Abstract

We propose an all-optical-control scheme to simultaneously realize parity-time (𝒫𝒯)-symmetric and 𝒫𝒯-antisymmetric susceptibilities along the propagation direction of light by applying an external magnetic field. Through the light-atom interaction within a double-Λ configuration, the resulting position-dependent susceptibilities for the interacting fields can be manipulated through the relative phase between them. In particular, for the probe field, one can switch its refractive index from the 𝒫𝒯-symmetry to 𝒫𝒯-antisymmetry by just varying the phase. Based on the quantum interference among transition channels in a closed loop, analytical formulas are also derived to illustrate the conditions for 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry.

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

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References

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  1. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having 𝒫𝒯-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
    [Crossref]
  2. Y.-C. Lee, M.-H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local 𝒫𝒯 symmetry violates the no-signaling principle,” Phys. Rev. Lett. 112, 130404 (2014).
    [Crossref]
  3. S. Longhi, “Parity-time symmetry meets photonics: A new twist in non-Hermitian optics,” Europhys. Lett. 120, 64001 (2017).
    [Crossref]
  4. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
    [Crossref]
  5. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
    [Crossref]
  6. Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015).
    [Crossref]
  7. J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015).
    [Crossref]
  8. X. Wang and J.-H.. Wu, “Optical 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry in coherently driven atomic lattices,” Opt. Express 24, 4289–4298 (2016).
    [Crossref] [PubMed]
  9. L. Praxmeyer, P. Yang, and R.-K. Lee, “Phase-space representation of a non-Hermitian system with 𝒫𝒯 symmetry,” Phys. Rev. A 93, 042122 (2016).
    [Crossref]
  10. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
    [Crossref] [PubMed]
  11. S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
    [Crossref]
  12. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
    [Crossref]
  13. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
    [Crossref]
  14. A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
    [Crossref] [PubMed]
  15. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
    [Crossref]
  16. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile 𝒫𝒯 symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
    [Crossref]
  17. C. Hang, Y. V. Kartashov, G. Huang, and V. V. Konotop, “Localization of light in a parity-time-symmetric quasi-periodic lattice,” Opt. Lett. 40, 2758–2761 (2015).
    [Crossref] [PubMed]
  18. S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
    [Crossref]
  19. Y.-M. Liu, F. Gao, C.-H. Fan, and J.-H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42, 4283–4286 (2017).
    [Crossref] [PubMed]
  20. B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
    [Crossref]
  21. M. G. Silveirinha, “Spontaneous parity-time-symmetry breaking in moving media,” Phys. Rev. A 90, 013842 (2014).
    [Crossref]
  22. J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011).
    [Crossref]
  23. W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).
  24. H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
    [Crossref]
  25. C. Hang, G. Huang, and V. V. Konotop, “𝒫𝒯 symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
    [Crossref]
  26. H.-J. Li, J.-P. Dou, and G. Huang, “PT symmetry via electromagnetically induced transparency,” Opt. Express 21, 32053–32062 (2013).
    [Crossref]
  27. J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “𝒫𝒯-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (2013).
    [Crossref]
  28. Ziauddin, Y.-L. Chuang, and R.-K. Lee, “Giant Goos-Hanchen shift using 𝒫𝒯 symmetry,” Phys. Rev. A 92, 013815 (2015).
    [Crossref]
  29. Ziauddin, Y.-L. Chaung, and R.-K. Lee, “𝒫𝒯-symmetry in Rydberg atoms,” Europhys. Lett. 115, 14005 (2016).
    [Crossref]
  30. Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
    [Crossref] [PubMed]
  31. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
    [Crossref] [PubMed]
  32. Y.-D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
    [Crossref] [PubMed]
  33. S. Longhi, “𝒫𝒯-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
    [Crossref]
  34. Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
    [Crossref]
  35. M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13, 3068–3078 (2005).
    [Crossref] [PubMed]
  36. S. Longhi, “Invisibility in 𝒫𝒯-symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
    [Crossref]
  37. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
    [Crossref]
  38. L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
    [Crossref]
  39. J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
    [Crossref] [PubMed]

2017 (2)

S. Longhi, “Parity-time symmetry meets photonics: A new twist in non-Hermitian optics,” Europhys. Lett. 120, 64001 (2017).
[Crossref]

Y.-M. Liu, F. Gao, C.-H. Fan, and J.-H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42, 4283–4286 (2017).
[Crossref] [PubMed]

2016 (5)

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).

Ziauddin, Y.-L. Chaung, and R.-K. Lee, “𝒫𝒯-symmetry in Rydberg atoms,” Europhys. Lett. 115, 14005 (2016).
[Crossref]

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

X. Wang and J.-H.. Wu, “Optical 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry in coherently driven atomic lattices,” Opt. Express 24, 4289–4298 (2016).
[Crossref] [PubMed]

L. Praxmeyer, P. Yang, and R.-K. Lee, “Phase-space representation of a non-Hermitian system with 𝒫𝒯 symmetry,” Phys. Rev. A 93, 042122 (2016).
[Crossref]

2015 (4)

Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015).
[Crossref]

C. Hang, Y. V. Kartashov, G. Huang, and V. V. Konotop, “Localization of light in a parity-time-symmetric quasi-periodic lattice,” Opt. Lett. 40, 2758–2761 (2015).
[Crossref] [PubMed]

Ziauddin, Y.-L. Chuang, and R.-K. Lee, “Giant Goos-Hanchen shift using 𝒫𝒯 symmetry,” Phys. Rev. A 92, 013815 (2015).
[Crossref]

2014 (6)

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

M. G. Silveirinha, “Spontaneous parity-time-symmetry breaking in moving media,” Phys. Rev. A 90, 013842 (2014).
[Crossref]

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Y.-C. Lee, M.-H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local 𝒫𝒯 symmetry violates the no-signaling principle,” Phys. Rev. Lett. 112, 130404 (2014).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

2013 (4)

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

C. Hang, G. Huang, and V. V. Konotop, “𝒫𝒯 symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

H.-J. Li, J.-P. Dou, and G. Huang, “PT symmetry via electromagnetically induced transparency,” Opt. Express 21, 32053–32062 (2013).
[Crossref]

J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “𝒫𝒯-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (2013).
[Crossref]

2012 (1)

A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

2011 (3)

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011).
[Crossref]

S. Longhi, “Invisibility in 𝒫𝒯-symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

2010 (3)

Y.-D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

S. Longhi, “𝒫𝒯-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

2009 (3)

S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile 𝒫𝒯 symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[Crossref]

2008 (4)

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[Crossref] [PubMed]

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

2007 (1)

2005 (1)

2003 (1)

H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
[Crossref]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having 𝒫𝒯-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Almeida, V. R.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Artoni, M.

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

Azaña, J.

Bélanger, N.

Bender, C. M.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having 𝒫𝒯-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Bendix, O.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile 𝒫𝒯 symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[Crossref]

Bersch, C.

A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having 𝒫𝒯-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Y.-D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Chang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Chaung, Y.-L.

Ziauddin, Y.-L. Chaung, and R.-K. Lee, “𝒫𝒯-symmetry in Rydberg atoms,” Europhys. Lett. 115, 14005 (2016).
[Crossref]

Chen, H.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

Chen, Y.-C.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Chen, Y.-F.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Chen, Y.-H.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Chong, Y.-D.

Y.-D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Christodoulides, D. N.

J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “𝒫𝒯-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (2013).
[Crossref]

A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Chuang, Y.-L.

Ziauddin, Y.-L. Chuang, and R.-K. Lee, “Giant Goos-Hanchen shift using 𝒫𝒯 symmetry,” Phys. Rev. A 92, 013815 (2015).
[Crossref]

Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015).
[Crossref]

Dou, J.-P.

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

El-Ganainy, R.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Ellis, F. M.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011).
[Crossref]

Fan, C.-H.

Fan, S.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Fegadolli, W. S.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Feng, L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Flammia, S. T.

Y.-C. Lee, M.-H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local 𝒫𝒯 symmetry violates the no-signaling principle,” Phys. Rev. Lett. 112, 130404 (2014).
[Crossref]

Fleischmann, R.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile 𝒫𝒯 symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[Crossref]

Gao, F.

Ge, L.

Y.-D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Gianfreda, M.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Gunther, U.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Hang, C.

C. Hang, Y. V. Kartashov, G. Huang, and V. V. Konotop, “Localization of light in a parity-time-symmetric quasi-periodic lattice,” Opt. Lett. 40, 2758–2761 (2015).
[Crossref] [PubMed]

C. Hang, G. Huang, and V. V. Konotop, “𝒫𝒯 symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

Hsieh, M.-H.

Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015).
[Crossref]

Y.-C. Lee, M.-H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local 𝒫𝒯 symmetry violates the no-signaling principle,” Phys. Rev. Lett. 112, 130404 (2014).
[Crossref]

Hua, S.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Huang, G.

Jiang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Jiang, X.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Jiang, Y.

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).

Kang, H.

H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
[Crossref]

Kartashov, Y. V.

Kip, D.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

Klaiman, S.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Konotop, V. V.

C. Hang, Y. V. Kartashov, G. Huang, and V. V. Konotop, “Localization of light in a parity-time-symmetric quasi-periodic lattice,” Opt. Lett. 40, 2758–2761 (2015).
[Crossref] [PubMed]

C. Hang, G. Huang, and V. V. Konotop, “𝒫𝒯 symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011).
[Crossref]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile 𝒫𝒯 symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[Crossref]

Kulishov, M.

La Rocca, G. C.

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

Landy, N. I.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[Crossref] [PubMed]

Laniel, J. M.

Lee, R.-K.

L. Praxmeyer, P. Yang, and R.-K. Lee, “Phase-space representation of a non-Hermitian system with 𝒫𝒯 symmetry,” Phys. Rev. A 93, 042122 (2016).
[Crossref]

Ziauddin, Y.-L. Chaung, and R.-K. Lee, “𝒫𝒯-symmetry in Rydberg atoms,” Europhys. Lett. 115, 14005 (2016).
[Crossref]

Ziauddin, Y.-L. Chuang, and R.-K. Lee, “Giant Goos-Hanchen shift using 𝒫𝒯 symmetry,” Phys. Rev. A 92, 013815 (2015).
[Crossref]

Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015).
[Crossref]

Y.-C. Lee, M.-H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local 𝒫𝒯 symmetry violates the no-signaling principle,” Phys. Rev. Lett. 112, 130404 (2014).
[Crossref]

Lee, Y.-C.

Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015).
[Crossref]

Y.-C. Lee, M.-H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local 𝒫𝒯 symmetry violates the no-signaling principle,” Phys. Rev. Lett. 112, 130404 (2014).
[Crossref]

Lei, F.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Li, A.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011).
[Crossref]

Li, C.

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).

Li, G.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Li, H. Q.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

Li, H.-J.

Li, J.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

Li, W.

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Liu, J.

Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015).
[Crossref]

Liu, Y.-M.

Liu, Z.-Y.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Lo, H.-Y.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Long, G. L.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Longhi, S.

S. Longhi, “Parity-time symmetry meets photonics: A new twist in non-Hermitian optics,” Europhys. Lett. 120, 64001 (2017).
[Crossref]

S. Longhi, “Invisibility in 𝒫𝒯-symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
[Crossref]

S. Longhi, “𝒫𝒯-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[Crossref]

S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
[Crossref]

Lu, M.-H.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Makris, K. G.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Miri, M. A

A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Miri, M. A.

J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “𝒫𝒯-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (2013).
[Crossref]

Mock, J. J.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[Crossref] [PubMed]

Moiseyev, N.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Monifi, F.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Nori, F.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Oliveira, J. E. B.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Onishchukov, G.

A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Ozdemir, S. K.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Padilla, W. J.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[Crossref] [PubMed]

Peng, B.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Peschel, U.

A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Plant, D. V.

Praxmeyer, L.

L. Praxmeyer, P. Yang, and R.-K. Lee, “Phase-space representation of a non-Hermitian system with 𝒫𝒯 symmetry,” Phys. Rev. A 93, 042122 (2016).
[Crossref]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Regensburger, A.

A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

Sajuyigbe, S.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[Crossref] [PubMed]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Scherer, A.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Schindler, J.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011).
[Crossref]

Segev, M.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

Shapiro, B.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile 𝒫𝒯 symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[Crossref]

Sheng, J.

J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “𝒫𝒯-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (2013).
[Crossref]

Silveirinha, M. G.

M. G. Silveirinha, “Spontaneous parity-time-symmetry breaking in moving media,” Phys. Rev. A 90, 013842 (2014).
[Crossref]

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Smith, D. R.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[Crossref] [PubMed]

Song, H.

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).

Stone, A. D.

Y.-D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Sun, Y.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

Tan, W.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

Tsai, P.-J.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Wang, G.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Wang, X.

Wen, J.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Wen, L.

H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
[Crossref]

Wu, J.-H.

Y.-M. Liu, F. Gao, C.-H. Fan, and J.-H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42, 4283–4286 (2017).
[Crossref] [PubMed]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

Wu, J.-H..

Xiao, M.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “𝒫𝒯-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (2013).
[Crossref]

Xu, Y.-L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Yang, C.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Yang, L.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Yang, P.

L. Praxmeyer, P. Yang, and R.-K. Lee, “Phase-space representation of a non-Hermitian system with 𝒫𝒯 symmetry,” Phys. Rev. A 93, 042122 (2016).
[Crossref]

Yu, I. A.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Zheng, M. C.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011).
[Crossref]

Zhu, Y.

H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
[Crossref]

Ziauddin,

Ziauddin, Y.-L. Chaung, and R.-K. Lee, “𝒫𝒯-symmetry in Rydberg atoms,” Europhys. Lett. 115, 14005 (2016).
[Crossref]

Ziauddin, Y.-L. Chuang, and R.-K. Lee, “Giant Goos-Hanchen shift using 𝒫𝒯 symmetry,” Phys. Rev. A 92, 013815 (2015).
[Crossref]

Europhys. Lett. (2)

S. Longhi, “Parity-time symmetry meets photonics: A new twist in non-Hermitian optics,” Europhys. Lett. 120, 64001 (2017).
[Crossref]

Ziauddin, Y.-L. Chaung, and R.-K. Lee, “𝒫𝒯-symmetry in Rydberg atoms,” Europhys. Lett. 115, 14005 (2016).
[Crossref]

J. Phys. A (1)

S. Longhi, “Invisibility in 𝒫𝒯-symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
[Crossref]

Nat. Photonics (1)

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Nature (1)

A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Nature Mater. (1)

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Nature Phys. (2)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. A (9)

S. Longhi, “𝒫𝒯-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[Crossref]

J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “𝒫𝒯-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (2013).
[Crossref]

Ziauddin, Y.-L. Chuang, and R.-K. Lee, “Giant Goos-Hanchen shift using 𝒫𝒯 symmetry,” Phys. Rev. A 92, 013815 (2015).
[Crossref]

H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
[Crossref]

M. G. Silveirinha, “Spontaneous parity-time-symmetry breaking in moving media,” Phys. Rev. A 90, 013842 (2014).
[Crossref]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011).
[Crossref]

L. Praxmeyer, P. Yang, and R.-K. Lee, “Phase-space representation of a non-Hermitian system with 𝒫𝒯 symmetry,” Phys. Rev. A 93, 042122 (2016).
[Crossref]

Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015).
[Crossref]

Phys. Rev. Lett. (15)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having 𝒫𝒯-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Y.-C. Lee, M.-H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local 𝒫𝒯 symmetry violates the no-signaling principle,” Phys. Rev. Lett. 112, 130404 (2014).
[Crossref]

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile 𝒫𝒯 symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[Crossref]

C. Hang, G. Huang, and V. V. Konotop, “𝒫𝒯 symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[Crossref] [PubMed]

Y.-D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Sci. Rep. (1)

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).

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Figures (5)

Fig. 1
Fig. 1 Our four-level atomic system in a double-Λ configuration. Here, the four fields are denoted as Ωs, Ωp, Ωc, and Ωd, with the corresponding Rabi frequencies for signal, probe, coupling, and driving fields, respectively. The one-photon detunings of these fields are characterized by Δs, Δp, Δc, and Δd, respectively.
Fig. 2
Fig. 2 Setup of our photon-light interaction scheme, where four fields propagate along the z-direction and the atomic ensemble having photon-atom interaction in a double-Λ configuration, as shown in Fig. 1. With a magnetic field B(z), which has its magnitude linearly increasing along the propagation z-direction, the resulting susceptibilities for probe and signal fields can support a ����-symmetry or ����-antisymmetry in the longitudinal z-direction.
Fig. 3
Fig. 3 Real (Left-column) and imaginary (Right-column) parts of the susceptibilities for probe and signal fields, depicted in Blue- and Red-curves, respectively. The relative phase difference ϕr is: (a–b) π/2, (c–d) π, (e–f) 3π/2, and (g–h) 2π. One can see that χp satisfies the ����-symmetric condition when ϕr = π/2, 3π/2; and satisfies the ����-antisymmetric condition when ϕr = π, 2π. However, χs only gives the ����-antisymmetric condition. Here, the parameters used are |Ωc| = |Ωd| = 20Γ, Ωp = 0.01Γ, and Ωs = 1Γ, respectively.
Fig. 4
Fig. 4 The deviation of symmetry on the probe susceptibility. For an even function, we calculate ξ(z) − ξ(Lz)/Θ; while for an odd function, we calculate [ξ (z) + ξ(Lz)] / Θ+. Here, ξ can be the real (in Blue-color) or imaginary (in Red-color) part of the susceptibility in the probe field, and Θ± is the normalized factor. All the parameters used are the same as those shown in Fig. 3, but the relative phases are choose for (a): ϕr = (2n) × π/2; and (b): ϕr = (2n + 1) × π/2.
Fig. 5
Fig. 5 Probe and signal field intensity verse the propagation distance with different phases: (a,b) ϕr = π/2, (c,d) ϕr = 2 × π/2, (e,f) ϕr = 3 × π/2, and (g,h) ϕr = 4 × π/2

Equations (20)

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H ^ = [ Δ p | 3 3 | + ( Δ p Δ c ) | 2 2 | + Δ s | 4 4 | ] 2 ( Ω p | 3 1 | + Ω c | 3 2 | + Ω s | 4 1 | + Ω d | 4 2 | + H . C . ) .
t ρ 21 = γ ˜ 21 ρ 21 + i 2 Ω c * ρ 31 + i 2 Ω d * ρ 41 ,
t ρ 31 = γ ˜ 31 ρ 31 + i 2 Ω c ρ 21 + i 2 Ω p ,
t ρ 41 = γ ˜ 41 ρ 41 + i 2 Ω d ρ 21 + i 2 Ω s .
χ p = χ p 0 [ i ( 4 γ ˜ 21 γ ˜ 41 Ω p + | Ω d | 2 Ω p Ω c Ω d * Ω s ) 4 γ ˜ 21 γ ˜ 31 γ ˜ 41 + γ ˜ 41 | Ω c | 2 + γ ˜ 31 | Ω d | 2 ] ,
χ s = χ s 0 [ i ( 4 γ ˜ 21 γ ˜ 31 Ω s + | Ω c | 2 Ω s Ω d Ω c * Ω p ) 4 γ ˜ 21 γ ˜ 31 γ ˜ 41 + γ ˜ 41 | Ω c | 2 + γ ˜ 31 | Ω d | 2 ] .
Δ s ( z ) = Δ s ( 0 ) g L m j μ B B ( z ) / .
B ( z ) = [ 2 Δ s ( 0 ) g L m j μ B L ] z .
Re χ p χ p 0 ( Δ s Ω p + 2 γ 31 Ω s Δ s 2 + 4 γ 31 2 ) ,
Im χ p χ p 0 ( 2 γ 31 Ω p Δ s Ω s Δ s 2 + 4 γ 31 2 ) .
Re χ p χ p 0 ( Δ s Ω p 2 γ 31 Ω s Δ s 2 + 4 γ 31 2 ) ,
Im χ p χ p 0 ( 2 γ 31 Ω p + Δ s Ω s Δ s 2 + 4 γ 31 2 ) .
Re χ p χ p 0 ( Δ s ( Ω p ± Ω s ) Δ s 2 + 4 γ 31 2 ) ,
Im χ p χ p 0 ( 2 γ 31 ( Ω p ± Ω s ) Δ s 2 + 4 γ 31 2 ) .
Re χ s χ s 0 ( Δ s Ω s ± 2 γ 31 Ω p Δ s 2 + 4 γ 31 2 ) ,
Im χ s χ s 0 ( 2 γ 31 Ω s ± Δ s Ω p ) Δ s 2 + 4 γ 31 2 ) .
Re χ s χ s 0 ( Δ s ( Ω s ± Ω p ) Δ s 2 ± 4 γ 31 2 ) ,
Im χ s χ s 0 ( 2 γ 31 ( Ω s ± Ω p ) Δ s 2 ± 4 γ 31 2 ) .
Ω p z = i γ 31 α 2 ρ 31 ,
Ω s z = i γ 41 α 2 ρ 41 .