treatment,” Opt. Express 17, 19629–19643 (2009).
[Crossref]

G. X. Li, J. Evers, and C. H. Keitel, “Spontaneous emission interference in negative-refractive-index waveguides,” Phys. Rev. B 80, 1–7 (2009).

P. Yao, C. P. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80, 195106 (2009).
[Crossref]

2008 (3)

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]

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its nonlinearity in a cavity QED system,” Nature 454, 315–318 (2008).
[Crossref]

Y. M. Niquet and D. C. Mojica, “Quantum dots and tunnel barriers in InAs InP nanowire heterostructures: electronic and optical properties,” Phys. Rev. B 77, 1–12 (2008).
[Crossref]

2007 (3)

V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
[Crossref]

D. P. Fussell and M. M. Dignam, “Quantum-dot photon dynamics in a coupled-cavity waveguide: observing band-edge quantum optics,” Phys. Rev. A 76, 053801 (2007).
[Crossref]

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
[Crossref]

2006 (1)

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006).
[Crossref]

2005 (3)

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005).
[Crossref]

Z. Yang, N. Kwong, R. Binder, and A. Smirl, “Stopping, storing and releasing light in quantum well Bragg structures,” J. Opt. B 22, 2144–2156 (2005).
[Crossref]

J. Shen and S. Fan, “Coherent photon transport from spontaneous emission in one-dimensional waveguides,” Opt. Lett. 30, 2001–2003 (2005).
[Crossref]

2004 (2)

L. G. Suttorp and A. J. V. Wonderen, “Fano diagonalization of a polariton model for an inhomogeneous absorptive dielectric,” Europhys. Lett. 67, 766–772 (2004).
[Crossref]

M. Tokushima, H. Yamada, and Y. Arakawa, “1.5-mm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab,” Appl. Phys. Lett. 84, 4298–4300 (2004).
[Crossref]

2003 (1)

A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91, 183901 (2003).
[Crossref]

2001 (1)

A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, T. Fujita, and T. Ishihara, “Polariton effect in distributed feedback microcavities,” J. Phys. Soc. Jpn. 70, 1137–1144 (2001).
[Crossref]

1995 (1)

1990 (1)

S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[Crossref]

Aassime, A.

M. Février, P. Gogol, A. Aassime, R. Mégy, C. Delacour, A. Chelnokov, A. Apuzzo, S. Blaize, J. M. Lourtioz, and B. Dagens, “Giant coupling effect between metal nanoparticle chain and optical waveguide,” Nano Lett. 12, 1032–1037 (2012).
[Crossref]

Akopian, N.

J.-C. Harmand, L. Liu, G. Patriarche, M. Tchernycheva, N. Akopian, U. Perinetti, and V. Zwiller, “Potential of semiconductor nanowires for single photon sources,” Proc. SPIE 7222, 722219 (2009).
[Crossref]

Androvitsaneas, P.

A. B. Young, A. C. T. Thijssen, D. M. Beggs, P. Androvitsaneas, L. Kuipers, J. G. Rarity, S. Hughes, and R. Oulton, “Polarization engineering in photonic crystal waveguides for spin-photon entanglers,” Phys. Rev. Lett. 115, 1–5 (2015).

Angelatos, G.

G. Angelatos and S. Hughes, “Entanglement dynamics and Mollow nonuplets between two coupled quantum dots in a nanowire photonic-crystal system,” Phys. Rev. A 91, 051803(R) (2015).
[Crossref]

G. Angelatos and S. Hughes, “Theory and design of quantum light sources from quantum dots embedded in semiconductor-nanowire photonic-crystal systems,” Phys. Rev. B 90, 205406 (2014).
[Crossref]

G. Angelatos, “Theory and applications of light-matter interactions in quantum dot nanowire photonic crystal systems,” Master’s thesis (Queen’s University, 2015).

Apuzzo, A.

M. Février, P. Gogol, A. Aassime, R. Mégy, C. Delacour, A. Chelnokov, A. Apuzzo, S. Blaize, J. M. Lourtioz, and B. Dagens, “Giant coupling effect between metal nanoparticle chain and optical waveguide,” Nano Lett. 12, 1032–1037 (2012).
[Crossref]

Arakawa, Y.

M. Tokushima, H. Yamada, and Y. Arakawa, “1.5-mm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab,” Appl. Phys. Lett. 84, 4298–4300 (2004).
[Crossref]

Atatüre, M.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
[Crossref]

Ba Hoang, T.

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H. J. Joyce, Q. Gao, H. Hoe Tan, C. Jagadish, Y. Kim, J. Zou, L. M. Smith, H. E. Jackson, J. M. Yarrison-Rice, P. Parkinson, and M. B. Johnston, “III-V semiconductor nanowires for optoelectronic device applications,” Prog. Quant. Electron. 35, 23–75 (2011).
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J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its nonlinearity in a cavity QED system,” Nature 454, 315–318 (2008).
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K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
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J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
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T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6, 607–611 (2012).
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S. L. Diedenhofen, O. T. A. Janssen, M. Hocevar, A. Pierret, E. P. A. M. Bakkers, H. P. Urbach, and J. Gómez Rivas, “Controlling the directional emission of light by periodic arrays of heterostructured semiconductor nanowires,” ACS Nano 5, 5830–5837 (2011).
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T. Ba Hoang, J. Beetz, L. Midolo, M. Skacel, M. Lermer, M. Kamp, S. Höfling, L. Balet, N. Chauvin, and A. Fiore, “Enhanced spontaneous emission from quantum dots in short photonic crystal waveguides,” Appl. Phys. Lett. 100, 061122 (2012).
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A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006).
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A. Goban, C. L. Hung, J. D. Hood, S. P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” Phys. Rev. Lett. 115, 1–5 (2015).
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Hu, E. L.

T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6, 607–611 (2012).
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K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
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J. Gao, S. Combrie, B. Liang, P. Schmitteckert, G. Lehoucq, S. Xavier, X. Xu, K. Busch, D. L. Huffaker, A. De Rossi, and C. W. Wong, “Strongly coupled slow-light polaritons in one-dimensional disordered localized states,” Sci. Rep. 3, 1994 (2013).

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P. Yao, C. P. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80, 195106 (2009).
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V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
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Skacel, M.

T. Ba Hoang, J. Beetz, L. Midolo, M. Skacel, M. Lermer, M. Kamp, S. Höfling, L. Balet, N. Chauvin, and A. Fiore, “Enhanced spontaneous emission from quantum dots in short photonic crystal waveguides,” Appl. Phys. Lett. 100, 061122 (2012).
[Crossref]

Skolnick, M. S.

M. N. Makhonin, A. P. Foster, A. B. Krysa, P. W. Fry, D. G. Davies, T. Grange, T. Walther, M. S. Skolnick, and L. R. Wilson, “Homogeneous array of nanowire-embedded quantum light emitters,” Nano Lett. 13, 861–865 (2013).
[Crossref]

Smirl, A.

Z. Yang, N. Kwong, R. Binder, and A. Smirl, “Stopping, storing and releasing light in quantum well Bragg structures,” J. Opt. B 22, 2144–2156 (2005).
[Crossref]

Smith, L. M.

H. J. Joyce, Q. Gao, H. Hoe Tan, C. Jagadish, Y. Kim, J. Zou, L. M. Smith, H. E. Jackson, J. M. Yarrison-Rice, P. Parkinson, and M. B. Johnston, “III-V semiconductor nanowires for optoelectronic device applications,” Prog. Quant. Electron. 35, 23–75 (2011).
[Crossref]

Söllner, I.

I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Deterministic photon-emitter coupling in chiral photonic circuits,” Nat. Nanotechnol. 10, 775–778 (2015).
[Crossref]

Solomon, G. S.

R. Bose, D. Sridharan, H. Kim, G. S. Solomon, and E. Waks, “Low-photon-number optical switching with a single quantum dot coupled to a photonic crystal cavity,” Phys. Rev. Lett. 108, 227402 (2012).
[Crossref]

Song, J. D.

I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Deterministic photon-emitter coupling in chiral photonic circuits,” Nat. Nanotechnol. 10, 775–778 (2015).
[Crossref]

Sridharan, D.

R. Bose, D. Sridharan, H. Kim, G. S. Solomon, and E. Waks, “Low-photon-number optical switching with a single quantum dot coupled to a photonic crystal cavity,” Phys. Rev. Lett. 108, 227402 (2012).
[Crossref]

Stobbe, S.

I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Deterministic photon-emitter coupling in chiral photonic circuits,” Nat. Nanotechnol. 10, 775–778 (2015).
[Crossref]

Suttorp, L. G.

L. G. Suttorp and A. J. V. Wonderen, “Fano diagonalization of a polariton model for an inhomogeneous absorptive dielectric,” Europhys. Lett. 67, 766–772 (2004).
[Crossref]

Tahan, C.

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006).
[Crossref]

Tchernycheva, M.

J.-C. Harmand, L. Liu, G. Patriarche, M. Tchernycheva, N. Akopian, U. Perinetti, and V. Zwiller, “Potential of semiconductor nanowires for single photon sources,” Proc. SPIE 7222, 722219 (2009).
[Crossref]

Thijssen, A. C. T.

A. B. Young, A. C. T. Thijssen, D. M. Beggs, P. Androvitsaneas, L. Kuipers, J. G. Rarity, S. Hughes, and R. Oulton, “Polarization engineering in photonic crystal waveguides for spin-photon entanglers,” Phys. Rev. Lett. 115, 1–5 (2015).

Tikhodeev, S. G.

A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91, 183901 (2003).
[Crossref]

A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, T. Fujita, and T. Ishihara, “Polariton effect in distributed feedback microcavities,” J. Phys. Soc. Jpn. 70, 1137–1144 (2001).
[Crossref]

Tokushima, M.

M. Tokushima, H. Yamada, and Y. Arakawa, “1.5-mm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab,” Appl. Phys. Lett. 84, 4298–4300 (2004).
[Crossref]

Tomás-Navarro, B.

F. J. Rodríguez-Fortuño, B. Tomás-Navarro, C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Zero-bandwidth mode in a split-ring-resonator-loaded one-dimensional photonic crystal,” Phys. Rev. B 81, 2–5 (2010).

Tran, N.-V.-Q.

M. Patterson, S. Hughes, S. Combrié, N.-V.-Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouën, “Disorder-induced coherent scattering in slow-light photonic crystal waveguides,” Phys. Rev. Lett. 102, 253903 (2009).
[Crossref]

Türeci, H. E.

D. O. Krimer, M. Liertzer, S. Rotter, and H. E. Türeci, “Route from spontaneous decay to complex multimode dynamics in cavity QED,” Phys. Rev. A 89, 1–8 (2014).
[Crossref]

Ulhaq, A.

S. Weiler, A. Ulhaq, S. M. Ulrich, D. Richter, M. Jetter, P. Michler, C. Roy, and S. Hughes, “Phonon-assisted incoherent excitation of a quantum dot and its emission properties,” Phys. Rev. B 86, 241304 (2012).
[Crossref]

Ulrich, S. M.

S. Weiler, A. Ulhaq, S. M. Ulrich, D. Richter, M. Jetter, P. Michler, C. Roy, and S. Hughes, “Phonon-assisted incoherent excitation of a quantum dot and its emission properties,” Phys. Rev. B 86, 241304 (2012).
[Crossref]

Urbach, H. P.

S. L. Diedenhofen, O. T. A. Janssen, M. Hocevar, A. Pierret, E. P. A. M. Bakkers, H. P. Urbach, and J. Gómez Rivas, “Controlling the directional emission of light by periodic arrays of heterostructured semiconductor nanowires,” ACS Nano 5, 5830–5837 (2011).
[Crossref]

Ustinov, V. M.

V. G. Dubrovskii, G. E. Cirlin, and V. M. Ustinov, “Semiconductor nanowhiskers: synthesis, properties, and applications,” Semiconductors 43, 1539–1584 (2009).
[Crossref]

Van Vlack, C. P.

P. Yao, C. P. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80, 195106 (2009).
[Crossref]

Volz, T.

T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6, 607–611 (2012).
[Crossref]

Waks, E.

R. Bose, D. Sridharan, H. Kim, G. S. Solomon, and E. Waks, “Low-photon-number optical switching with a single quantum dot coupled to a photonic crystal cavity,” Phys. Rev. Lett. 108, 227402 (2012).
[Crossref]

Wallraff, A.

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its nonlinearity in a cavity QED system,” Nature 454, 315–318 (2008).
[Crossref]

Walther, T.

M. N. Makhonin, A. P. Foster, A. B. Krysa, P. W. Fry, D. G. Davies, T. Grange, T. Walther, M. S. Skolnick, and L. R. Wilson, “Homogeneous array of nanowire-embedded quantum light emitters,” Nano Lett. 13, 861–865 (2013).
[Crossref]

Wang, J.

S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[Crossref]

Weiler, S.

S. Weiler, A. Ulhaq, S. M. Ulrich, D. Richter, M. Jetter, P. Michler, C. Roy, and S. Hughes, “Phonon-assisted incoherent excitation of a quantum dot and its emission properties,” Phys. Rev. B 86, 241304 (2012).
[Crossref]

Wilson, L. R.

M. N. Makhonin, A. P. Foster, A. B. Krysa, P. W. Fry, D. G. Davies, T. Grange, T. Walther, M. S. Skolnick, and L. R. Wilson, “Homogeneous array of nanowire-embedded quantum light emitters,” Nano Lett. 13, 861–865 (2013).
[Crossref]

Winger, M.

T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6, 607–611 (2012).
[Crossref]

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
[Crossref]

Wonderen, A. J. V.

L. G. Suttorp and A. J. V. Wonderen, “Fano diagonalization of a polariton model for an inhomogeneous absorptive dielectric,” Europhys. Lett. 67, 766–772 (2004).
[Crossref]

Wong, C. W.

J. Gao, S. Combrie, B. Liang, P. Schmitteckert, G. Lehoucq, S. Xavier, X. Xu, K. Busch, D. L. Huffaker, A. De Rossi, and C. W. Wong, “Strongly coupled slow-light polaritons in one-dimensional disordered localized states,” Sci. Rep. 3, 1994 (2013).

Xavier, S.

J. Gao, S. Combrie, B. Liang, P. Schmitteckert, G. Lehoucq, S. Xavier, X. Xu, K. Busch, D. L. Huffaker, A. De Rossi, and C. W. Wong, “Strongly coupled slow-light polaritons in one-dimensional disordered localized states,” Sci. Rep. 3, 1994 (2013).

Xu, X.

J. Gao, S. Combrie, B. Liang, P. Schmitteckert, G. Lehoucq, S. Xavier, X. Xu, K. Busch, D. L. Huffaker, A. De Rossi, and C. W. Wong, “Strongly coupled slow-light polaritons in one-dimensional disordered localized states,” Sci. Rep. 3, 1994 (2013).

Yablonskii, A. L.

A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, T. Fujita, and T. Ishihara, “Polariton effect in distributed feedback microcavities,” J. Phys. Soc. Jpn. 70, 1137–1144 (2001).
[Crossref]

Yamada, H.

M. Tokushima, H. Yamada, and Y. Arakawa, “1.5-mm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab,” Appl. Phys. Lett. 84, 4298–4300 (2004).
[Crossref]

Yang, Z.

Z. Yang, N. Kwong, R. Binder, and A. Smirl, “Stopping, storing and releasing light in quantum well Bragg structures,” J. Opt. B 22, 2144–2156 (2005).
[Crossref]

Yao, P.

P. Yao, V. S. C. Manga Rao, and S. Hughes, “On-chip single photon sources using planar photonic crystals and single quantum dots,” Laser Photon. Rev. 4, 499–516 (2010).
[Crossref]

P. Yao, C. P. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80, 195106 (2009).
[Crossref]

Yarrison-Rice, J. M.

H. J. Joyce, Q. Gao, H. Hoe Tan, C. Jagadish, Y. Kim, J. Zou, L. M. Smith, H. E. Jackson, J. M. Yarrison-Rice, P. Parkinson, and M. B. Johnston, “III-V semiconductor nanowires for optoelectronic device applications,” Prog. Quant. Electron. 35, 23–75 (2011).
[Crossref]

Young, A. B.

A. B. Young, A. C. T. Thijssen, D. M. Beggs, P. Androvitsaneas, L. Kuipers, J. G. Rarity, S. Hughes, and R. Oulton, “Polarization engineering in photonic crystal waveguides for spin-photon entanglers,” Phys. Rev. Lett. 115, 1–5 (2015).

Young, J. F.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005).
[Crossref]

Yu, S. P.

A. Goban, C. L. Hung, J. D. Hood, S. P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” Phys. Rev. Lett. 115, 1–5 (2015).
[Crossref]

S. P. Yu, J. D. Hood, J. A. Muniz, M. J. Martin, R. Norte, C. L. Hung, S. M. Meenehan, J. D. Cohen, O. Painter, and H. J. Kimble, “Nanowire photonic crystal waveguides for single-atom trapping and strong light-matter interactions,” Appl. Phys. Lett. 104, 2012–2017 (2014).

Zou, J.

H. J. Joyce, Q. Gao, H. Hoe Tan, C. Jagadish, Y. Kim, J. Zou, L. M. Smith, H. E. Jackson, J. M. Yarrison-Rice, P. Parkinson, and M. B. Johnston, “III-V semiconductor nanowires for optoelectronic device applications,” Prog. Quant. Electron. 35, 23–75 (2011).
[Crossref]

Zwiller, V.

J.-C. Harmand, L. Liu, G. Patriarche, M. Tchernycheva, N. Akopian, U. Perinetti, and V. Zwiller, “Potential of semiconductor nanowires for single photon sources,” Proc. SPIE 7222, 722219 (2009).
[Crossref]

ACS Nano (1)

S. L. Diedenhofen, O. T. A. Janssen, M. Hocevar, A. Pierret, E. P. A. M. Bakkers, H. P. Urbach, and J. Gómez Rivas, “Controlling the directional emission of light by periodic arrays of heterostructured semiconductor nanowires,” ACS Nano 5, 5830–5837 (2011).
[Crossref]

Appl. Phys. Lett. (3)

S. P. Yu, J. D. Hood, J. A. Muniz, M. J. Martin, R. Norte, C. L. Hung, S. M. Meenehan, J. D. Cohen, O. Painter, and H. J. Kimble, “Nanowire photonic crystal waveguides for single-atom trapping and strong light-matter interactions,” Appl. Phys. Lett. 104, 2012–2017 (2014).

M. Tokushima, H. Yamada, and Y. Arakawa, “1.5-mm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab,” Appl. Phys. Lett. 84, 4298–4300 (2004).
[Crossref]

T. Ba Hoang, J. Beetz, L. Midolo, M. Skacel, M. Lermer, M. Kamp, S. Höfling, L. Balet, N. Chauvin, and A. Fiore, “Enhanced spontaneous emission from quantum dots in short photonic crystal waveguides,” Appl. Phys. Lett. 100, 061122 (2012).
[Crossref]

Europhys. Lett. (1)

L. G. Suttorp and A. J. V. Wonderen, “Fano diagonalization of a polariton model for an inhomogeneous absorptive dielectric,” Europhys. Lett. 67, 766–772 (2004).
[Crossref]

J. Opt. B (1)

Z. Yang, N. Kwong, R. Binder, and A. Smirl, “Stopping, storing and releasing light in quantum well Bragg structures,” J. Opt. B 22, 2144–2156 (2005).
[Crossref]

J. Phys. Soc. Jpn. (1)

A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, T. Fujita, and T. Ishihara, “Polariton effect in distributed feedback microcavities,” J. Phys. Soc. Jpn. 70, 1137–1144 (2001).
[Crossref]

Laser Photon. Rev. (1)

P. Yao, V. S. C. Manga Rao, and S. Hughes, “On-chip single photon sources using planar photonic crystals and single quantum dots,” Laser Photon. Rev. 4, 499–516 (2010).
[Crossref]

Nano Lett. (2)

M. N. Makhonin, A. P. Foster, A. B. Krysa, P. W. Fry, D. G. Davies, T. Grange, T. Walther, M. S. Skolnick, and L. R. Wilson, “Homogeneous array of nanowire-embedded quantum light emitters,” Nano Lett. 13, 861–865 (2013).
[Crossref]

M. Février, P. Gogol, A. Aassime, R. Mégy, C. Delacour, A. Chelnokov, A. Apuzzo, S. Blaize, J. M. Lourtioz, and B. Dagens, “Giant coupling effect between metal nanoparticle chain and optical waveguide,” Nano Lett. 12, 1032–1037 (2012).
[Crossref]

Nanotechnology (1)

N. Dhindsa, A. Chia, J. Boulanger, I. Khodadad, R. LaPierre, and S. S. Saini, “Highly ordered vertical GaAs nanowire arrays with dry etching and their optical properties,” Nanotechnology 25, 305303 (2014).
[Crossref]

Nat. Commun. (1)

B. le Feber, N. Rotenberg, and L. Kuipers, “Nanophotonic control of circular dipole emission,” Nat. Commun. 6, 6695 (2015).
[Crossref]

Nat. Nanotechnol. (1)

I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Deterministic photon-emitter coupling in chiral photonic circuits,” Nat. Nanotechnol. 10, 775–778 (2015).
[Crossref]

Nat. Photonics (2)

T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6, 607–611 (2012).
[Crossref]

J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
[Crossref]

Nat. Phys. (1)

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006).
[Crossref]

Nature (2)

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its nonlinearity in a cavity QED system,” Nature 454, 315–318 (2008).
[Crossref]

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (3)

D. O. Krimer, M. Liertzer, S. Rotter, and H. E. Türeci, “Route from spontaneous decay to complex multimode dynamics in cavity QED,” Phys. Rev. A 89, 1–8 (2014).
[Crossref]

G. Angelatos and S. Hughes, “Entanglement dynamics and Mollow nonuplets between two coupled quantum dots in a nanowire photonic-crystal system,” Phys. Rev. A 91, 051803(R) (2015).
[Crossref]

D. P. Fussell and M. M. Dignam, “Quantum-dot photon dynamics in a coupled-cavity waveguide: observing band-edge quantum optics,” Phys. Rev. A 76, 053801 (2007).
[Crossref]

Phys. Rev. B (10)

F. J. Rodríguez-Fortuño, B. Tomás-Navarro, C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Zero-bandwidth mode in a split-ring-resonator-loaded one-dimensional photonic crystal,” Phys. Rev. B 81, 2–5 (2010).

S. Weiler, A. Ulhaq, S. M. Ulrich, D. Richter, M. Jetter, P. Michler, C. Roy, and S. Hughes, “Phonon-assisted incoherent excitation of a quantum dot and its emission properties,” Phys. Rev. B 86, 241304 (2012).
[Crossref]

Y. M. Niquet and D. C. Mojica, “Quantum dots and tunnel barriers in InAs InP nanowire heterostructures: electronic and optical properties,” Phys. Rev. B 77, 1–12 (2008).
[Crossref]

M. Patterson and S. Hughes, “Interplay between disorder-induced scattering and local field effects in photonic crystal waveguides,” Phys. Rev. B 81, 245321 (2010).
[Crossref]

V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
[Crossref]

P. T. Kristensen, J. Mørk, P. Lodahl, and S. Hughes, “Decay dynamics of radiatively coupled quantum dots in photonic crystal slabs,” Phys. Rev. B 83, 075305 (2011).
[Crossref]

G. Angelatos and S. Hughes, “Theory and design of quantum light sources from quantum dots embedded in semiconductor-nanowire photonic-crystal systems,” Phys. Rev. B 90, 205406 (2014).
[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]

G. X. Li, J. Evers, and C. H. Keitel, “Spontaneous emission interference in negative-refractive-index waveguides,” Phys. Rev. B 80, 1–7 (2009).

P. Yao, C. P. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80, 195106 (2009).
[Crossref]

Phys. Rev. Lett. (7)

A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91, 183901 (2003).
[Crossref]

A. B. Young, A. C. T. Thijssen, D. M. Beggs, P. Androvitsaneas, L. Kuipers, J. G. Rarity, S. Hughes, and R. Oulton, “Polarization engineering in photonic crystal waveguides for spin-photon entanglers,” Phys. Rev. Lett. 115, 1–5 (2015).

R. Bose, D. Sridharan, H. Kim, G. S. Solomon, and E. Waks, “Low-photon-number optical switching with a single quantum dot coupled to a photonic crystal cavity,” Phys. Rev. Lett. 108, 227402 (2012).
[Crossref]

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005).
[Crossref]

M. Patterson, S. Hughes, S. Combrié, N.-V.-Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouën, “Disorder-induced coherent scattering in slow-light photonic crystal waveguides,” Phys. Rev. Lett. 102, 253903 (2009).
[Crossref]

S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[Crossref]

A. Goban, C. L. Hung, J. D. Hood, S. P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” Phys. Rev. Lett. 115, 1–5 (2015).
[Crossref]

Proc. SPIE (1)

J.-C. Harmand, L. Liu, G. Patriarche, M. Tchernycheva, N. Akopian, U. Perinetti, and V. Zwiller, “Potential of semiconductor nanowires for single photon sources,” Proc. SPIE 7222, 722219 (2009).
[Crossref]

Prog. Quant. Electron. (1)

H. J. Joyce, Q. Gao, H. Hoe Tan, C. Jagadish, Y. Kim, J. Zou, L. M. Smith, H. E. Jackson, J. M. Yarrison-Rice, P. Parkinson, and M. B. Johnston, “III-V semiconductor nanowires for optoelectronic device applications,” Prog. Quant. Electron. 35, 23–75 (2011).
[Crossref]

Sci. Rep. (1)

J. Gao, S. Combrie, B. Liang, P. Schmitteckert, G. Lehoucq, S. Xavier, X. Xu, K. Busch, D. L. Huffaker, A. De Rossi, and C. W. Wong, “Strongly coupled slow-light polaritons in one-dimensional disordered localized states,” Sci. Rep. 3, 1994 (2013).

Semiconductors (1)

V. G. Dubrovskii, G. E. Cirlin, and V. M. Ustinov, “Semiconductor nanowhiskers: synthesis, properties, and applications,” Semiconductors 43, 1539–1584 (2009).
[Crossref]

Other (4)

G. Angelatos, “Theory and applications of light-matter interactions in quantum dot nanowire photonic crystal systems,” Master’s thesis (Queen’s University, 2015).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

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

P. Meystre and M. Sargent, Elements of Quantum Optics (Springer, 1999).

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Figures (6)

Fig. 1.
Fig. 1.

(a) Proposed polariton waveguide, with yellow QDs embedded inside the PC waveguide channel, and a red external QD coupling to the structure. Red arrows denote the waveguide direction ( e x ) and NWs are cut out so embedded QDs can be seen. (b) Slow-light region of the band structure on top, and Im { G ( r n , r n ) } in units of ρ h (see text) on bottom; the infinite polariton waveguide in solid blue is compared with the original PC waveguide in dashed orange.

Fig. 2.
Fig. 2.

(a) Imaginary and real components of G ( N ) ( r t , r t ; ω ) on right and left, respectively. G ( 101 ) ( r 51 , r 51 ; ω ) in solid blue is compared with G ( 51 ) ( r 26 , r 26 ; ω ) in solid orange, G ( 21 ) ( r 11 , r 11 ; ω ) in dashed red, and N = 0 results in dash-dotted gray. (b)  Im { G ( 101 ) ( r n , r n ; ω ) } near ω 0 , plotted on a logarithmic scale due to the narrowness of the primary resonance. Δ x = n 51 and the gray dashed lines denote the terminus of the QD array at Δ x = ± 50 . All values are in units of ρ h ( ω ) .

Fig. 3.
Fig. 3.

| G ( 101 ) ( r n , r 51 ; ω ) | in units of ρ h ( ω ) for the N = 101 polariton waveguide, where the termination of the QD array at Δ x = ± 50 is again denoted with dashed gray lines.

Fig. 4.
Fig. 4.

Im { G ( 101 ) ( r n , r n ; ω ) } in units of ρ h ( ω ) under various instances of disorder compared with the disorder-free result in dashed–dotted gray. On left, sample results for coupling-strength reductions σ g = 5 % and 10% are plotted in dashed red and solid blue. On right, the influence of fluctuations in ω 0 is seen, with a σ ω 0 = 5    μeV result in solid orange compared with a σ ω 0 = 10    μeV result in solid blue. An instance of the same set of σ ω 0 = 10    μeV frequencies and random σ g = 10 % reductions is plotted in dashed red.

Fig. 5.
Fig. 5.

S 0 ( ω ) in dashed blue and S ( r D , ω ) in solid orange for ω t = ω 1 , with d t = 30    D in (a) and ω t = ω FP with d t = 60    D in (b). Im { G ( r t , r t ; ω ) } in arbitrary units is shown in dash-dotted gray.

Fig. 6.
Fig. 6.

Peaks in S 0 ( ω ) in blue as a function of ω t and d t . Results for d t = 10 , 30, and 60 D are presented from bottom to top, with ω t near ω 1 and in the FP region on the left and right, respectively. Crosses and circles denote peaks of Figs. 5(a) and 5(b), ω t is in dashed red, and LDOS peaks are in dashed gray.

Equations (7)

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[ × × ω 2 c 2 ε ( r ) ] G ( r , r ; ω ) = ω 2 c 2 1 δ ( r r ) ,
ω k , ± = ω 0 + ω k 2 ± 1 2 ( ω 0 ω k ) 2 + 4 g k 2 ,
Γ k , ± = Γ 0 ω k , ± 2 ω k 2 ( ω k , ± 2 ω k 2 ) + ( ω k , ± 2 ω 0 2 ) .
G P ( r , r ; ω ) = i a ω 2 v ˜ g [ Θ ( x x ) u k ω ( r ) u k ω * ( r ) e ( i k ω κ ω ) ( x x ) + Θ ( x x ) u k ω * ( r ) u k ω ( r ) e ( i k ω κ ω ) ( x x ) ] ,
G w ( r , r ; ω ) = i a ω 2 v g [ Θ ( x x ) u k ω ( r ) u k ω * ( r ) e i k ω ( x x ) + Θ ( x x ) u k ω * ( r ) u k ω ( r ) e i k ω ( x x ) ] .
H = ω t σ ^ + σ ^ + d r 0 d ω l ω l b ^ ( r ; ω l ) · b ^ ( r ; ω l ) ( σ ^ + + σ ^ ) ( d t · E ^ ( r t ) + H.c. ) ,
S ( r D , ω ) = | ( ω t + ω ) G ( r D , r t ; ω ) · d / ε 0 ω t 2 ω 2 ω Σ ( ω ) i ω Γ t | 2 .

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