Abstract

The single photon scattering in one-dimensional waveguide coupled to two separated atoms is investigated. The first atom is considered as a Λ system and the second one is taken as V -type configuration. The analytical expressions of the single photon scattering spectra are obtained. The calculated results show that the polarization conversion of single photon can be realized by controlling the distance between the two atoms due to the interference effects. The conversion efficiency can reach unit in the ideal case. Furthermore, the polarization conversion of the single photon also depends on the initial state of the Λ system. The influences of dissipations on the single photon polarization conversion are also shown.

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

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  1. 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, 402-406 (2007).
    [Crossref] [PubMed]
  2. M. A. M. Versteegh, M. E. Reimer, K. D. Jöns, D. Dalacu, P. J. Poole, A. Gulinatti, A. Giudice, and V. Zwiller, “Observation of strongly entangled photon pairs from a nanowire quantum dot,” Nat. Commun. 5, 5298 (2014).
    [Crossref] [PubMed]
  3. M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
    [Crossref] [PubMed]
  4. G. Reithmaier, M. Kaniber, F. Flassig, S. Lichtmannecker, K. Müller, A. Andrejew, J. Vučković, R. Gross, and J. J. Finley, “On-chip generation, routing, and detection of resonance fluorescence,” Nano Lett. 15, 5208-5213 (2015).
    [Crossref] [PubMed]
  5. J. T. Shen and S. Fan, “Coherent photon transport from spontaneous emission in one-dimensional waveguide,” Opt. Lett. 30, 2001-2003 (2005).
    [Crossref] [PubMed]
  6. D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807-812 (2009).
    [Crossref]
  7. J. T. Shen and S. Fan, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system,” Phys. Rev. Lett. 98, 153003 (2007).
    [Crossref] [PubMed]
  8. L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett. 101, 100501 (2008).
    [Crossref] [PubMed]
  9. P. Longo, P. Schmitteckert, and K. Busch, “Few-photon transport in low-dimensional systems: interaction-induced radiation trapping,” Phys. Rev. Lett. 104, 023602 (2010).
    [Crossref] [PubMed]
  10. D. Roy, “Two-photon scattering by a driven three-level emitter in a one-dimensional waveguide and electromagnetically induced transparency,” Phys. Rev. Lett. 106, 053601 (2011).
    [Crossref] [PubMed]
  11. P. Kolchin, R. F. Oulton, and X. Zhang, “Nonlinear quantum optics in a waveguide: distinct single photons strongly interacting at the single atom level,” Phys. Rev. Lett. 106, 113601 (2011).
    [Crossref] [PubMed]
  12. M. Bradford, K. C. Obi, and J. T. Shen, “Efficient single-photon frequency conversion using a sagnac interferometer,” Phys. Rev. Lett. 108, 103902 (2012).
    [Crossref] [PubMed]
  13. L. Neumeier, M. Leib, and M. J. Hartmann, “Single-photon transistor in circuit quantum electrodynamics,” Phys. Rev. Lett. 111, 063601 (2013).
    [Crossref] [PubMed]
  14. L. Zhou, L. P. Yang, Y. Li, and C. P. Sun, “Quantum routing of single photons with a cyclic three-level system,” Phys. Rev. Lett. 111, 103604 (2013).
    [Crossref]
  15. E. Sanchez-Burillo, D. Zueco, J. J. Garcia-Ripoll, and L. Martin-Moreno, “Scattering in the ultrastrong regime: nonlinear Optics with One Photon,” Phys. Rev. Lett. 113, 263604 (2014).
    [Crossref]
  16. 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]
  17. L. Yuan, S. Xu, and S. Fan, “Achieving nonreciprocal unidirectional single-photon quantum transport using the photonic Aharonov-Bohm effect,” Opt. Lett. 40, 5140-5142 (2015).
    [Crossref] [PubMed]
  18. Z. Liao, H. Nha, and M. S. Zubairy, “Dynamical theory of single-photon transport in a one-dimensional waveguide coupled to identical and nonidentical emitters,” Phys. Rev. A 94, 053842 (2016).
    [Crossref]
  19. Z. Liao, M. Al-Amri, and M. S. Zubairy, “Measurement of deep-subwavelength emitter separation in a waveguide-QED system,” Opt. Express 25, 31997-32009 (2017).
    [Crossref] [PubMed]
  20. I. C. Hoi, C. M. Wilson, G. Johansson, T. Palomaki, B. Peropadre, and P. Delsing, “Demonstration of a single-photon router in the microwave regime,” Phys. Rev. Lett. 107, 073601 (2011).
    [Crossref] [PubMed]
  21. D. Roy, “Few-photon optical diode,” Phys. Rev. B 81, 155117 (2010).
    [Crossref]
  22. M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577-1580 (2016).
    [Crossref] [PubMed]
  23. H. Zheng, D. J. Gauthier, and H. U. Baranger, “Waveguide-QED-Based photonic quantum computation,” Phys. Rev. Lett. 111, 090502 (2013).
    [Crossref] [PubMed]
  24. H. Pichler and P. Zoller, “Photonic circuits with time delays and quantum feedback,” Phys. Rev. Lett. 116, 093601 (2016).
    [Crossref] [PubMed]
  25. D. Roy, C. M. Wilson, and O. Firstenberg, “Colloquium: Strongly interacting photons in one-dimensional continuum,” Rev. Mod. Phys. 89, 021001 (2017).
    [Crossref]
  26. Z. Liao, X. Zeng, H. Nha, and M. S. Zubairy, “Photon transport in a one-dimensional nanophotonic waveguide QED system,” Phys. Scr. 91, 063004 (2016).
    [Crossref]
  27. S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393 (1991).
    [Crossref]
  28. J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99, 063908 (2007).
    [Crossref] [PubMed]
  29. L. Cong, W. Cao, X. Zhang, Z. Tian, J. Cu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103, 171107 (2013).
    [Crossref]
  30. J. Sun, C. Cu, X. Chen, Z. Li, L. Liu, and F. Martin, “Ultra-wideband and broad-angle linear polarization conversion metasurface,” J. Appl. Phys. 121, 174902 (2017).
    [Crossref]
  31. A. Cerjan and S. Fan, “Achieving arbitrary control over pairs of polarization states using complex birefringent metamaterials,” Phys. Rev. Lett. 118, 253902 (2017).
    [Crossref] [PubMed]
  32. N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science,  340, 1304-1307 (2013).
    [Crossref] [PubMed]
  33. Y. Guo, M. Xiao, and S. Fan, “Topologically protected complete polarization conversion,” Phys. Rev. Lett. 119, 167401 (2017).
    [Crossref] [PubMed]
  34. T. S. Tsoi and C. K. Law, “Single-photon scattering on Λ-type three-level atoms in a one-dimensional waveguide,” Phys. Rev. A 80,033823 (2009).
    [Crossref]
  35. K. C. Obi and J. T. Shen, “Perturbative and iterative methods for photon transport in one-dimensional waveguides,” Opt. Commun. 343, 135-139 (2015).
    [Crossref]
  36. Z. Y. Zhang, Y. L. Dong, S. L. Zhang, and S. Q. Zhu, “Polarization-dependent photon switch in a one-dimensional coupled-r esonator waveguide,” Opt. Express 21, 20786-20799 (2013).
    [Crossref] [PubMed]
  37. M. X. Li, J. Yang, G. W. Lin, Y. P. Niu, and S. Q. Gong, “Scattering of a single photon in a one-dimensional coupled resonator waveguide with a Λ-type emitter assisted by an additional cavity,” Chin. Phys. B 27, 054206 (2018).
    [Crossref]
  38. J. T. Shen and S. Fan, “Thoery of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom,” Phys. Rev. A 79, 023837 (2009).
    [Crossref]

2018 (1)

M. X. Li, J. Yang, G. W. Lin, Y. P. Niu, and S. Q. Gong, “Scattering of a single photon in a one-dimensional coupled resonator waveguide with a Λ-type emitter assisted by an additional cavity,” Chin. Phys. B 27, 054206 (2018).
[Crossref]

2017 (5)

Z. Liao, M. Al-Amri, and M. S. Zubairy, “Measurement of deep-subwavelength emitter separation in a waveguide-QED system,” Opt. Express 25, 31997-32009 (2017).
[Crossref] [PubMed]

D. Roy, C. M. Wilson, and O. Firstenberg, “Colloquium: Strongly interacting photons in one-dimensional continuum,” Rev. Mod. Phys. 89, 021001 (2017).
[Crossref]

J. Sun, C. Cu, X. Chen, Z. Li, L. Liu, and F. Martin, “Ultra-wideband and broad-angle linear polarization conversion metasurface,” J. Appl. Phys. 121, 174902 (2017).
[Crossref]

A. Cerjan and S. Fan, “Achieving arbitrary control over pairs of polarization states using complex birefringent metamaterials,” Phys. Rev. Lett. 118, 253902 (2017).
[Crossref] [PubMed]

Y. Guo, M. Xiao, and S. Fan, “Topologically protected complete polarization conversion,” Phys. Rev. Lett. 119, 167401 (2017).
[Crossref] [PubMed]

2016 (4)

H. Pichler and P. Zoller, “Photonic circuits with time delays and quantum feedback,” Phys. Rev. Lett. 116, 093601 (2016).
[Crossref] [PubMed]

Z. Liao, X. Zeng, H. Nha, and M. S. Zubairy, “Photon transport in a one-dimensional nanophotonic waveguide QED system,” Phys. Scr. 91, 063004 (2016).
[Crossref]

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577-1580 (2016).
[Crossref] [PubMed]

Z. Liao, H. Nha, and M. S. Zubairy, “Dynamical theory of single-photon transport in a one-dimensional waveguide coupled to identical and nonidentical emitters,” Phys. Rev. A 94, 053842 (2016).
[Crossref]

2015 (4)

K. C. Obi and J. T. Shen, “Perturbative and iterative methods for photon transport in one-dimensional waveguides,” Opt. Commun. 343, 135-139 (2015).
[Crossref]

G. Reithmaier, M. Kaniber, F. Flassig, S. Lichtmannecker, K. Müller, A. Andrejew, J. Vučković, R. Gross, and J. J. Finley, “On-chip generation, routing, and detection of resonance fluorescence,” Nano Lett. 15, 5208-5213 (2015).
[Crossref] [PubMed]

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]

L. Yuan, S. Xu, and S. Fan, “Achieving nonreciprocal unidirectional single-photon quantum transport using the photonic Aharonov-Bohm effect,” Opt. Lett. 40, 5140-5142 (2015).
[Crossref] [PubMed]

2014 (3)

M. A. M. Versteegh, M. E. Reimer, K. D. Jöns, D. Dalacu, P. J. Poole, A. Gulinatti, A. Giudice, and V. Zwiller, “Observation of strongly entangled photon pairs from a nanowire quantum dot,” Nat. Commun. 5, 5298 (2014).
[Crossref] [PubMed]

M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref] [PubMed]

E. Sanchez-Burillo, D. Zueco, J. J. Garcia-Ripoll, and L. Martin-Moreno, “Scattering in the ultrastrong regime: nonlinear Optics with One Photon,” Phys. Rev. Lett. 113, 263604 (2014).
[Crossref]

2013 (6)

H. Zheng, D. J. Gauthier, and H. U. Baranger, “Waveguide-QED-Based photonic quantum computation,” Phys. Rev. Lett. 111, 090502 (2013).
[Crossref] [PubMed]

Z. Y. Zhang, Y. L. Dong, S. L. Zhang, and S. Q. Zhu, “Polarization-dependent photon switch in a one-dimensional coupled-r esonator waveguide,” Opt. Express 21, 20786-20799 (2013).
[Crossref] [PubMed]

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science,  340, 1304-1307 (2013).
[Crossref] [PubMed]

L. Neumeier, M. Leib, and M. J. Hartmann, “Single-photon transistor in circuit quantum electrodynamics,” Phys. Rev. Lett. 111, 063601 (2013).
[Crossref] [PubMed]

L. Zhou, L. P. Yang, Y. Li, and C. P. Sun, “Quantum routing of single photons with a cyclic three-level system,” Phys. Rev. Lett. 111, 103604 (2013).
[Crossref]

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Cu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103, 171107 (2013).
[Crossref]

2012 (1)

M. Bradford, K. C. Obi, and J. T. Shen, “Efficient single-photon frequency conversion using a sagnac interferometer,” Phys. Rev. Lett. 108, 103902 (2012).
[Crossref] [PubMed]

2011 (3)

D. Roy, “Two-photon scattering by a driven three-level emitter in a one-dimensional waveguide and electromagnetically induced transparency,” Phys. Rev. Lett. 106, 053601 (2011).
[Crossref] [PubMed]

P. Kolchin, R. F. Oulton, and X. Zhang, “Nonlinear quantum optics in a waveguide: distinct single photons strongly interacting at the single atom level,” Phys. Rev. Lett. 106, 113601 (2011).
[Crossref] [PubMed]

I. C. Hoi, C. M. Wilson, G. Johansson, T. Palomaki, B. Peropadre, and P. Delsing, “Demonstration of a single-photon router in the microwave regime,” Phys. Rev. Lett. 107, 073601 (2011).
[Crossref] [PubMed]

2010 (2)

D. Roy, “Few-photon optical diode,” Phys. Rev. B 81, 155117 (2010).
[Crossref]

P. Longo, P. Schmitteckert, and K. Busch, “Few-photon transport in low-dimensional systems: interaction-induced radiation trapping,” Phys. Rev. Lett. 104, 023602 (2010).
[Crossref] [PubMed]

2009 (3)

D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807-812 (2009).
[Crossref]

T. S. Tsoi and C. K. Law, “Single-photon scattering on Λ-type three-level atoms in a one-dimensional waveguide,” Phys. Rev. A 80,033823 (2009).
[Crossref]

J. T. Shen and S. Fan, “Thoery of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom,” Phys. Rev. A 79, 023837 (2009).
[Crossref]

2008 (1)

L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett. 101, 100501 (2008).
[Crossref] [PubMed]

2007 (3)

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, 402-406 (2007).
[Crossref] [PubMed]

J. T. Shen and S. Fan, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system,” Phys. Rev. Lett. 98, 153003 (2007).
[Crossref] [PubMed]

J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99, 063908 (2007).
[Crossref] [PubMed]

2005 (1)

1991 (1)

S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393 (1991).
[Crossref]

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, 402-406 (2007).
[Crossref] [PubMed]

Al-Amri, M.

Andrejew, A.

G. Reithmaier, M. Kaniber, F. Flassig, S. Lichtmannecker, K. Müller, A. Andrejew, J. Vučković, R. Gross, and J. J. Finley, “On-chip generation, routing, and detection of resonance fluorescence,” Nano Lett. 15, 5208-5213 (2015).
[Crossref] [PubMed]

Arcari, M.

M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref] [PubMed]

Azad, A. K.

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science,  340, 1304-1307 (2013).
[Crossref] [PubMed]

Baranger, H. U.

H. Zheng, D. J. Gauthier, and H. U. Baranger, “Waveguide-QED-Based photonic quantum computation,” Phys. Rev. Lett. 111, 090502 (2013).
[Crossref] [PubMed]

Bradford, M.

M. Bradford, K. C. Obi, and J. T. Shen, “Efficient single-photon frequency conversion using a sagnac interferometer,” Phys. Rev. Lett. 108, 103902 (2012).
[Crossref] [PubMed]

Bryan-Brown, G. P.

S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393 (1991).
[Crossref]

Busch, K.

P. Longo, P. Schmitteckert, and K. Busch, “Few-photon transport in low-dimensional systems: interaction-induced radiation trapping,” Phys. Rev. Lett. 104, 023602 (2010).
[Crossref] [PubMed]

Cao, W.

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Cu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103, 171107 (2013).
[Crossref]

Cerjan, A.

A. Cerjan and S. Fan, “Achieving arbitrary control over pairs of polarization states using complex birefringent metamaterials,” Phys. Rev. Lett. 118, 253902 (2017).
[Crossref] [PubMed]

Chan, C. T.

J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99, 063908 (2007).
[Crossref] [PubMed]

Chang, D. E.

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]

D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807-812 (2009).
[Crossref]

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, 402-406 (2007).
[Crossref] [PubMed]

Chen, H. T.

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science,  340, 1304-1307 (2013).
[Crossref] [PubMed]

Chen, X.

J. Sun, C. Cu, X. Chen, Z. Li, L. Liu, and F. Martin, “Ultra-wideband and broad-angle linear polarization conversion metasurface,” J. Appl. Phys. 121, 174902 (2017).
[Crossref]

Chowdhury, D. R.

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science,  340, 1304-1307 (2013).
[Crossref] [PubMed]

Cong, L.

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Cu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103, 171107 (2013).
[Crossref]

Cu, C.

J. Sun, C. Cu, X. Chen, Z. Li, L. Liu, and F. Martin, “Ultra-wideband and broad-angle linear polarization conversion metasurface,” J. Appl. Phys. 121, 174902 (2017).
[Crossref]

Cu, J.

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Cu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103, 171107 (2013).
[Crossref]

Dalacu, D.

M. A. M. Versteegh, M. E. Reimer, K. D. Jöns, D. Dalacu, P. J. Poole, A. Gulinatti, A. Giudice, and V. Zwiller, “Observation of strongly entangled photon pairs from a nanowire quantum dot,” Nat. Commun. 5, 5298 (2014).
[Crossref] [PubMed]

Dalvit, D. A. R.

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science,  340, 1304-1307 (2013).
[Crossref] [PubMed]

Delsing, P.

I. C. Hoi, C. M. Wilson, G. Johansson, T. Palomaki, B. Peropadre, and P. Delsing, “Demonstration of a single-photon router in the microwave regime,” Phys. Rev. Lett. 107, 073601 (2011).
[Crossref] [PubMed]

Demler, E. A.

D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807-812 (2009).
[Crossref]

Dong, Y. L.

Douglas, J. S.

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]

Elston, S. J.

S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393 (1991).
[Crossref]

Fan, S.

A. Cerjan and S. Fan, “Achieving arbitrary control over pairs of polarization states using complex birefringent metamaterials,” Phys. Rev. Lett. 118, 253902 (2017).
[Crossref] [PubMed]

Y. Guo, M. Xiao, and S. Fan, “Topologically protected complete polarization conversion,” Phys. Rev. Lett. 119, 167401 (2017).
[Crossref] [PubMed]

L. Yuan, S. Xu, and S. Fan, “Achieving nonreciprocal unidirectional single-photon quantum transport using the photonic Aharonov-Bohm effect,” Opt. Lett. 40, 5140-5142 (2015).
[Crossref] [PubMed]

J. T. Shen and S. Fan, “Thoery of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom,” Phys. Rev. A 79, 023837 (2009).
[Crossref]

J. T. Shen and S. Fan, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system,” Phys. Rev. Lett. 98, 153003 (2007).
[Crossref] [PubMed]

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

Finley, J. J.

G. Reithmaier, M. Kaniber, F. Flassig, S. Lichtmannecker, K. Müller, A. Andrejew, J. Vučković, R. Gross, and J. J. Finley, “On-chip generation, routing, and detection of resonance fluorescence,” Nano Lett. 15, 5208-5213 (2015).
[Crossref] [PubMed]

Firstenberg, O.

D. Roy, C. M. Wilson, and O. Firstenberg, “Colloquium: Strongly interacting photons in one-dimensional continuum,” Rev. Mod. Phys. 89, 021001 (2017).
[Crossref]

Flassig, F.

G. Reithmaier, M. Kaniber, F. Flassig, S. Lichtmannecker, K. Müller, A. Andrejew, J. Vučković, R. Gross, and J. J. Finley, “On-chip generation, routing, and detection of resonance fluorescence,” Nano Lett. 15, 5208-5213 (2015).
[Crossref] [PubMed]

Garcia-Ripoll, J. J.

E. Sanchez-Burillo, D. Zueco, J. J. Garcia-Ripoll, and L. Martin-Moreno, “Scattering in the ultrastrong regime: nonlinear Optics with One Photon,” Phys. Rev. Lett. 113, 263604 (2014).
[Crossref]

Gauthier, D. J.

H. Zheng, D. J. Gauthier, and H. U. Baranger, “Waveguide-QED-Based photonic quantum computation,” Phys. Rev. Lett. 111, 090502 (2013).
[Crossref] [PubMed]

Giudice, A.

M. A. M. Versteegh, M. E. Reimer, K. D. Jöns, D. Dalacu, P. J. Poole, A. Gulinatti, A. Giudice, and V. Zwiller, “Observation of strongly entangled photon pairs from a nanowire quantum dot,” Nat. Commun. 5, 5298 (2014).
[Crossref] [PubMed]

Gong, S. Q.

M. X. Li, J. Yang, G. W. Lin, Y. P. Niu, and S. Q. Gong, “Scattering of a single photon in a one-dimensional coupled resonator waveguide with a Λ-type emitter assisted by an additional cavity,” Chin. Phys. B 27, 054206 (2018).
[Crossref]

Gong, Z. R.

L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett. 101, 100501 (2008).
[Crossref] [PubMed]

Gorshkov, A. V.

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]

Grady, N. K.

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science,  340, 1304-1307 (2013).
[Crossref] [PubMed]

Gross, R.

G. Reithmaier, M. Kaniber, F. Flassig, S. Lichtmannecker, K. Müller, A. Andrejew, J. Vučković, R. Gross, and J. J. Finley, “On-chip generation, routing, and detection of resonance fluorescence,” Nano Lett. 15, 5208-5213 (2015).
[Crossref] [PubMed]

Gulinatti, A.

M. A. M. Versteegh, M. E. Reimer, K. D. Jöns, D. Dalacu, P. J. Poole, A. Gulinatti, A. Giudice, and V. Zwiller, “Observation of strongly entangled photon pairs from a nanowire quantum dot,” Nat. Commun. 5, 5298 (2014).
[Crossref] [PubMed]

Guo, Y.

Y. Guo, M. Xiao, and S. Fan, “Topologically protected complete polarization conversion,” Phys. Rev. Lett. 119, 167401 (2017).
[Crossref] [PubMed]

Habibian, H.

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]

Han, J.

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Cu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103, 171107 (2013).
[Crossref]

Hansen, S. L.

M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref] [PubMed]

Hao, J.

J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99, 063908 (2007).
[Crossref] [PubMed]

Hartmann, M. J.

L. Neumeier, M. Leib, and M. J. Hartmann, “Single-photon transistor in circuit quantum electrodynamics,” Phys. Rev. Lett. 111, 063601 (2013).
[Crossref] [PubMed]

Hemmer, P. R.

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, 402-406 (2007).
[Crossref] [PubMed]

Heyes, J. E.

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science,  340, 1304-1307 (2013).
[Crossref] [PubMed]

Hilico, A.

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577-1580 (2016).
[Crossref] [PubMed]

Hoi, I. C.

I. C. Hoi, C. M. Wilson, G. Johansson, T. Palomaki, B. Peropadre, and P. Delsing, “Demonstration of a single-photon router in the microwave regime,” Phys. Rev. Lett. 107, 073601 (2011).
[Crossref] [PubMed]

Hung, C. L.

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]

Javadi, A.

M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref] [PubMed]

Jiang, T.

J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99, 063908 (2007).
[Crossref] [PubMed]

Johansson, G.

I. C. Hoi, C. M. Wilson, G. Johansson, T. Palomaki, B. Peropadre, and P. Delsing, “Demonstration of a single-photon router in the microwave regime,” Phys. Rev. Lett. 107, 073601 (2011).
[Crossref] [PubMed]

Jöns, K. D.

M. A. M. Versteegh, M. E. Reimer, K. D. Jöns, D. Dalacu, P. J. Poole, A. Gulinatti, A. Giudice, and V. Zwiller, “Observation of strongly entangled photon pairs from a nanowire quantum dot,” Nat. Commun. 5, 5298 (2014).
[Crossref] [PubMed]

Kaniber, M.

G. Reithmaier, M. Kaniber, F. Flassig, S. Lichtmannecker, K. Müller, A. Andrejew, J. Vučković, R. Gross, and J. J. Finley, “On-chip generation, routing, and detection of resonance fluorescence,” Nano Lett. 15, 5208-5213 (2015).
[Crossref] [PubMed]

Kimble, H. J.

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]

Kolchin, P.

P. Kolchin, R. F. Oulton, and X. Zhang, “Nonlinear quantum optics in a waveguide: distinct single photons strongly interacting at the single atom level,” Phys. Rev. Lett. 106, 113601 (2011).
[Crossref] [PubMed]

Kong, J. A.

J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99, 063908 (2007).
[Crossref] [PubMed]

Law, C. K.

T. S. Tsoi and C. K. Law, “Single-photon scattering on Λ-type three-level atoms in a one-dimensional waveguide,” Phys. Rev. A 80,033823 (2009).
[Crossref]

Lee, E. H.

M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref] [PubMed]

Leib, M.

L. Neumeier, M. Leib, and M. J. Hartmann, “Single-photon transistor in circuit quantum electrodynamics,” Phys. Rev. Lett. 111, 063601 (2013).
[Crossref] [PubMed]

Li, M. X.

M. X. Li, J. Yang, G. W. Lin, Y. P. Niu, and S. Q. Gong, “Scattering of a single photon in a one-dimensional coupled resonator waveguide with a Λ-type emitter assisted by an additional cavity,” Chin. Phys. B 27, 054206 (2018).
[Crossref]

Li, Y.

L. Zhou, L. P. Yang, Y. Li, and C. P. Sun, “Quantum routing of single photons with a cyclic three-level system,” Phys. Rev. Lett. 111, 103604 (2013).
[Crossref]

Li, Z.

J. Sun, C. Cu, X. Chen, Z. Li, L. Liu, and F. Martin, “Ultra-wideband and broad-angle linear polarization conversion metasurface,” J. Appl. Phys. 121, 174902 (2017).
[Crossref]

Liao, Z.

Z. Liao, M. Al-Amri, and M. S. Zubairy, “Measurement of deep-subwavelength emitter separation in a waveguide-QED system,” Opt. Express 25, 31997-32009 (2017).
[Crossref] [PubMed]

Z. Liao, X. Zeng, H. Nha, and M. S. Zubairy, “Photon transport in a one-dimensional nanophotonic waveguide QED system,” Phys. Scr. 91, 063004 (2016).
[Crossref]

Z. Liao, H. Nha, and M. S. Zubairy, “Dynamical theory of single-photon transport in a one-dimensional waveguide coupled to identical and nonidentical emitters,” Phys. Rev. A 94, 053842 (2016).
[Crossref]

Lichtmannecker, S.

G. Reithmaier, M. Kaniber, F. Flassig, S. Lichtmannecker, K. Müller, A. Andrejew, J. Vučković, R. Gross, and J. J. Finley, “On-chip generation, routing, and detection of resonance fluorescence,” Nano Lett. 15, 5208-5213 (2015).
[Crossref] [PubMed]

Lin, G. W.

M. X. Li, J. Yang, G. W. Lin, Y. P. Niu, and S. Q. Gong, “Scattering of a single photon in a one-dimensional coupled resonator waveguide with a Λ-type emitter assisted by an additional cavity,” Chin. Phys. B 27, 054206 (2018).
[Crossref]

Liu, J.

M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref] [PubMed]

Liu, L.

J. Sun, C. Cu, X. Chen, Z. Li, L. Liu, and F. Martin, “Ultra-wideband and broad-angle linear polarization conversion metasurface,” J. Appl. Phys. 121, 174902 (2017).
[Crossref]

Liu, Y. X.

L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett. 101, 100501 (2008).
[Crossref] [PubMed]

Lodahl, P.

M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref] [PubMed]

Longo, P.

P. Longo, P. Schmitteckert, and K. Busch, “Few-photon transport in low-dimensional systems: interaction-induced radiation trapping,” Phys. Rev. Lett. 104, 023602 (2010).
[Crossref] [PubMed]

Lukin, M. D.

D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807-812 (2009).
[Crossref]

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, 402-406 (2007).
[Crossref] [PubMed]

Mahmoodian, S.

M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref] [PubMed]

Martin, F.

J. Sun, C. Cu, X. Chen, Z. Li, L. Liu, and F. Martin, “Ultra-wideband and broad-angle linear polarization conversion metasurface,” J. Appl. Phys. 121, 174902 (2017).
[Crossref]

Martin-Moreno, L.

E. Sanchez-Burillo, D. Zueco, J. J. Garcia-Ripoll, and L. Martin-Moreno, “Scattering in the ultrastrong regime: nonlinear Optics with One Photon,” Phys. Rev. Lett. 113, 263604 (2014).
[Crossref]

Mukherjee, A.

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, 402-406 (2007).
[Crossref] [PubMed]

Müller, K.

G. Reithmaier, M. Kaniber, F. Flassig, S. Lichtmannecker, K. Müller, A. Andrejew, J. Vučković, R. Gross, and J. J. Finley, “On-chip generation, routing, and detection of resonance fluorescence,” Nano Lett. 15, 5208-5213 (2015).
[Crossref] [PubMed]

Neumeier, L.

L. Neumeier, M. Leib, and M. J. Hartmann, “Single-photon transistor in circuit quantum electrodynamics,” Phys. Rev. Lett. 111, 063601 (2013).
[Crossref] [PubMed]

Nha, H.

Z. Liao, H. Nha, and M. S. Zubairy, “Dynamical theory of single-photon transport in a one-dimensional waveguide coupled to identical and nonidentical emitters,” Phys. Rev. A 94, 053842 (2016).
[Crossref]

Z. Liao, X. Zeng, H. Nha, and M. S. Zubairy, “Photon transport in a one-dimensional nanophotonic waveguide QED system,” Phys. Scr. 91, 063004 (2016).
[Crossref]

Niu, Y. P.

M. X. Li, J. Yang, G. W. Lin, Y. P. Niu, and S. Q. Gong, “Scattering of a single photon in a one-dimensional coupled resonator waveguide with a Λ-type emitter assisted by an additional cavity,” Chin. Phys. B 27, 054206 (2018).
[Crossref]

Nori, F.

L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett. 101, 100501 (2008).
[Crossref] [PubMed]

Obi, K. C.

K. C. Obi and J. T. Shen, “Perturbative and iterative methods for photon transport in one-dimensional waveguides,” Opt. Commun. 343, 135-139 (2015).
[Crossref]

M. Bradford, K. C. Obi, and J. T. Shen, “Efficient single-photon frequency conversion using a sagnac interferometer,” Phys. Rev. Lett. 108, 103902 (2012).
[Crossref] [PubMed]

Oulton, R. F.

P. Kolchin, R. F. Oulton, and X. Zhang, “Nonlinear quantum optics in a waveguide: distinct single photons strongly interacting at the single atom level,” Phys. Rev. Lett. 106, 113601 (2011).
[Crossref] [PubMed]

Palomaki, T.

I. C. Hoi, C. M. Wilson, G. Johansson, T. Palomaki, B. Peropadre, and P. Delsing, “Demonstration of a single-photon router in the microwave regime,” Phys. Rev. Lett. 107, 073601 (2011).
[Crossref] [PubMed]

Park, H.

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, 402-406 (2007).
[Crossref] [PubMed]

Peropadre, B.

I. C. Hoi, C. M. Wilson, G. Johansson, T. Palomaki, B. Peropadre, and P. Delsing, “Demonstration of a single-photon router in the microwave regime,” Phys. Rev. Lett. 107, 073601 (2011).
[Crossref] [PubMed]

Pichler, H.

H. Pichler and P. Zoller, “Photonic circuits with time delays and quantum feedback,” Phys. Rev. Lett. 116, 093601 (2016).
[Crossref] [PubMed]

Poole, P. J.

M. A. M. Versteegh, M. E. Reimer, K. D. Jöns, D. Dalacu, P. J. Poole, A. Gulinatti, A. Giudice, and V. Zwiller, “Observation of strongly entangled photon pairs from a nanowire quantum dot,” Nat. Commun. 5, 5298 (2014).
[Crossref] [PubMed]

Ran, L.

J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99, 063908 (2007).
[Crossref] [PubMed]

Rauschenbeutel, A.

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577-1580 (2016).
[Crossref] [PubMed]

Reimer, M. E.

M. A. M. Versteegh, M. E. Reimer, K. D. Jöns, D. Dalacu, P. J. Poole, A. Gulinatti, A. Giudice, and V. Zwiller, “Observation of strongly entangled photon pairs from a nanowire quantum dot,” Nat. Commun. 5, 5298 (2014).
[Crossref] [PubMed]

Reiten, M. T.

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science,  340, 1304-1307 (2013).
[Crossref] [PubMed]

Reithmaier, G.

G. Reithmaier, M. Kaniber, F. Flassig, S. Lichtmannecker, K. Müller, A. Andrejew, J. Vučković, R. Gross, and J. J. Finley, “On-chip generation, routing, and detection of resonance fluorescence,” Nano Lett. 15, 5208-5213 (2015).
[Crossref] [PubMed]

Roy, D.

D. Roy, C. M. Wilson, and O. Firstenberg, “Colloquium: Strongly interacting photons in one-dimensional continuum,” Rev. Mod. Phys. 89, 021001 (2017).
[Crossref]

D. Roy, “Two-photon scattering by a driven three-level emitter in a one-dimensional waveguide and electromagnetically induced transparency,” Phys. Rev. Lett. 106, 053601 (2011).
[Crossref] [PubMed]

D. Roy, “Few-photon optical diode,” Phys. Rev. B 81, 155117 (2010).
[Crossref]

Sambles, J. R.

S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393 (1991).
[Crossref]

Sanchez-Burillo, E.

E. Sanchez-Burillo, D. Zueco, J. J. Garcia-Ripoll, and L. Martin-Moreno, “Scattering in the ultrastrong regime: nonlinear Optics with One Photon,” Phys. Rev. Lett. 113, 263604 (2014).
[Crossref]

Scheucher, M.

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577-1580 (2016).
[Crossref] [PubMed]

Schmitteckert, P.

P. Longo, P. Schmitteckert, and K. Busch, “Few-photon transport in low-dimensional systems: interaction-induced radiation trapping,” Phys. Rev. Lett. 104, 023602 (2010).
[Crossref] [PubMed]

Shen, J. T.

K. C. Obi and J. T. Shen, “Perturbative and iterative methods for photon transport in one-dimensional waveguides,” Opt. Commun. 343, 135-139 (2015).
[Crossref]

M. Bradford, K. C. Obi, and J. T. Shen, “Efficient single-photon frequency conversion using a sagnac interferometer,” Phys. Rev. Lett. 108, 103902 (2012).
[Crossref] [PubMed]

J. T. Shen and S. Fan, “Thoery of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom,” Phys. Rev. A 79, 023837 (2009).
[Crossref]

J. T. Shen and S. Fan, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system,” Phys. Rev. Lett. 98, 153003 (2007).
[Crossref] [PubMed]

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

Singh, R.

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Cu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103, 171107 (2013).
[Crossref]

Söllner, I.

M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref] [PubMed]

Song, J. D.

M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref] [PubMed]

Sørensen, A. S.

D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807-812 (2009).
[Crossref]

Stobbe, S.

M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref] [PubMed]

Sun, C. P.

L. Zhou, L. P. Yang, Y. Li, and C. P. Sun, “Quantum routing of single photons with a cyclic three-level system,” Phys. Rev. Lett. 111, 103604 (2013).
[Crossref]

L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett. 101, 100501 (2008).
[Crossref] [PubMed]

Sun, J.

J. Sun, C. Cu, X. Chen, Z. Li, L. Liu, and F. Martin, “Ultra-wideband and broad-angle linear polarization conversion metasurface,” J. Appl. Phys. 121, 174902 (2017).
[Crossref]

Taylor, A. J.

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science,  340, 1304-1307 (2013).
[Crossref] [PubMed]

Thyrrestrup, H.

M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref] [PubMed]

Tian, Z.

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Cu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103, 171107 (2013).
[Crossref]

Tsoi, T. S.

T. S. Tsoi and C. K. Law, “Single-photon scattering on Λ-type three-level atoms in a one-dimensional waveguide,” Phys. Rev. A 80,033823 (2009).
[Crossref]

Versteegh, M. A. M.

M. A. M. Versteegh, M. E. Reimer, K. D. Jöns, D. Dalacu, P. J. Poole, A. Gulinatti, A. Giudice, and V. Zwiller, “Observation of strongly entangled photon pairs from a nanowire quantum dot,” Nat. Commun. 5, 5298 (2014).
[Crossref] [PubMed]

Volz, J.

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577-1580 (2016).
[Crossref] [PubMed]

Vuckovic, J.

G. Reithmaier, M. Kaniber, F. Flassig, S. Lichtmannecker, K. Müller, A. Andrejew, J. Vučković, R. Gross, and J. J. Finley, “On-chip generation, routing, and detection of resonance fluorescence,” Nano Lett. 15, 5208-5213 (2015).
[Crossref] [PubMed]

Will, E.

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577-1580 (2016).
[Crossref] [PubMed]

Wilson, C. M.

D. Roy, C. M. Wilson, and O. Firstenberg, “Colloquium: Strongly interacting photons in one-dimensional continuum,” Rev. Mod. Phys. 89, 021001 (2017).
[Crossref]

I. C. Hoi, C. M. Wilson, G. Johansson, T. Palomaki, B. Peropadre, and P. Delsing, “Demonstration of a single-photon router in the microwave regime,” Phys. Rev. Lett. 107, 073601 (2011).
[Crossref] [PubMed]

Xiao, M.

Y. Guo, M. Xiao, and S. Fan, “Topologically protected complete polarization conversion,” Phys. Rev. Lett. 119, 167401 (2017).
[Crossref] [PubMed]

Xu, S.

Yang, J.

M. X. Li, J. Yang, G. W. Lin, Y. P. Niu, and S. Q. Gong, “Scattering of a single photon in a one-dimensional coupled resonator waveguide with a Λ-type emitter assisted by an additional cavity,” Chin. Phys. B 27, 054206 (2018).
[Crossref]

Yang, L. P.

L. Zhou, L. P. Yang, Y. Li, and C. P. Sun, “Quantum routing of single photons with a cyclic three-level system,” Phys. Rev. Lett. 111, 103604 (2013).
[Crossref]

Yu, C. L.

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, 402-406 (2007).
[Crossref] [PubMed]

Yuan, L.

Yuan, Y.

J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99, 063908 (2007).
[Crossref] [PubMed]

Zeng, X.

Z. Liao, X. Zeng, H. Nha, and M. S. Zubairy, “Photon transport in a one-dimensional nanophotonic waveguide QED system,” Phys. Scr. 91, 063004 (2016).
[Crossref]

Zeng, Y.

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science,  340, 1304-1307 (2013).
[Crossref] [PubMed]

Zhang, S. L.

Zhang, W.

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Cu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103, 171107 (2013).
[Crossref]

Zhang, X.

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Cu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103, 171107 (2013).
[Crossref]

P. Kolchin, R. F. Oulton, and X. Zhang, “Nonlinear quantum optics in a waveguide: distinct single photons strongly interacting at the single atom level,” Phys. Rev. Lett. 106, 113601 (2011).
[Crossref] [PubMed]

Zhang, Z. Y.

Zheng, H.

H. Zheng, D. J. Gauthier, and H. U. Baranger, “Waveguide-QED-Based photonic quantum computation,” Phys. Rev. Lett. 111, 090502 (2013).
[Crossref] [PubMed]

Zhou, L.

L. Zhou, L. P. Yang, Y. Li, and C. P. Sun, “Quantum routing of single photons with a cyclic three-level system,” Phys. Rev. Lett. 111, 103604 (2013).
[Crossref]

L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett. 101, 100501 (2008).
[Crossref] [PubMed]

J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99, 063908 (2007).
[Crossref] [PubMed]

Zhu, S. Q.

Zibrov, A. S.

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, 402-406 (2007).
[Crossref] [PubMed]

Zoller, P.

H. Pichler and P. Zoller, “Photonic circuits with time delays and quantum feedback,” Phys. Rev. Lett. 116, 093601 (2016).
[Crossref] [PubMed]

Zubairy, M. S.

Z. Liao, M. Al-Amri, and M. S. Zubairy, “Measurement of deep-subwavelength emitter separation in a waveguide-QED system,” Opt. Express 25, 31997-32009 (2017).
[Crossref] [PubMed]

Z. Liao, H. Nha, and M. S. Zubairy, “Dynamical theory of single-photon transport in a one-dimensional waveguide coupled to identical and nonidentical emitters,” Phys. Rev. A 94, 053842 (2016).
[Crossref]

Z. Liao, X. Zeng, H. Nha, and M. S. Zubairy, “Photon transport in a one-dimensional nanophotonic waveguide QED system,” Phys. Scr. 91, 063004 (2016).
[Crossref]

Zueco, D.

E. Sanchez-Burillo, D. Zueco, J. J. Garcia-Ripoll, and L. Martin-Moreno, “Scattering in the ultrastrong regime: nonlinear Optics with One Photon,” Phys. Rev. Lett. 113, 263604 (2014).
[Crossref]

Zwiller, V.

M. A. M. Versteegh, M. E. Reimer, K. D. Jöns, D. Dalacu, P. J. Poole, A. Gulinatti, A. Giudice, and V. Zwiller, “Observation of strongly entangled photon pairs from a nanowire quantum dot,” Nat. Commun. 5, 5298 (2014).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Cu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103, 171107 (2013).
[Crossref]

Chin. Phys. B (1)

M. X. Li, J. Yang, G. W. Lin, Y. P. Niu, and S. Q. Gong, “Scattering of a single photon in a one-dimensional coupled resonator waveguide with a Λ-type emitter assisted by an additional cavity,” Chin. Phys. B 27, 054206 (2018).
[Crossref]

J. Appl. Phys. (1)

J. Sun, C. Cu, X. Chen, Z. Li, L. Liu, and F. Martin, “Ultra-wideband and broad-angle linear polarization conversion metasurface,” J. Appl. Phys. 121, 174902 (2017).
[Crossref]

Nano Lett. (1)

G. Reithmaier, M. Kaniber, F. Flassig, S. Lichtmannecker, K. Müller, A. Andrejew, J. Vučković, R. Gross, and J. J. Finley, “On-chip generation, routing, and detection of resonance fluorescence,” Nano Lett. 15, 5208-5213 (2015).
[Crossref] [PubMed]

Nat. Commun. (1)

M. A. M. Versteegh, M. E. Reimer, K. D. Jöns, D. Dalacu, P. J. Poole, A. Gulinatti, A. Giudice, and V. Zwiller, “Observation of strongly entangled photon pairs from a nanowire quantum dot,” Nat. Commun. 5, 5298 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

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)

D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807-812 (2009).
[Crossref]

Nature (1)

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, 402-406 (2007).
[Crossref] [PubMed]

Opt. Commun. (1)

K. C. Obi and J. T. Shen, “Perturbative and iterative methods for photon transport in one-dimensional waveguides,” Opt. Commun. 343, 135-139 (2015).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (3)

J. T. Shen and S. Fan, “Thoery of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom,” Phys. Rev. A 79, 023837 (2009).
[Crossref]

Z. Liao, H. Nha, and M. S. Zubairy, “Dynamical theory of single-photon transport in a one-dimensional waveguide coupled to identical and nonidentical emitters,” Phys. Rev. A 94, 053842 (2016).
[Crossref]

T. S. Tsoi and C. K. Law, “Single-photon scattering on Λ-type three-level atoms in a one-dimensional waveguide,” Phys. Rev. A 80,033823 (2009).
[Crossref]

Phys. Rev. B (2)

D. Roy, “Few-photon optical diode,” Phys. Rev. B 81, 155117 (2010).
[Crossref]

S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393 (1991).
[Crossref]

Phys. Rev. Lett. (16)

J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett. 99, 063908 (2007).
[Crossref] [PubMed]

I. C. Hoi, C. M. Wilson, G. Johansson, T. Palomaki, B. Peropadre, and P. Delsing, “Demonstration of a single-photon router in the microwave regime,” Phys. Rev. Lett. 107, 073601 (2011).
[Crossref] [PubMed]

H. Zheng, D. J. Gauthier, and H. U. Baranger, “Waveguide-QED-Based photonic quantum computation,” Phys. Rev. Lett. 111, 090502 (2013).
[Crossref] [PubMed]

H. Pichler and P. Zoller, “Photonic circuits with time delays and quantum feedback,” Phys. Rev. Lett. 116, 093601 (2016).
[Crossref] [PubMed]

J. T. Shen and S. Fan, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system,” Phys. Rev. Lett. 98, 153003 (2007).
[Crossref] [PubMed]

L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett. 101, 100501 (2008).
[Crossref] [PubMed]

P. Longo, P. Schmitteckert, and K. Busch, “Few-photon transport in low-dimensional systems: interaction-induced radiation trapping,” Phys. Rev. Lett. 104, 023602 (2010).
[Crossref] [PubMed]

D. Roy, “Two-photon scattering by a driven three-level emitter in a one-dimensional waveguide and electromagnetically induced transparency,” Phys. Rev. Lett. 106, 053601 (2011).
[Crossref] [PubMed]

P. Kolchin, R. F. Oulton, and X. Zhang, “Nonlinear quantum optics in a waveguide: distinct single photons strongly interacting at the single atom level,” Phys. Rev. Lett. 106, 113601 (2011).
[Crossref] [PubMed]

M. Bradford, K. C. Obi, and J. T. Shen, “Efficient single-photon frequency conversion using a sagnac interferometer,” Phys. Rev. Lett. 108, 103902 (2012).
[Crossref] [PubMed]

L. Neumeier, M. Leib, and M. J. Hartmann, “Single-photon transistor in circuit quantum electrodynamics,” Phys. Rev. Lett. 111, 063601 (2013).
[Crossref] [PubMed]

L. Zhou, L. P. Yang, Y. Li, and C. P. Sun, “Quantum routing of single photons with a cyclic three-level system,” Phys. Rev. Lett. 111, 103604 (2013).
[Crossref]

E. Sanchez-Burillo, D. Zueco, J. J. Garcia-Ripoll, and L. Martin-Moreno, “Scattering in the ultrastrong regime: nonlinear Optics with One Photon,” Phys. Rev. Lett. 113, 263604 (2014).
[Crossref]

Y. Guo, M. Xiao, and S. Fan, “Topologically protected complete polarization conversion,” Phys. Rev. Lett. 119, 167401 (2017).
[Crossref] [PubMed]

A. Cerjan and S. Fan, “Achieving arbitrary control over pairs of polarization states using complex birefringent metamaterials,” Phys. Rev. Lett. 118, 253902 (2017).
[Crossref] [PubMed]

M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref] [PubMed]

Phys. Scr. (1)

Z. Liao, X. Zeng, H. Nha, and M. S. Zubairy, “Photon transport in a one-dimensional nanophotonic waveguide QED system,” Phys. Scr. 91, 063004 (2016).
[Crossref]

Rev. Mod. Phys. (1)

D. Roy, C. M. Wilson, and O. Firstenberg, “Colloquium: Strongly interacting photons in one-dimensional continuum,” Rev. Mod. Phys. 89, 021001 (2017).
[Crossref]

Science (2)

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577-1580 (2016).
[Crossref] [PubMed]

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science,  340, 1304-1307 (2013).
[Crossref] [PubMed]

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Figures (5)

Fig. 1
Fig. 1 The system considered in the manuscript. Two atoms couple to an one-dimensional waveguide. The energy-level configurations of the two atoms are taken as Λ and V -type systems, respectively. The distance between the two atoms is L.
Fig. 2
Fig. 2 Single photon scattering probabilities varying with θ and Δ without dissipations. Panels (a), (b), (c) and (d) denote Th, Tv, Rh and Rv, respectively. In the calculations, G = Gc = 10−5ωe.
Fig. 3
Fig. 3 The reflected probabilities of the single photon with h -polarization and v-polarization as a function of θ. In the calculations, Δ = 0, G = Gc = 10−5ωe.
Fig. 4
Fig. 4 Fh (a), Fv (b), Fr (c) and Fl (d) as a function of Δ and Gc/G. In (a), (b), (c) and (d), θ is taken as π, 0.5π, 3π/4 and π/4, respectively. G = 10−5ωe in all the calculations.
Fig. 5
Fig. 5 Fh (a), Fv (b), Fl (c) and Fr (d) as a function of θ and γ/G. In the calculations, G = Gc = 10−5ωe, Δ = 0.

Equations (15)

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

H = H w + H a + H i n t ,
H w = p = h , v [ i v g d x C R p ( x ) x C R p ( x ) + i v g d x C L p ( x ) x C L p ( x ) ]
H a = ( ω e i γ e / 2 ) σ e e + p = h , v ( ω e p i γ e p / 2 ) σ e p e p ,
H i n t = p = h d = R , L { J p d x δ ( x ) [ C d p ( x ) σ g p e + H . c . ] + J c p d x δ ( x L ) [ C d p ( x ) σ g e p + H . C . ] } ,
| Ψ = p = h , v d = R , L [ d x ϕ d p ( x ) C d p | 0 , g , g p + u p | 0 , e p ] + u e | 0 , e ,
ϕ R h ( x ) = e i k x [ θ ( x ) + a h θ ( x ) θ ( L x ) + t h θ ( x L ) ] ,
ϕ L h ( x ) = e i k x [ r h θ ( x ) + b h θ ( x ) θ ( L x ) ] ,
ϕ R v ( x ) = e i k x [ a v θ ( x ) θ ( L x ) + t v θ ( x L ) ] ,
ϕ L v ( x ) = e i k x [ r v θ ( x ) + b v θ ( x ) θ ( L x ) ] ,
t h = ( 1 + e 2 i θ ) Δ G G c + i Δ 2 ( G + G c ) + Δ 3 2 i ( 1 + e 2 i θ ) G G c 2 + [ 2 ( 1 + e 2 i θ ) G G c 2 G G c G c 2 ] Δ + 2 i Δ 2 ( G + G c ) + Δ 3 ,
t v = ( 1 + e 2 i θ ) Δ G G c + i G Δ 2 2 i ( 1 + e 2 i θ ) G G c 2 + [ 2 ( 1 + e 2 i θ ) G G c 2 G G c G c 2 ] Δ + 2 i Δ 2 ( G + G c ) + Δ 3 ,
r h = ( 1 + e 4 i θ ) G G c 2 2 i G G c Δ i e 2 i θ G c 2 Δ G Δ 2 e 2 i θ G c Δ 2 2 i ( 1 + e 2 i θ ) G G c 2 + [ 2 ( 1 + e 2 i θ ) G G c 2 G G c G c 2 ] Δ + 2 i Δ 2 ( G + G c ) + Δ 3 ,
r v = i ( 1 + e 2 i θ ) 2 G G c 2 2 ( 1 + e 2 i θ ) G G c Δ i G Δ 2 2 i ( 1 + e 2 i θ ) G G c 2 + [ 2 ( 1 + e 2 i θ ) G G c 2 G G c G c 2 ] Δ + 2 i Δ 2 ( G + G c ) + Δ 3 ,
r h = ( 1 e 2 i θ ) / 2 ,
r v = ( 1 + e 2 i θ ) / 2.

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