Abstract

We study an ensemble of ultracold atoms trapped in the one-dimensional optical lattices under the N level configuration to examine the absorption and dispersion properties modulated into different ways along the lattice direction z. We find these trapped atoms with a Gaussian density distribution in each period may exhibit either symmetry χp(z)=χp*(z) or antisymmetry χp(z)=χp*(z) of parity-time (𝒫𝒯) in terms of the probe susceptibility. Such intriguing twofold modulations of real (χ′p) and imaginary (χ″p) susceptibilities with a π/2 phase shift are attained by spatially modulating intensities or frequencies of one driving field in a suitable way. The 𝒫𝒯-symmetric or 𝒫𝒯-antisymmetric atomic lattices correspond in fact to complex photonic crystals and may be extended to develop functional devices like photonic diodes and transistors, a task impossible for real photonic crystals.

© 2016 Optical Society of America

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    [Crossref]
  2. C. M. Bender, S. Boettcher, and P. N. Meisinger, “𝒫𝒯-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
    [Crossref]
  3. A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of 𝒫𝒯-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
    [Crossref]
  4. 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]
  5. 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]
  6. S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
    [Crossref]
  7. S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
    [Crossref]
  8. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [Crossref]
  9. L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
    [Crossref] [PubMed]
  10. 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]
  11. 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]
  12. S. Longhi, “𝒫𝒯-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
    [Crossref]
  13. 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]
  14. 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]
  15. S. Longhi, “Invisibility in 𝒫𝒯-symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
    [Crossref]
  16. 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]
  17. 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,” Nat. Mater. 12, 108–113 (2013).
    [Crossref]
  18. B. He, S.-B Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
    [Crossref]
  19. H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic 𝒫𝒯 symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91, 053825 (2015).
    [Crossref]
  20. A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “𝒫𝒯-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84, 021806 (2011).
    [Crossref]
  21. M. Turduev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
    [Crossref]
  22. B. Peng, S. 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,” Nat. Phys. 10, 394–398 (2014).
    [Crossref]
  23. H. Jing, S. K. Ozdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “𝒫𝒯-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
    [Crossref]
  24. X.-Y Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
    [Crossref]
  25. X.-W. Xu, Y.-X. Liu, C.-P. Sun, and Y. Li, “Mechanical 𝒫𝒯 symmetry in coupled optomechanical systems,” Phys. Rev. A 92, 013852 (2015).
    [Crossref]
  26. C. Hang, G.-X. Huang, and V. V. Konotop, “𝒫𝒯 symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
    [Crossref]
  27. J.-T. 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. H.-J. Li, J.-P. Dou, and G.-X. Huang, “𝒫𝒯 symmetry via electromagnetically induced transparency,” Opt. Express 21, 32053–32062 (2013).
    [Crossref]
  29. 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]
  30. J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015).
    [Crossref]
  31. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “𝒫𝒯-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
    [Crossref]
  32. 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]
  33. 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]
  34. G. Della Valle and S. Longhi, “Spectral and transport properties of time-periodic 𝒫𝒯-symmetric tight-binding lattices,” Phys. Rev. A 87, 022119 (2013).
    [Crossref]
  35. S. Nixon and J. Yang, “Light propagation in periodically modulated complex waveguides,” Phys. Rev. A 91, 033807 (2015).
    [Crossref]
  36. S. A. Horsley, M. Artoni, and G. C. La Rocca, “Spatial Kramers-Kronig relations and the reflection of waves,” Nat. Photonics 9, 436–439 (2015).
    [Crossref]

2015 (8)

B. He, S.-B Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
[Crossref]

H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic 𝒫𝒯 symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91, 053825 (2015).
[Crossref]

X.-Y Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

X.-W. Xu, Y.-X. Liu, C.-P. Sun, and Y. Li, “Mechanical 𝒫𝒯 symmetry in coupled optomechanical systems,” Phys. Rev. A 92, 013852 (2015).
[Crossref]

M. Turduev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (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]

S. Nixon and J. Yang, “Light propagation in periodically modulated complex waveguides,” Phys. Rev. A 91, 033807 (2015).
[Crossref]

S. A. Horsley, M. Artoni, and G. C. La Rocca, “Spatial Kramers-Kronig relations and the reflection of waves,” Nat. Photonics 9, 436–439 (2015).
[Crossref]

2014 (4)

B. Peng, S. 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,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

H. Jing, S. K. Ozdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “𝒫𝒯-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (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]

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]

2013 (5)

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,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

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

J.-T. 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]

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

G. Della Valle and S. Longhi, “Spectral and transport properties of time-periodic 𝒫𝒯-symmetric tight-binding lattices,” Phys. Rev. A 87, 022119 (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 (4)

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]

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “𝒫𝒯-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84, 021806 (2011).
[Crossref]

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref] [PubMed]

2010 (4)

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,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “𝒫𝒯-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
[Crossref]

2009 (2)

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]

2008 (4)

S. Klaiman, U. Günther, 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]

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]

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]

2005 (2)

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]

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of 𝒫𝒯-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[Crossref]

1999 (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “𝒫𝒯-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[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,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Artoni, M.

S. A. Horsley, M. Artoni, and G. C. La Rocca, “Spatial Kramers-Kronig relations and the reflection of waves,” Nat. Photonics 9, 436–439 (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]

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]

Ayache, M.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref] [PubMed]

Azaña, J.

Bahat-Treidel, O.

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “𝒫𝒯-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84, 021806 (2011).
[Crossref]

Bélanger, N.

Bender, C. M.

B. Peng, S. 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,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

C. M. Bender, S. Boettcher, and P. N. Meisinger, “𝒫𝒯-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[Crossref]

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

Benisty, H.

H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic 𝒫𝒯 symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91, 053825 (2015).
[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, S. Boettcher, and P. N. Meisinger, “𝒫𝒯-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[Crossref]

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

Botey, M.

M. Turduev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
[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]

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. F.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref] [PubMed]

Chen, Y.-F.

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,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

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.-T. 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,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “𝒫𝒯-symmetric optical lattices,” Phys. Rev. A 81, 063807 (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]

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]

Degiron, A.

H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic 𝒫𝒯 symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91, 053825 (2015).
[Crossref]

Delgado, F.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of 𝒫𝒯-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[Crossref]

Della Valle, G.

G. Della Valle and S. Longhi, “Spectral and transport properties of time-periodic 𝒫𝒯-symmetric tight-binding lattices,” Phys. Rev. A 87, 022119 (2013).
[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).
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K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “𝒫𝒯-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
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C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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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).
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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).
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L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
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B. Peng, S. 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,” Nat. Phys. 10, 394–398 (2014).
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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,” Nat. Mater. 12, 108–113 (2013).
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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,” Nat. Mater. 12, 108–113 (2013).
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L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
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Y.-D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
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B. Peng, S. 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,” Nat. Phys. 10, 394–398 (2014).
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M. Turduev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
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S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
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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).
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C. Hang, G.-X. Huang, and V. V. Konotop, “𝒫𝒯 symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
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B. He, S.-B Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
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M. Turduev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
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S. A. Horsley, M. Artoni, and G. C. La Rocca, “Spatial Kramers-Kronig relations and the reflection of waves,” Nat. Photonics 9, 436–439 (2015).
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C. Hang, G.-X. Huang, and V. V. Konotop, “𝒫𝒯 symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
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H.-J. Li, J.-P. Dou, and G.-X. Huang, “𝒫𝒯 symmetry via electromagnetically induced transparency,” Opt. Express 21, 32053–32062 (2013).
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L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
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X.-Y Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
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H. Jing, S. K. Ozdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “𝒫𝒯-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
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C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
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C. Hang, G.-X. Huang, and V. V. Konotop, “𝒫𝒯 symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
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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).
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Kurt, H.

M. Turduev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
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S. A. Horsley, M. Artoni, and G. C. La Rocca, “Spatial Kramers-Kronig relations and the reflection of waves,” Nat. Photonics 9, 436–439 (2015).
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J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015).
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J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
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N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
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Lei, F.

B. Peng, S. 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,” Nat. Phys. 10, 394–398 (2014).
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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).
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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).
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X.-W. Xu, Y.-X. Liu, C.-P. Sun, and Y. Li, “Mechanical 𝒫𝒯 symmetry in coupled optomechanical systems,” Phys. Rev. A 92, 013852 (2015).
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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).
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X.-W. Xu, Y.-X. Liu, C.-P. Sun, and Y. Li, “Mechanical 𝒫𝒯 symmetry in coupled optomechanical systems,” Phys. Rev. A 92, 013852 (2015).
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B. Peng, S. 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,” Nat. Phys. 10, 394–398 (2014).
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L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
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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,” Nat. Mater. 12, 108–113 (2013).
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X.-Y Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
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Lü, X.-Y.

H. Jing, S. K. Ozdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “𝒫𝒯-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
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H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic 𝒫𝒯 symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91, 053825 (2015).
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X.-Y Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
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C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “𝒫𝒯-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
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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).
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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).
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J.-T. 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).
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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).
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N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
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S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
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B. Peng, S. 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,” Nat. Phys. 10, 394–398 (2014).
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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).
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A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of 𝒫𝒯-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
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K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “𝒫𝒯-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
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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).
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H. Jing, S. K. Ozdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “𝒫𝒯-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
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B. Peng, S. 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,” Nat. Phys. 10, 394–398 (2014).
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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,” Nat. Mater. 12, 108–113 (2013).
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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).
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M. Turduev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
[Crossref]

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B. Peng, S. 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,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

H. Jing, S. K. Ozdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “𝒫𝒯-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref]

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N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
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B. Peng, S. 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,” Nat. Phys. 10, 394–398 (2014).
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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.

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]

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A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “𝒫𝒯-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84, 021806 (2011).
[Crossref]

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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]

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A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of 𝒫𝒯-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[Crossref]

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,” Nat. 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]

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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,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref] [PubMed]

Segev, M.

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “𝒫𝒯-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84, 021806 (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,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

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J.-T. 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]

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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]

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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]

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M. Turduev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
[Crossref]

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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]

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X.-W. Xu, Y.-X. Liu, C.-P. Sun, and Y. Li, “Mechanical 𝒫𝒯 symmetry in coupled optomechanical systems,” Phys. Rev. A 92, 013852 (2015).
[Crossref]

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]

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A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “𝒫𝒯-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84, 021806 (2011).
[Crossref]

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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).
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M. Turduev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
[Crossref]

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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, J.

B. He, S.-B Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
[Crossref]

Wu, J.-H.

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015).
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X.-Y Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
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B. He, S.-B Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
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J.-T. 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).
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X.-W. Xu, Y.-X. Liu, C.-P. Sun, and Y. Li, “Mechanical 𝒫𝒯 symmetry in coupled optomechanical systems,” Phys. Rev. A 92, 013852 (2015).
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Xu, Y. L.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref] [PubMed]

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,” Nat. Mater. 12, 108–113 (2013).
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Yan, S.-B

B. He, S.-B Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91, 053832 (2015).
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S. Nixon and J. Yang, “Light propagation in periodically modulated complex waveguides,” Phys. Rev. A 91, 033807 (2015).
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H. Jing, S. K. Ozdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “𝒫𝒯-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
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Nat. 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,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Nat. Photonics (1)

S. A. Horsley, M. Artoni, and G. C. La Rocca, “Spatial Kramers-Kronig relations and the reflection of waves,” Nat. Photonics 9, 436–439 (2015).
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Nat. Phys. (2)

B. Peng, S. 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,” Nat. Phys. 10, 394–398 (2014).
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C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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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).
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Opt. Express (2)

Phys. Rev. A (11)

X.-W. Xu, Y.-X. Liu, C.-P. Sun, and Y. Li, “Mechanical 𝒫𝒯 symmetry in coupled optomechanical systems,” Phys. Rev. A 92, 013852 (2015).
[Crossref]

J.-T. 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]

G. Della Valle and S. Longhi, “Spectral and transport properties of time-periodic 𝒫𝒯-symmetric tight-binding lattices,” Phys. Rev. A 87, 022119 (2013).
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S. Nixon and J. Yang, “Light propagation in periodically modulated complex waveguides,” Phys. Rev. A 91, 033807 (2015).
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J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015).
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S. Longhi, “𝒫𝒯-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
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H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic 𝒫𝒯 symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91, 053825 (2015).
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H. Jing, S. K. Ozdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “𝒫𝒯-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref]

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[Crossref]

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Science (1)

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref] [PubMed]

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Figures (6)

Fig. 1
Fig. 1 (a) Diagram of a four-level N configuration of ultracold atoms, which are driven by a weak probe field Ω p , a strong coupling field Ω c , and a strong driving field Ω d . (b) An ensemble of N-type ultracold atoms trapped at the bottoms of dipole traps forming 1D optical lattices in the z direction. The atomic density in each dipole trap of spatial period a is assumed to exhibit a Gaussian distribution.
Fig. 2
Fig. 2 Imaginary (a) and real (b) parts of probe susceptibility χp vs. probe detuning Δ p for a homogeneous sample with atomic density N0 = 5.0 × 1012 cm−3 and dipole moment d14 = 2.0 × 10−29 C·m. Other parameters are Γ31 = Γ32 = Γ41 = Γ42 = 3.0 MHz, γ12 = 2.0 kHz, Δ c = Δ d = 0, Ω d = 2.0 MHz, and Ω c = 2.0 (black); 2.5 (blue); 3.0 (red).
Fig. 3
Fig. 3 Imaginary (a) and real (b) parts of probe susceptibility χp vs. lattice position (zzi)/a and probe detuning Δ p ; imaginary (c) and real (d) parts of probe susceptibility χp vs. lattice position (zzi)/a with probe detuning Δ p = 3.284 MHz. The coupling field is modulated as in Eq. (5) with Ωc0 = 3.0 MHz and δΩ c = 0.3 MHz. Other parameters used in the calculations are the same as in Fig. 2 except N0 = 5.0 × 1012 cm−3 and σ = 0.2a for the Gaussian density distribution in Eq. (5).
Fig. 4
Fig. 4 Imaginary (a) and real (b) parts of probe susceptibility χp vs. lattice position (zzi)/a for Δ p = 3.284 MHz and δΩ c = 0.3 MHz (red); δΩ c = 0.6 MHz (blue); δΩ c = 0.9 MHz (black) with other parameters being the same as in Fig. 3.
Fig. 5
Fig. 5 Imaginary (a) and real (b) parts of probe susceptibility χp vs. lattice position (zzi)/a and probe detuning Δ p ; imaginary (c) and real (d) parts of probe susceptibility χp vs. lattice position (zzi)/a with probe detuning Δ p = 0.0 MHz. The driving field is modulated as in Eq. (7) with Δd0 = 1.0 MHz. Other parameters used in the calculations are the same as in Fig. 2 except Ω c = 2.0 MHz as well as N0 = 5.0 × 1012 cm−3 and σ = 0.2a for the Gaussian density distribution in Eq. (5).
Fig. 6
Fig. 6 Imaginary (a) and real (b) parts of probe susceptibility χp vs. lattice position (zzi)/a for Δ p = 0.0 MHz and Δd0 = 1.0 MHz (red); Δd0 = 1.5 MHz (blue); Δd0 = 2.0 MHz (black) with other parameters being the same as in Fig. 5.

Equations (7)

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t ρ 11 = + Γ 31 ρ 33 + Γ 41 ρ 44 + i Ω c * ρ 31 i Ω c ρ 13 + i Ω p * ρ 41 i Ω p ρ 14 t ρ 22 = + Γ 32 ρ 33 + Γ 42 ρ 44 + i Ω d * ρ 42 i Ω d ρ 24 t ρ 33 = Γ 31 ρ 33 Γ 32 ρ 33 + i Ω c ρ 13 i Ω c * ρ 31 t ρ 12 = γ 12 ρ 12 + i Ω c * ρ 32 + i Ω p * ρ 42 i Ω d ρ 14 t ρ 13 = γ 13 ρ 13 + i Ω p * ρ 43 + i Ω c * ( ρ 33 ρ 11 ) t ρ 14 = γ 14 ρ 14 + i Ω c * ρ 34 + i Ω p * ( ρ 44 ρ 11 ) i Ω d * ρ 12 t ρ 23 = γ 23 ρ 23 + i Ω d * ρ 43 i Ω c * ρ 21 t ρ 24 = γ 24 ρ 24 i Ω p * ρ 21 + i Ω d * ( ρ 44 ρ 22 ) t ρ 34 = γ 34 ρ 34 + i Ω c ρ 14 i Ω p * ρ 31 i Ω d * ρ 32 ,
ρ 41 ( 1 ) = Ω p γ 23 ( Ω c 2 + γ 21 γ 23 ) [ ( Ω c 2 Ω d 2 + γ 21 γ 23 ) Ω c ρ 13 ( 0 ) + ( Ω d 2 Ω c 2 + γ 43 γ 23 ) Ω d ρ 24 ( 0 ) ] ( γ 23 Ω d 2 + γ 41 Ω c 2 + γ 21 γ 23 γ 41 ) ( γ 21 γ 23 Ω d 2 + γ 43 γ 23 Ω c 2 + γ 21 γ 43 γ 23 2 ) + ( Ω c 2 Ω d 2 + γ 21 γ 23 ) 2 γ 23 Ω c 2 + Ω p i ( γ 21 γ 23 Ω d 2 + γ 43 γ 23 Ω c 2 + γ 21 γ 43 γ 23 2 ) ( Ω c 2 + γ 21 γ 23 ) ( ρ 44 ( 0 ) ρ 11 ( 0 ) ) ( γ 23 Ω d 2 + γ 41 Ω c 2 + γ 21 γ 23 γ 41 ) ( γ 21 γ 23 Ω d 2 + γ 43 γ 23 Ω c 2 + γ 21 γ 43 γ 23 2 ) + ( Ω c 2 Ω d 2 + γ 21 γ 23 ) 2 γ 23 Ω c 2
ρ 13 ( 0 ) = Ω c * ( Δ c + i γ ) Ω d 2 Ω d 2 ( 2 Ω c 2 + Δ c 2 + γ 2 ) + Ω c 2 ( 2 Ω d 2 + Δ d 2 + γ 2 ) ρ 24 ( 0 ) = Ω d * ( Δ d + i γ ) Ω c 2 Ω d 2 ( 2 Ω c 2 + Δ c 2 + γ 2 ) + Ω c 2 ( 2 Ω d 2 + Δ d 2 + γ 2 ) ρ 11 ( 0 ) = ( Ω c 2 + Δ c 2 + γ 2 ) Ω d 2 Ω d 2 ( 2 Ω c 2 + Δ c 2 + γ 2 ) + Ω c 2 ( 2 Ω d 2 + Δ d 2 + γ 2 ) ρ 44 ( 0 ) = Ω c 2 Ω d 2 Ω d 2 ( 2 Ω c 2 + Δ c 2 + γ 2 ) + Ω c 2 ( 2 Ω d 2 + Δ d 2 + γ 2 )
χ p = χ p + i χ p = N 0 d 14 2 2 ε 0 h ¯ ρ 41 ( 1 )
N i ( z ) = N 0 e ( z z i ) 2 / σ 2 , z ( z i a / 2 , z i + a / 2 )
Ω c ( z ) = Ω c 0 + δ Ω c sin [ 2 π ( z z 0 ) / a ]
Δ d ( z ) = Δ d 0 sin [ 2 π ( z z 0 ) / a ]

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