Abstract

We theoretically study the transport properties in a one-dimensional photonic lattice influenced by the presence of side-coupled $\cal {PT}$-symmetric non-Hermitian defects. The $\cal {PT}$ symmetry is manifested as the complex potentials on the defects and the complex defect-lattice couplings, respectively. These two mechanisms are found to induce the Fano effect in the transport processes, with the different characteristics of it. Next, if the complex potentials and defect-lattice couplings co-exist, the Fano effect will be achieved more efficiently. However, further enhancing either of them can weaken the Fano interference seriously. Our findings reveal the physical essence of the Fano effect on the $\cal {PT}$-symmetric non-Hermitian defects, and the results can provide insights into the engineering and dynamical control of Fano resonances in non-Hermitian photonic structures.

© 2020 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 $\cal {PT}$PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
    [Crossref]
  2. S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of Branch Points in $\cal {PT}$PT-Symmetric Waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
    [Crossref]
  3. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in $\cal {PT}$PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
    [Crossref]
  4. A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in $\cal {PT}$PT-symmetric optical couplers,” Phys. Rev. A 82(4), 043818 (2010).
    [Crossref]
  5. H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “$\cal {PT}$PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012).
    [Crossref]
  6. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in $\cal {PT}$PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
    [Crossref]
  7. X. B. Luo, J. H. Huang, H. H. Zhong, X. Z. Qin, Q. T. Xie, Y. S. Kivshar, and C. H. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110(24), 243902 (2013).
    [Crossref]
  8. Y. C. Hu and T. L. Hughes, “Absence of topological insulator phases in non-Hermitian $\cal {PT}$PT-symmetric Hamiltonians,” Phys. Rev. B 84(15), 153101 (2011).
    [Crossref]
  9. B. G. Zhu, R. Lü, and S. Chen, “$\cal {PT}$PT symmetry in the non-Hermitian Su-Schrieffer-Heeger model with complex boundary potentials,” Phys. Rev. A 89(6), 062102 (2014).
    [Crossref]
  10. I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A: Math. Theor. 42(15), 153001 (2009).
    [Crossref]
  11. I. Y. Goldsheid and B. A. Khoruzhenko, “Distribution of eigenvalues in non-Hermitian Anderson models,” Phys. Rev. Lett. 80(13), 2897–2900 (1998).
    [Crossref]
  12. J. Heinrichs, “Eigenvalues in the non-Hermitian Anderson model,” Phys. Rev. B 63(16), 165108 (2001).
    [Crossref]
  13. L. G. Molinari, “Non-Hermitian spectra and Anderson localization,” J. Phys. A: Math. Theor. 42(26), 265204 (2009).
    [Crossref]
  14. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of $\cal {PT}$PT-Symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
    [Crossref]
  15. 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(3), 192–195 (2010).
    [Crossref]
  16. A. Regensburger, M. A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in $\cal {PT}$PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
    [Crossref]
  17. C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
    [Crossref]
  18. A. Mostafazadeh, “Pseudo-Hermiticity versus symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43(1), 205–214 (2002).
    [Crossref]
  19. C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of $\cal {PT}$PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70(2), 025001 (2004).
    [Crossref]
  20. H. F. Jones, “Equivalent Hamiltonians for $\cal {PT}$PT-symmetric versions of dual 2D field theories,” J. Phys. A: Math. Theor. 39(32), 10123–10132 (2006).
    [Crossref]
  21. T. Kottos, “Optical physics: Broken symmetry makes light work,” Nat. Phys. 6(3), 166–167 (2010).
    [Crossref]
  22. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical $\cal {PT}$PT-symmetric structures,” Opt. Lett. 32(17), 2632 (2007).
    [Crossref]
  23. L. Jin and Z. Song, “Physics counterpart of the $\cal {PT}$PT non-Hermitian tight-binding chain,” Phys. Rev. A 81(3), 032109 (2010).
    [Crossref]
  24. L. Jin and Z. Song, “Hermitian scattering behavior for a non-Hermitian scattering center,” Phys. Rev. A 85(1), 012111 (2012).
    [Crossref]
  25. H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with PT symmetry,” Phys. Rev. A 85(1), 013818 (2012).
    [Crossref]
  26. N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
    [Crossref]
  27. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
    [Crossref]
  28. B. Peng, S. K. Özdemir, 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(5), 394–398 (2014).
    [Crossref]
  29. 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(7), 524–529 (2014).
    [Crossref]
  30. L. F. Xue, Z. R. Gong, H. B. Zhu, and Z. H. Wang, “Symmetric phase transition and photonic transmission in an optical trimer system,” Opt. Express 25(15), 17249 (2017).
    [Crossref]
  31. C. M. Bender and M. Gianfreda, “Scattering off $\cal {PT}$PT-symmetric upside-down potentials,” Phys. Rev. A 98(5), 052118 (2018).
    [Crossref]
  32. B. Zhu, R. Lü, and S. Chen, “Interplay between Fano resonance and $\cal {PT}$PT-symmetry in non-Hermitian discrete systems,” Phys. Rev. A 91(4), 042131 (2015).
    [Crossref]
  33. A. A. Gorbatsevich and N. M. Shubin, “$\cal {PT}$PT-symmetry breaking in resonant tunneling heterostructures,” JETP Lett. 103(12), 769–773 (2016).
    [Crossref]
  34. A. A. Gorbatsevich and N. M. Shubin, “Coalescence of resonances in dissipationless resonant tunneling structures and $\cal {PT}$PT-symmetry breaking,” Ann. Phys. 376, 353–371 (2017).
    [Crossref]
  35. L. L. Zhang, G. H. Zhan, Z. Z. Li, and W. J. Gong, “Effect of $\cal {PT}$PT symmetry in a parallel double-quantum-dot structure,” Phys. Rev. A 96(6), 062133 (2017).
    [Crossref]
  36. L. L. Zhang and W. J. Gong, “Transport properties in a non-Hermitian triple-quantum-dot structure,” Phys. Rev. A 95(6), 062123 (2017).
    [Crossref]
  37. L. L. Zhang, Z. Z. Li, G. H. Zhan, G. Y. Yi, and W. J. Gong, “Eigenenergies and quantum transport properties in a non-Hermitian quantum-dot chain with side-coupled dots,” Phys. Rev. A 99(3), 032119 (2019).
    [Crossref]
  38. V. Achilleos, Y. Auregan, and V. Pagneux, “Scattering by Finite Periodic $\cal {PT}$PT-Symmetric Structures,” Phys. Rev. Lett. 119(24), 243904 (2017).
    [Crossref]
  39. S. Longhi, “Non-Hermitian tight-binding network engineering,” Phys. Rev. A 93(2), 022102 (2016).
    [Crossref]
  40. F. Tellander and K. F. Berggren, “Spectra, current flow, and wave-function morphology in a model $\cal {PT}$PT-symmetric quantum dot with external interactions,” Phys. Rev. A 95(4), 042115 (2017).
    [Crossref]
  41. L. Du, J. H. Wu, M. Artoni, and G. C. La Rocca, “Phase-dependent topological interface state and spatial adiabatic passage in a generalized Su-Schrieffer-Heeger model,” Phys. Rev. A 100(1), 012112 (2019).
    [Crossref]
  42. K. Kawabata, S. Higashikawa, Z. Gong, Y. Ashida, and M. Ueda, “Topological unification of time-reversal and particle-hole symmetries in non-Hermitian physics,” Nat. Commun. 10(1), 297 (2019).
    [Crossref]
  43. N. P. Robins, C. Figl, M. Jeppesen, G. R. Dennis, and J. D. Close, “A pumped atom laser,” Nat. Phys. 4(9), 731–736 (2008).
    [Crossref]
  44. J. A. S. Lourenço, R. L. Eneias, and R. G. Pereira, “Kondo effect in a $\cal {PT}$PT-symmetric non-Hermitian Hamiltonian,” Phys. Rev. B 98(8), 085126 (2018).
    [Crossref]
  45. W. Gong, Y. Zheng, Y. Liu, and T. Lü, “A Feynman path analysis of the Fano effect in electronic transport through a parallel double quantum dot structure,” Phys. E 40(3), 618–626 (2008).
    [Crossref]
  46. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124(6), 1866–1878 (1961).
    [Crossref]
  47. A. E. Miroschnichenko, S. Flach, and Y. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
    [Crossref]
  48. B. Lukyanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
    [Crossref]
  49. N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sonnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
    [Crossref]
  50. M. Rahmani, B. Lukyanchuk, and M. Hong, “Fano resonance in novel plasmonic nanostructures,” Laser Photonics Rev. 7(3), 329–349 (2013).
    [Crossref]
  51. K. Nozaki, A. Shinya, S. Matsuo, T. Sato, E. Kuramochi, and M. Notomi, “Ultralow-energy and high-contrast all-optical switch involving Fano resonance based on coupled photonic crystal nanocavities,” Opt. Express 21(10), 11877 (2013).
    [Crossref]
  52. W. Zhou, D. Zhao, Y. C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38(1), 1–74 (2014).
    [Crossref]

2019 (3)

L. L. Zhang, Z. Z. Li, G. H. Zhan, G. Y. Yi, and W. J. Gong, “Eigenenergies and quantum transport properties in a non-Hermitian quantum-dot chain with side-coupled dots,” Phys. Rev. A 99(3), 032119 (2019).
[Crossref]

L. Du, J. H. Wu, M. Artoni, and G. C. La Rocca, “Phase-dependent topological interface state and spatial adiabatic passage in a generalized Su-Schrieffer-Heeger model,” Phys. Rev. A 100(1), 012112 (2019).
[Crossref]

K. Kawabata, S. Higashikawa, Z. Gong, Y. Ashida, and M. Ueda, “Topological unification of time-reversal and particle-hole symmetries in non-Hermitian physics,” Nat. Commun. 10(1), 297 (2019).
[Crossref]

2018 (2)

J. A. S. Lourenço, R. L. Eneias, and R. G. Pereira, “Kondo effect in a $\cal {PT}$PT-symmetric non-Hermitian Hamiltonian,” Phys. Rev. B 98(8), 085126 (2018).
[Crossref]

C. M. Bender and M. Gianfreda, “Scattering off $\cal {PT}$PT-symmetric upside-down potentials,” Phys. Rev. A 98(5), 052118 (2018).
[Crossref]

2017 (6)

A. A. Gorbatsevich and N. M. Shubin, “Coalescence of resonances in dissipationless resonant tunneling structures and $\cal {PT}$PT-symmetry breaking,” Ann. Phys. 376, 353–371 (2017).
[Crossref]

L. L. Zhang, G. H. Zhan, Z. Z. Li, and W. J. Gong, “Effect of $\cal {PT}$PT symmetry in a parallel double-quantum-dot structure,” Phys. Rev. A 96(6), 062133 (2017).
[Crossref]

L. L. Zhang and W. J. Gong, “Transport properties in a non-Hermitian triple-quantum-dot structure,” Phys. Rev. A 95(6), 062123 (2017).
[Crossref]

L. F. Xue, Z. R. Gong, H. B. Zhu, and Z. H. Wang, “Symmetric phase transition and photonic transmission in an optical trimer system,” Opt. Express 25(15), 17249 (2017).
[Crossref]

V. Achilleos, Y. Auregan, and V. Pagneux, “Scattering by Finite Periodic $\cal {PT}$PT-Symmetric Structures,” Phys. Rev. Lett. 119(24), 243904 (2017).
[Crossref]

F. Tellander and K. F. Berggren, “Spectra, current flow, and wave-function morphology in a model $\cal {PT}$PT-symmetric quantum dot with external interactions,” Phys. Rev. A 95(4), 042115 (2017).
[Crossref]

2016 (2)

A. A. Gorbatsevich and N. M. Shubin, “$\cal {PT}$PT-symmetry breaking in resonant tunneling heterostructures,” JETP Lett. 103(12), 769–773 (2016).
[Crossref]

S. Longhi, “Non-Hermitian tight-binding network engineering,” Phys. Rev. A 93(2), 022102 (2016).
[Crossref]

2015 (1)

B. Zhu, R. Lü, and S. Chen, “Interplay between Fano resonance and $\cal {PT}$PT-symmetry in non-Hermitian discrete systems,” Phys. Rev. A 91(4), 042131 (2015).
[Crossref]

2014 (4)

B. Peng, S. K. Özdemir, 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(5), 394–398 (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(7), 524–529 (2014).
[Crossref]

B. G. Zhu, R. Lü, and S. Chen, “$\cal {PT}$PT symmetry in the non-Hermitian Su-Schrieffer-Heeger model with complex boundary potentials,” Phys. Rev. A 89(6), 062102 (2014).
[Crossref]

W. Zhou, D. Zhao, Y. C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38(1), 1–74 (2014).
[Crossref]

2013 (5)

M. Rahmani, B. Lukyanchuk, and M. Hong, “Fano resonance in novel plasmonic nanostructures,” Laser Photonics Rev. 7(3), 329–349 (2013).
[Crossref]

K. Nozaki, A. Shinya, S. Matsuo, T. Sato, E. Kuramochi, and M. Notomi, “Ultralow-energy and high-contrast all-optical switch involving Fano resonance based on coupled photonic crystal nanocavities,” Opt. Express 21(10), 11877 (2013).
[Crossref]

A. Regensburger, M. A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in $\cal {PT}$PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref]

X. B. Luo, J. H. Huang, H. H. Zhong, X. Z. Qin, Q. T. Xie, Y. S. Kivshar, and C. H. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110(24), 243902 (2013).
[Crossref]

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref]

2012 (3)

L. Jin and Z. Song, “Hermitian scattering behavior for a non-Hermitian scattering center,” Phys. Rev. A 85(1), 012111 (2012).
[Crossref]

H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with PT symmetry,” Phys. Rev. A 85(1), 013818 (2012).
[Crossref]

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “$\cal {PT}$PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012).
[Crossref]

2011 (1)

Y. C. Hu and T. L. Hughes, “Absence of topological insulator phases in non-Hermitian $\cal {PT}$PT-symmetric Hamiltonians,” Phys. Rev. B 84(15), 153101 (2011).
[Crossref]

2010 (8)

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in $\cal {PT}$PT-symmetric optical couplers,” Phys. Rev. A 82(4), 043818 (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(3), 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(6), 063807 (2010).
[Crossref]

T. Kottos, “Optical physics: Broken symmetry makes light work,” Nat. Phys. 6(3), 166–167 (2010).
[Crossref]

L. Jin and Z. Song, “Physics counterpart of the $\cal {PT}$PT non-Hermitian tight-binding chain,” Phys. Rev. A 81(3), 032109 (2010).
[Crossref]

A. E. Miroschnichenko, S. Flach, and Y. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
[Crossref]

B. Lukyanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[Crossref]

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sonnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref]

2009 (3)

I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A: Math. Theor. 42(15), 153001 (2009).
[Crossref]

L. G. Molinari, “Non-Hermitian spectra and Anderson localization,” J. Phys. A: Math. Theor. 42(26), 265204 (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 $\cal {PT}$PT-Symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

2008 (5)

S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of Branch Points in $\cal {PT}$PT-Symmetric Waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in $\cal {PT}$PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in $\cal {PT}$PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref]

W. Gong, Y. Zheng, Y. Liu, and T. Lü, “A Feynman path analysis of the Fano effect in electronic transport through a parallel double quantum dot structure,” Phys. E 40(3), 618–626 (2008).
[Crossref]

N. P. Robins, C. Figl, M. Jeppesen, G. R. Dennis, and J. D. Close, “A pumped atom laser,” Nat. Phys. 4(9), 731–736 (2008).
[Crossref]

2007 (1)

2006 (1)

H. F. Jones, “Equivalent Hamiltonians for $\cal {PT}$PT-symmetric versions of dual 2D field theories,” J. Phys. A: Math. Theor. 39(32), 10123–10132 (2006).
[Crossref]

2004 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of $\cal {PT}$PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70(2), 025001 (2004).
[Crossref]

2002 (2)

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref]

A. Mostafazadeh, “Pseudo-Hermiticity versus symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43(1), 205–214 (2002).
[Crossref]

2001 (1)

J. Heinrichs, “Eigenvalues in the non-Hermitian Anderson model,” Phys. Rev. B 63(16), 165108 (2001).
[Crossref]

1998 (2)

I. Y. Goldsheid and B. A. Khoruzhenko, “Distribution of eigenvalues in non-Hermitian Anderson models,” Phys. Rev. Lett. 80(13), 2897–2900 (1998).
[Crossref]

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having $\cal {PT}$PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

1961 (1)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124(6), 1866–1878 (1961).
[Crossref]

Achilleos, V.

V. Achilleos, Y. Auregan, and V. Pagneux, “Scattering by Finite Periodic $\cal {PT}$PT-Symmetric Structures,” Phys. Rev. Lett. 119(24), 243904 (2017).
[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 $\cal {PT}$PT-Symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

Artoni, M.

L. Du, J. H. Wu, M. Artoni, and G. C. La Rocca, “Phase-dependent topological interface state and spatial adiabatic passage in a generalized Su-Schrieffer-Heeger model,” Phys. Rev. A 100(1), 012112 (2019).
[Crossref]

Ashida, Y.

K. Kawabata, S. Higashikawa, Z. Gong, Y. Ashida, and M. Ueda, “Topological unification of time-reversal and particle-hole symmetries in non-Hermitian physics,” Nat. Commun. 10(1), 297 (2019).
[Crossref]

Auregan, Y.

V. Achilleos, Y. Auregan, and V. Pagneux, “Scattering by Finite Periodic $\cal {PT}$PT-Symmetric Structures,” Phys. Rev. Lett. 119(24), 243904 (2017).
[Crossref]

Bender, C. M.

C. M. Bender and M. Gianfreda, “Scattering off $\cal {PT}$PT-symmetric upside-down potentials,” Phys. Rev. A 98(5), 052118 (2018).
[Crossref]

B. Peng, S. K. Özdemir, 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(5), 394–398 (2014).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of $\cal {PT}$PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70(2), 025001 (2004).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref]

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having $\cal {PT}$PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Bender, N.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref]

Berggren, K. F.

F. Tellander and K. F. Berggren, “Spectra, current flow, and wave-function morphology in a model $\cal {PT}$PT-symmetric quantum dot with external interactions,” Phys. Rev. A 95(4), 042115 (2017).
[Crossref]

Bersch, C.

A. Regensburger, M. A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in $\cal {PT}$PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref]

Bodyfelt, J. D.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having $\cal {PT}$PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of $\cal {PT}$PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70(2), 025001 (2004).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref]

Chadha, A.

W. Zhou, D. Zhao, Y. C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38(1), 1–74 (2014).
[Crossref]

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(7), 524–529 (2014).
[Crossref]

Chen, S.

B. Zhu, R. Lü, and S. Chen, “Interplay between Fano resonance and $\cal {PT}$PT-symmetry in non-Hermitian discrete systems,” Phys. Rev. A 91(4), 042131 (2015).
[Crossref]

B. G. Zhu, R. Lü, and S. Chen, “$\cal {PT}$PT symmetry in the non-Hermitian Su-Schrieffer-Heeger model with complex boundary potentials,” Phys. Rev. A 89(6), 062102 (2014).
[Crossref]

Chong, C. T.

B. Lukyanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[Crossref]

Christodoulides, D. N.

A. Regensburger, M. A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in $\cal {PT}$PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref]

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref]

H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with PT symmetry,” Phys. Rev. A 85(1), 013818 (2012).
[Crossref]

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “$\cal {PT}$PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012).
[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(3), 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(6), 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 $\cal {PT}$PT-Symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in $\cal {PT}$PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in $\cal {PT}$PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical $\cal {PT}$PT-symmetric structures,” Opt. Lett. 32(17), 2632 (2007).
[Crossref]

Chuwongin, S.

W. Zhou, D. Zhao, Y. C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38(1), 1–74 (2014).
[Crossref]

Close, J. D.

N. P. Robins, C. Figl, M. Jeppesen, G. R. Dennis, and J. D. Close, “A pumped atom laser,” Nat. Phys. 4(9), 731–736 (2008).
[Crossref]

Dennis, G. R.

N. P. Robins, C. Figl, M. Jeppesen, G. R. Dennis, and J. D. Close, “A pumped atom laser,” Nat. Phys. 4(9), 731–736 (2008).
[Crossref]

Du, L.

L. Du, J. H. Wu, M. Artoni, and G. C. La Rocca, “Phase-dependent topological interface state and spatial adiabatic passage in a generalized Su-Schrieffer-Heeger model,” Phys. Rev. A 100(1), 012112 (2019).
[Crossref]

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 $\cal {PT}$PT-Symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

Eigenthaler, U.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sonnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[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,” Nat. Phys. 6(3), 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(6), 063807 (2010).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in $\cal {PT}$PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in $\cal {PT}$PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical $\cal {PT}$PT-symmetric structures,” Opt. Lett. 32(17), 2632 (2007).
[Crossref]

Ellis, F. M.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref]

Eneias, R. L.

J. A. S. Lourenço, R. L. Eneias, and R. G. Pereira, “Kondo effect in a $\cal {PT}$PT-symmetric non-Hermitian Hamiltonian,” Phys. Rev. B 98(8), 085126 (2018).
[Crossref]

Factor, S.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref]

Fan, S.

B. Peng, S. K. Özdemir, 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(5), 394–398 (2014).
[Crossref]

W. Zhou, D. Zhao, Y. C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38(1), 1–74 (2014).
[Crossref]

Fano, U.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124(6), 1866–1878 (1961).
[Crossref]

Figl, C.

N. P. Robins, C. Figl, M. Jeppesen, G. R. Dennis, and J. D. Close, “A pumped atom laser,” Nat. Phys. 4(9), 731–736 (2008).
[Crossref]

Flach, S.

A. E. Miroschnichenko, S. Flach, and Y. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
[Crossref]

Gianfreda, M.

C. M. Bender and M. Gianfreda, “Scattering off $\cal {PT}$PT-symmetric upside-down potentials,” Phys. Rev. A 98(5), 052118 (2018).
[Crossref]

B. Peng, S. K. Özdemir, 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(5), 394–398 (2014).
[Crossref]

Giessen, H.

B. Lukyanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[Crossref]

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sonnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref]

Goldsheid, I. Y.

I. Y. Goldsheid and B. A. Khoruzhenko, “Distribution of eigenvalues in non-Hermitian Anderson models,” Phys. Rev. Lett. 80(13), 2897–2900 (1998).
[Crossref]

Gong, W.

W. Gong, Y. Zheng, Y. Liu, and T. Lü, “A Feynman path analysis of the Fano effect in electronic transport through a parallel double quantum dot structure,” Phys. E 40(3), 618–626 (2008).
[Crossref]

Gong, W. J.

L. L. Zhang, Z. Z. Li, G. H. Zhan, G. Y. Yi, and W. J. Gong, “Eigenenergies and quantum transport properties in a non-Hermitian quantum-dot chain with side-coupled dots,” Phys. Rev. A 99(3), 032119 (2019).
[Crossref]

L. L. Zhang, G. H. Zhan, Z. Z. Li, and W. J. Gong, “Effect of $\cal {PT}$PT symmetry in a parallel double-quantum-dot structure,” Phys. Rev. A 96(6), 062133 (2017).
[Crossref]

L. L. Zhang and W. J. Gong, “Transport properties in a non-Hermitian triple-quantum-dot structure,” Phys. Rev. A 95(6), 062123 (2017).
[Crossref]

Gong, Z.

K. Kawabata, S. Higashikawa, Z. Gong, Y. Ashida, and M. Ueda, “Topological unification of time-reversal and particle-hole symmetries in non-Hermitian physics,” Nat. Commun. 10(1), 297 (2019).
[Crossref]

Gong, Z. R.

Gorbatsevich, A. A.

A. A. Gorbatsevich and N. M. Shubin, “Coalescence of resonances in dissipationless resonant tunneling structures and $\cal {PT}$PT-symmetry breaking,” Ann. Phys. 376, 353–371 (2017).
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A. A. Gorbatsevich and N. M. Shubin, “$\cal {PT}$PT-symmetry breaking in resonant tunneling heterostructures,” JETP Lett. 103(12), 769–773 (2016).
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Günther, U.

S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of Branch Points in $\cal {PT}$PT-Symmetric Waveguides,” Phys. Rev. Lett. 101(8), 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 $\cal {PT}$PT-Symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

Halas, N. J.

B. Lukyanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
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J. Heinrichs, “Eigenvalues in the non-Hermitian Anderson model,” Phys. Rev. B 63(16), 165108 (2001).
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K. Kawabata, S. Higashikawa, Z. Gong, Y. Ashida, and M. Ueda, “Topological unification of time-reversal and particle-hole symmetries in non-Hermitian physics,” Nat. Commun. 10(1), 297 (2019).
[Crossref]

Hirscher, M.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sonnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref]

Hong, M.

M. Rahmani, B. Lukyanchuk, and M. Hong, “Fano resonance in novel plasmonic nanostructures,” Laser Photonics Rev. 7(3), 329–349 (2013).
[Crossref]

Hu, Y. C.

Y. C. Hu and T. L. Hughes, “Absence of topological insulator phases in non-Hermitian $\cal {PT}$PT-symmetric Hamiltonians,” Phys. Rev. B 84(15), 153101 (2011).
[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(7), 524–529 (2014).
[Crossref]

Huang, J. H.

X. B. Luo, J. H. Huang, H. H. Zhong, X. Z. Qin, Q. T. Xie, Y. S. Kivshar, and C. H. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110(24), 243902 (2013).
[Crossref]

Hughes, T. L.

Y. C. Hu and T. L. Hughes, “Absence of topological insulator phases in non-Hermitian $\cal {PT}$PT-symmetric Hamiltonians,” Phys. Rev. B 84(15), 153101 (2011).
[Crossref]

Jeppesen, M.

N. P. Robins, C. Figl, M. Jeppesen, G. R. Dennis, and J. D. Close, “A pumped atom laser,” Nat. Phys. 4(9), 731–736 (2008).
[Crossref]

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(7), 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(7), 524–529 (2014).
[Crossref]

Jin, L.

L. Jin and Z. Song, “Hermitian scattering behavior for a non-Hermitian scattering center,” Phys. Rev. A 85(1), 012111 (2012).
[Crossref]

L. Jin and Z. Song, “Physics counterpart of the $\cal {PT}$PT non-Hermitian tight-binding chain,” Phys. Rev. A 81(3), 032109 (2010).
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Jones, H. F.

H. F. Jones, “Equivalent Hamiltonians for $\cal {PT}$PT-symmetric versions of dual 2D field theories,” J. Phys. A: Math. Theor. 39(32), 10123–10132 (2006).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of $\cal {PT}$PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70(2), 025001 (2004).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref]

Kawabata, K.

K. Kawabata, S. Higashikawa, Z. Gong, Y. Ashida, and M. Ueda, “Topological unification of time-reversal and particle-hole symmetries in non-Hermitian physics,” Nat. Commun. 10(1), 297 (2019).
[Crossref]

Khoruzhenko, B. A.

I. Y. Goldsheid and B. A. Khoruzhenko, “Distribution of eigenvalues in non-Hermitian Anderson models,” Phys. Rev. Lett. 80(13), 2897–2900 (1998).
[Crossref]

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,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Kivshar, Y.

A. E. Miroschnichenko, S. Flach, and Y. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
[Crossref]

Kivshar, Y. S.

X. B. Luo, J. H. Huang, H. H. Zhong, X. Z. Qin, Q. T. Xie, Y. S. Kivshar, and C. H. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110(24), 243902 (2013).
[Crossref]

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in $\cal {PT}$PT-symmetric optical couplers,” Phys. Rev. A 82(4), 043818 (2010).
[Crossref]

Klaiman, S.

S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of Branch Points in $\cal {PT}$PT-Symmetric Waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref]

Kottos, T.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref]

H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with PT symmetry,” Phys. Rev. A 85(1), 013818 (2012).
[Crossref]

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “$\cal {PT}$PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012).
[Crossref]

T. Kottos, “Optical physics: Broken symmetry makes light work,” Nat. Phys. 6(3), 166–167 (2010).
[Crossref]

Kovanis, V.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “$\cal {PT}$PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012).
[Crossref]

H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with PT symmetry,” Phys. Rev. A 85(1), 013818 (2012).
[Crossref]

Kuramochi, E.

La Rocca, G. C.

L. Du, J. H. Wu, M. Artoni, and G. C. La Rocca, “Phase-dependent topological interface state and spatial adiabatic passage in a generalized Su-Schrieffer-Heeger model,” Phys. Rev. A 100(1), 012112 (2019).
[Crossref]

Langguth, L.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sonnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref]

Lee, C. H.

X. B. Luo, J. H. Huang, H. H. Zhong, X. Z. Qin, Q. T. Xie, Y. S. Kivshar, and C. H. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110(24), 243902 (2013).
[Crossref]

Lei, F.

B. Peng, S. K. Özdemir, 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(5), 394–398 (2014).
[Crossref]

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(7), 524–529 (2014).
[Crossref]

Li, Z. Z.

L. L. Zhang, Z. Z. Li, G. H. Zhan, G. Y. Yi, and W. J. Gong, “Eigenenergies and quantum transport properties in a non-Hermitian quantum-dot chain with side-coupled dots,” Phys. Rev. A 99(3), 032119 (2019).
[Crossref]

L. L. Zhang, G. H. Zhan, Z. Z. Li, and W. J. Gong, “Effect of $\cal {PT}$PT symmetry in a parallel double-quantum-dot structure,” Phys. Rev. A 96(6), 062133 (2017).
[Crossref]

Liu, N.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sonnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref]

Liu, V.

W. Zhou, D. Zhao, Y. C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38(1), 1–74 (2014).
[Crossref]

Liu, Y.

W. Gong, Y. Zheng, Y. Liu, and T. Lü, “A Feynman path analysis of the Fano effect in electronic transport through a parallel double quantum dot structure,” Phys. E 40(3), 618–626 (2008).
[Crossref]

Long, G. L.

B. Peng, S. K. Özdemir, 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(5), 394–398 (2014).
[Crossref]

Longhi, S.

S. Longhi, “Non-Hermitian tight-binding network engineering,” Phys. Rev. A 93(2), 022102 (2016).
[Crossref]

Lourenço, J. A. S.

J. A. S. Lourenço, R. L. Eneias, and R. G. Pereira, “Kondo effect in a $\cal {PT}$PT-symmetric non-Hermitian Hamiltonian,” Phys. Rev. B 98(8), 085126 (2018).
[Crossref]

Lü, R.

B. Zhu, R. Lü, and S. Chen, “Interplay between Fano resonance and $\cal {PT}$PT-symmetry in non-Hermitian discrete systems,” Phys. Rev. A 91(4), 042131 (2015).
[Crossref]

B. G. Zhu, R. Lü, and S. Chen, “$\cal {PT}$PT symmetry in the non-Hermitian Su-Schrieffer-Heeger model with complex boundary potentials,” Phys. Rev. A 89(6), 062102 (2014).
[Crossref]

Lü, T.

W. Gong, Y. Zheng, Y. Liu, and T. Lü, “A Feynman path analysis of the Fano effect in electronic transport through a parallel double quantum dot structure,” Phys. E 40(3), 618–626 (2008).
[Crossref]

Lukyanchuk, B.

M. Rahmani, B. Lukyanchuk, and M. Hong, “Fano resonance in novel plasmonic nanostructures,” Laser Photonics Rev. 7(3), 329–349 (2013).
[Crossref]

B. Lukyanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[Crossref]

Luo, X. B.

X. B. Luo, J. H. Huang, H. H. Zhong, X. Z. Qin, Q. T. Xie, Y. S. Kivshar, and C. H. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110(24), 243902 (2013).
[Crossref]

Ma, Z.

W. Zhou, D. Zhao, Y. C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38(1), 1–74 (2014).
[Crossref]

Maier, S. A.

B. Lukyanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[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,” Nat. Phys. 6(3), 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(6), 063807 (2010).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in $\cal {PT}$PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in $\cal {PT}$PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical $\cal {PT}$PT-symmetric structures,” Opt. Lett. 32(17), 2632 (2007).
[Crossref]

Matsuo, S.

Mesch, M.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sonnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref]

Miri, M. A.

A. Regensburger, M. A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in $\cal {PT}$PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref]

Miroschnichenko, A. E.

A. E. Miroschnichenko, S. Flach, and Y. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
[Crossref]

Moiseyev, N.

S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of Branch Points in $\cal {PT}$PT-Symmetric Waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref]

Molinari, L. G.

L. G. Molinari, “Non-Hermitian spectra and Anderson localization,” J. Phys. A: Math. Theor. 42(26), 265204 (2009).
[Crossref]

Monifi, F.

B. Peng, S. K. Özdemir, 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(5), 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 $\cal {PT}$PT-Symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

Mostafazadeh, A.

A. Mostafazadeh, “Pseudo-Hermiticity versus symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43(1), 205–214 (2002).
[Crossref]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in $\cal {PT}$PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in $\cal {PT}$PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical $\cal {PT}$PT-symmetric structures,” Opt. Lett. 32(17), 2632 (2007).
[Crossref]

Näger, J.

A. Regensburger, M. A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in $\cal {PT}$PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref]

Nordlander, P.

B. Lukyanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[Crossref]

Nori, F.

B. Peng, S. K. Özdemir, 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(5), 394–398 (2014).
[Crossref]

Notomi, M.

Nozaki, K.

Onishchukov, G.

A. Regensburger, M. A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in $\cal {PT}$PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref]

Özdemir, S. K.

B. Peng, S. K. Özdemir, 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(5), 394–398 (2014).
[Crossref]

Pagneux, V.

V. Achilleos, Y. Auregan, and V. Pagneux, “Scattering by Finite Periodic $\cal {PT}$PT-Symmetric Structures,” Phys. Rev. Lett. 119(24), 243904 (2017).
[Crossref]

Peng, B.

B. Peng, S. K. Özdemir, 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(5), 394–398 (2014).
[Crossref]

Pereira, R. G.

J. A. S. Lourenço, R. L. Eneias, and R. G. Pereira, “Kondo effect in a $\cal {PT}$PT-symmetric non-Hermitian Hamiltonian,” Phys. Rev. B 98(8), 085126 (2018).
[Crossref]

Peschel, U.

A. Regensburger, M. A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in $\cal {PT}$PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref]

Qin, X. Z.

X. B. Luo, J. H. Huang, H. H. Zhong, X. Z. Qin, Q. T. Xie, Y. S. Kivshar, and C. H. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110(24), 243902 (2013).
[Crossref]

Rahmani, M.

M. Rahmani, B. Lukyanchuk, and M. Hong, “Fano resonance in novel plasmonic nanostructures,” Laser Photonics Rev. 7(3), 329–349 (2013).
[Crossref]

Ramezani, H.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
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H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with PT symmetry,” Phys. Rev. A 85(1), 013818 (2012).
[Crossref]

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “$\cal {PT}$PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012).
[Crossref]

Regensburger, A.

A. Regensburger, M. A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in $\cal {PT}$PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref]

Robins, N. P.

N. P. Robins, C. Figl, M. Jeppesen, G. R. Dennis, and J. D. Close, “A pumped atom laser,” Nat. Phys. 4(9), 731–736 (2008).
[Crossref]

Rotter, I.

I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A: Math. Theor. 42(15), 153001 (2009).
[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(3), 192–195 (2010).
[Crossref]

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 $\cal {PT}$PT-Symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

Sato, T.

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,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Seo, J. H.

W. Zhou, D. Zhao, Y. C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38(1), 1–74 (2014).
[Crossref]

Shinya, A.

Shuai, Y. C.

W. Zhou, D. Zhao, Y. C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38(1), 1–74 (2014).
[Crossref]

Shubin, N. M.

A. A. Gorbatsevich and N. M. Shubin, “Coalescence of resonances in dissipationless resonant tunneling structures and $\cal {PT}$PT-symmetry breaking,” Ann. Phys. 376, 353–371 (2017).
[Crossref]

A. A. Gorbatsevich and N. M. Shubin, “$\cal {PT}$PT-symmetry breaking in resonant tunneling heterostructures,” JETP Lett. 103(12), 769–773 (2016).
[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 $\cal {PT}$PT-Symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

Song, Z.

L. Jin and Z. Song, “Hermitian scattering behavior for a non-Hermitian scattering center,” Phys. Rev. A 85(1), 012111 (2012).
[Crossref]

L. Jin and Z. Song, “Physics counterpart of the $\cal {PT}$PT non-Hermitian tight-binding chain,” Phys. Rev. A 81(3), 032109 (2010).
[Crossref]

Sonnichsen, C.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sonnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref]

Sukhorukov, A. A.

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in $\cal {PT}$PT-symmetric optical couplers,” Phys. Rev. A 82(4), 043818 (2010).
[Crossref]

Tellander, F.

F. Tellander and K. F. Berggren, “Spectra, current flow, and wave-function morphology in a model $\cal {PT}$PT-symmetric quantum dot with external interactions,” Phys. Rev. A 95(4), 042115 (2017).
[Crossref]

Ueda, M.

K. Kawabata, S. Higashikawa, Z. Gong, Y. Ashida, and M. Ueda, “Topological unification of time-reversal and particle-hole symmetries in non-Hermitian physics,” Nat. Commun. 10(1), 297 (2019).
[Crossref]

Vitebskiy, I.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “$\cal {PT}$PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012).
[Crossref]

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 $\cal {PT}$PT-Symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 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(7), 524–529 (2014).
[Crossref]

Wang, K. X.

W. Zhou, D. Zhao, Y. C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38(1), 1–74 (2014).
[Crossref]

Wang, Z. H.

Weiss, T.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sonnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref]

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(7), 524–529 (2014).
[Crossref]

Wu, J. H.

L. Du, J. H. Wu, M. Artoni, and G. C. La Rocca, “Phase-dependent topological interface state and spatial adiabatic passage in a generalized Su-Schrieffer-Heeger model,” Phys. Rev. A 100(1), 012112 (2019).
[Crossref]

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(7), 524–529 (2014).
[Crossref]

Xie, Q. T.

X. B. Luo, J. H. Huang, H. H. Zhong, X. Z. Qin, Q. T. Xie, Y. S. Kivshar, and C. H. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110(24), 243902 (2013).
[Crossref]

Xu, Z.

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in $\cal {PT}$PT-symmetric optical couplers,” Phys. Rev. A 82(4), 043818 (2010).
[Crossref]

Xue, L. F.

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(7), 524–529 (2014).
[Crossref]

Yang, H.

W. Zhou, D. Zhao, Y. C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38(1), 1–74 (2014).
[Crossref]

Yang, L.

B. Peng, S. K. Özdemir, 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(5), 394–398 (2014).
[Crossref]

Yi, G. Y.

L. L. Zhang, Z. Z. Li, G. H. Zhan, G. Y. Yi, and W. J. Gong, “Eigenenergies and quantum transport properties in a non-Hermitian quantum-dot chain with side-coupled dots,” Phys. Rev. A 99(3), 032119 (2019).
[Crossref]

Zhan, G. H.

L. L. Zhang, Z. Z. Li, G. H. Zhan, G. Y. Yi, and W. J. Gong, “Eigenenergies and quantum transport properties in a non-Hermitian quantum-dot chain with side-coupled dots,” Phys. Rev. A 99(3), 032119 (2019).
[Crossref]

L. L. Zhang, G. H. Zhan, Z. Z. Li, and W. J. Gong, “Effect of $\cal {PT}$PT symmetry in a parallel double-quantum-dot structure,” Phys. Rev. A 96(6), 062133 (2017).
[Crossref]

Zhang, L. L.

L. L. Zhang, Z. Z. Li, G. H. Zhan, G. Y. Yi, and W. J. Gong, “Eigenenergies and quantum transport properties in a non-Hermitian quantum-dot chain with side-coupled dots,” Phys. Rev. A 99(3), 032119 (2019).
[Crossref]

L. L. Zhang and W. J. Gong, “Transport properties in a non-Hermitian triple-quantum-dot structure,” Phys. Rev. A 95(6), 062123 (2017).
[Crossref]

L. L. Zhang, G. H. Zhan, Z. Z. Li, and W. J. Gong, “Effect of $\cal {PT}$PT symmetry in a parallel double-quantum-dot structure,” Phys. Rev. A 96(6), 062133 (2017).
[Crossref]

Zhao, D.

W. Zhou, D. Zhao, Y. C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38(1), 1–74 (2014).
[Crossref]

Zheludev, N. I.

B. Lukyanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[Crossref]

Zheng, Y.

W. Gong, Y. Zheng, Y. Liu, and T. Lü, “A Feynman path analysis of the Fano effect in electronic transport through a parallel double quantum dot structure,” Phys. E 40(3), 618–626 (2008).
[Crossref]

Zhong, H. H.

X. B. Luo, J. H. Huang, H. H. Zhong, X. Z. Qin, Q. T. Xie, Y. S. Kivshar, and C. H. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110(24), 243902 (2013).
[Crossref]

Zhou, W.

W. Zhou, D. Zhao, Y. C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38(1), 1–74 (2014).
[Crossref]

Zhu, B.

B. Zhu, R. Lü, and S. Chen, “Interplay between Fano resonance and $\cal {PT}$PT-symmetry in non-Hermitian discrete systems,” Phys. Rev. A 91(4), 042131 (2015).
[Crossref]

Zhu, B. G.

B. G. Zhu, R. Lü, and S. Chen, “$\cal {PT}$PT symmetry in the non-Hermitian Su-Schrieffer-Heeger model with complex boundary potentials,” Phys. Rev. A 89(6), 062102 (2014).
[Crossref]

Zhu, H. B.

Ann. Phys. (1)

A. A. Gorbatsevich and N. M. Shubin, “Coalescence of resonances in dissipationless resonant tunneling structures and $\cal {PT}$PT-symmetry breaking,” Ann. Phys. 376, 353–371 (2017).
[Crossref]

J. Math. Phys. (1)

A. Mostafazadeh, “Pseudo-Hermiticity versus symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43(1), 205–214 (2002).
[Crossref]

J. Phys. A: Math. Theor. (3)

I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A: Math. Theor. 42(15), 153001 (2009).
[Crossref]

L. G. Molinari, “Non-Hermitian spectra and Anderson localization,” J. Phys. A: Math. Theor. 42(26), 265204 (2009).
[Crossref]

H. F. Jones, “Equivalent Hamiltonians for $\cal {PT}$PT-symmetric versions of dual 2D field theories,” J. Phys. A: Math. Theor. 39(32), 10123–10132 (2006).
[Crossref]

JETP Lett. (1)

A. A. Gorbatsevich and N. M. Shubin, “$\cal {PT}$PT-symmetry breaking in resonant tunneling heterostructures,” JETP Lett. 103(12), 769–773 (2016).
[Crossref]

Laser Photonics Rev. (1)

M. Rahmani, B. Lukyanchuk, and M. Hong, “Fano resonance in novel plasmonic nanostructures,” Laser Photonics Rev. 7(3), 329–349 (2013).
[Crossref]

Nano Lett. (1)

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sonnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref]

Nat. Commun. (1)

K. Kawabata, S. Higashikawa, Z. Gong, Y. Ashida, and M. Ueda, “Topological unification of time-reversal and particle-hole symmetries in non-Hermitian physics,” Nat. Commun. 10(1), 297 (2019).
[Crossref]

Nat. Mater. (1)

B. Lukyanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[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(7), 524–529 (2014).
[Crossref]

Nat. Phys. (4)

B. Peng, S. K. Özdemir, 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(5), 394–398 (2014).
[Crossref]

N. P. Robins, C. Figl, M. Jeppesen, G. R. Dennis, and J. D. Close, “A pumped atom laser,” Nat. Phys. 4(9), 731–736 (2008).
[Crossref]

T. Kottos, “Optical physics: Broken symmetry makes light work,” Nat. Phys. 6(3), 166–167 (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(3), 192–195 (2010).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. E (1)

W. Gong, Y. Zheng, Y. Liu, and T. Lü, “A Feynman path analysis of the Fano effect in electronic transport through a parallel double quantum dot structure,” Phys. E 40(3), 618–626 (2008).
[Crossref]

Phys. Rev. (1)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124(6), 1866–1878 (1961).
[Crossref]

Phys. Rev. A (14)

S. Longhi, “Non-Hermitian tight-binding network engineering,” Phys. Rev. A 93(2), 022102 (2016).
[Crossref]

F. Tellander and K. F. Berggren, “Spectra, current flow, and wave-function morphology in a model $\cal {PT}$PT-symmetric quantum dot with external interactions,” Phys. Rev. A 95(4), 042115 (2017).
[Crossref]

L. Du, J. H. Wu, M. Artoni, and G. C. La Rocca, “Phase-dependent topological interface state and spatial adiabatic passage in a generalized Su-Schrieffer-Heeger model,” Phys. Rev. A 100(1), 012112 (2019).
[Crossref]

C. M. Bender and M. Gianfreda, “Scattering off $\cal {PT}$PT-symmetric upside-down potentials,” Phys. Rev. A 98(5), 052118 (2018).
[Crossref]

B. Zhu, R. Lü, and S. Chen, “Interplay between Fano resonance and $\cal {PT}$PT-symmetry in non-Hermitian discrete systems,” Phys. Rev. A 91(4), 042131 (2015).
[Crossref]

L. L. Zhang, G. H. Zhan, Z. Z. Li, and W. J. Gong, “Effect of $\cal {PT}$PT symmetry in a parallel double-quantum-dot structure,” Phys. Rev. A 96(6), 062133 (2017).
[Crossref]

L. L. Zhang and W. J. Gong, “Transport properties in a non-Hermitian triple-quantum-dot structure,” Phys. Rev. A 95(6), 062123 (2017).
[Crossref]

L. L. Zhang, Z. Z. Li, G. H. Zhan, G. Y. Yi, and W. J. Gong, “Eigenenergies and quantum transport properties in a non-Hermitian quantum-dot chain with side-coupled dots,” Phys. Rev. A 99(3), 032119 (2019).
[Crossref]

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in $\cal {PT}$PT-symmetric optical couplers,” Phys. Rev. A 82(4), 043818 (2010).
[Crossref]

B. G. Zhu, R. Lü, and S. Chen, “$\cal {PT}$PT symmetry in the non-Hermitian Su-Schrieffer-Heeger model with complex boundary potentials,” Phys. Rev. A 89(6), 062102 (2014).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
[Crossref]

L. Jin and Z. Song, “Physics counterpart of the $\cal {PT}$PT non-Hermitian tight-binding chain,” Phys. Rev. A 81(3), 032109 (2010).
[Crossref]

L. Jin and Z. Song, “Hermitian scattering behavior for a non-Hermitian scattering center,” Phys. Rev. A 85(1), 012111 (2012).
[Crossref]

H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with PT symmetry,” Phys. Rev. A 85(1), 013818 (2012).
[Crossref]

Phys. Rev. B (3)

Y. C. Hu and T. L. Hughes, “Absence of topological insulator phases in non-Hermitian $\cal {PT}$PT-symmetric Hamiltonians,” Phys. Rev. B 84(15), 153101 (2011).
[Crossref]

J. Heinrichs, “Eigenvalues in the non-Hermitian Anderson model,” Phys. Rev. B 63(16), 165108 (2001).
[Crossref]

J. A. S. Lourenço, R. L. Eneias, and R. G. Pereira, “Kondo effect in a $\cal {PT}$PT-symmetric non-Hermitian Hamiltonian,” Phys. Rev. B 98(8), 085126 (2018).
[Crossref]

Phys. Rev. D (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of $\cal {PT}$PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70(2), 025001 (2004).
[Crossref]

Phys. Rev. Lett. (12)

A. Regensburger, M. A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in $\cal {PT}$PT-symmetric optical lattices,” Phys. Rev. Lett. 110(22), 223902 (2013).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of $\cal {PT}$PT-Symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
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I. Y. Goldsheid and B. A. Khoruzhenko, “Distribution of eigenvalues in non-Hermitian Anderson models,” Phys. Rev. Lett. 80(13), 2897–2900 (1998).
[Crossref]

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having $\cal {PT}$PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
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S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of Branch Points in $\cal {PT}$PT-Symmetric Waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in $\cal {PT}$PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref]

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “$\cal {PT}$PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in $\cal {PT}$PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
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X. B. Luo, J. H. Huang, H. H. Zhong, X. Z. Qin, Q. T. Xie, Y. S. Kivshar, and C. H. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. 110(24), 243902 (2013).
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V. Achilleos, Y. Auregan, and V. Pagneux, “Scattering by Finite Periodic $\cal {PT}$PT-Symmetric Structures,” Phys. Rev. Lett. 119(24), 243904 (2017).
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N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
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Prog. Quantum Electron. (1)

W. Zhou, D. Zhao, Y. C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38(1), 1–74 (2014).
[Crossref]

Rev. Mod. Phys. (1)

A. E. Miroschnichenko, S. Flach, and Y. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. Schematic of a 1D photonic lattice which is side-coupled to two defects with $\cal {PT}$-symmetric complex potentials, i.e., $\varepsilon _1=\varepsilon _0+i\gamma$ and $\varepsilon _2=\varepsilon _0-i\gamma$. Besides, $\cal {PT}$-symmetric complex defect-lattice couplings are taken into account with $\tau _1=\nu _0+\lambda e^{i\theta }$ and $\tau _2=\nu _0+\lambda e^{-i\theta }$.
Fig. 2.
Fig. 2. TF curves affected by the $\cal {PT}$-symmetric complex potentials or defect-lattice coupling, respectively. (a) Influence of the complex potentials. (b)-(c) Changes of the TF spectra due to the presence of $\cal {PT}$-symmetric defect-lattice coupling.
Fig. 3.
Fig. 3. TF influenced by the interplay between $\cal {PT}$-symmetric complex potentials and defect-lattice couplings, in the case of left-right asymmetry. (a)-(c) $\gamma =0.2$, 0.5, and 1.0, respectively. (d)-(f) $\theta =0.2\pi$, $0.5\pi$, and $0.9\pi$.
Fig. 4.
Fig. 4. Spectra of the TF due to the interplay between $\cal {PT}$-symmetric complex potentials and defect-lattice couplings, when upper-down asymmetry is considered. (a)-(c) $\gamma =0.2$, $0.5$, and $1.0$, respectively. (d)-(f) $\theta =0.2\pi$, $0.5\pi$, and $0.9\pi$.
Fig. 5.
Fig. 5. Diagrams of $\Delta$’s phases, i.e., $+$ or $-$, with the adjustment of $\theta$ and $\gamma$. (a) Case of left-right asymmetry. (b) Case of upper-down asymmetry.
Fig. 6.
Fig. 6. TF spectra due to the parameter changes of the $\cal {PT}$-symmetric defect-lattice couplings, in the case of $\theta =0.2\pi$ and $\gamma =0.2$. (a)-(b) Case of left-right asymmetry. (c)-(d) Case of upper-down asymmetry.

Equations (23)

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H = H C + H D + H T ,
H C = n = 2 t 0 c n c n + 1 + t 0 c 1 c 1 + n = 1 t 0 c n c n + 1 + h . c . , H D = ε 1 d 1 d 1 + ε 2 d 2 d 2 , H T = τ L 1 ( c 1 d 1 + h . c . ) + τ L 2 ( c 1 d 2 + h . c . ) + τ R 1 ( c 1 d 1 + h . c . ) + τ R 2 ( c 1 d 2 + h . c . ) .
T ( ω ) = ( μ R L + t 0 ) 2 | D μ | 2 ( 4 t 0 2 ω 2 ) ,
ω = ε 0 1 t 0 ( τ 2 ± τ 4 t 0 2 γ 2 ) .
ω = ε 0 2 | τ 1 | 2 t 0 = ε 0 2 t 0 ( ν 0 2 + λ 2 + 2 ν 0 λ cos θ ) .
ω = ε 0 2 R e ( τ 1 2 ) t 0 = ε 0 2 t 0 ( ν 0 2 + λ 2 cos 2 θ + 2 ν 0 λ cos θ ) .
ω = ε 0 1 t 0 ( | τ 1 | 2 ± | τ 1 | 4 t 0 2 γ 2 ) ,
ω = ε 0 1 t 0 { Re ( τ 1 2 ) ± [ Re ( τ 1 2 ) ] 2 2 Im ( τ 1 2 ) t 0 γ t 0 2 γ 2 } .
T ( ω ) = | e + q | 2 e 2 + 1 ,
e = 4 t 0 2 ω 2 A 2 + B ( μ R L + t 0 )
e = μ 0 + t 0 μ 0 ( ω + 2 t 0 ) 4 t 0 2 ω 2
e = 4 t 0 2 ω 2 ( μ 2 + t 0 ) ( μ 2 + t 0 ) 2 + μ 1 ω μ 1 2 t 0 2 .
i c ˙ n = t 0 c n 1 ( 1 δ n , 1 ) + t 0 c n + 1 ( 1 δ n , 1 ) + t 0 c n 2 δ n , 1 + t 0 c n + 2 δ n , 1 + ( τ L 1 d 1 + τ L 2 d 2 ) δ n , 1 + ( τ R 1 d 1 + τ R 2 d 2 ) δ n , 1 ,
i d ˙ 1 = ε 1 d 1 + τ L 1 c 1 + τ R 1 c 1 ,
i d ˙ 2 = ε 2 d 2 + τ R 2 c 1 + τ L 2 c 1 ,
ω A n = t 0 A n 1 ( 1 δ n , 1 ) + t 0 A n + 1 ( 1 δ n , 1 ) + t 0 A n 2 δ n , 1 + t 0 A n + 2 δ n , 1 + ( τ L 1 B 1 + τ L 2 B 2 ) δ n , 1 + ( τ R 1 B 1 + τ R 2 B 2 ) δ n , 1 ,
ω B 1 = ε 1 B 1 + τ L 1 A 1 + τ R 1 A 1 ,
ω B 2 = ε 2 B 2 + τ R 2 A 1 + τ L 2 A 1 ,
( ω μ L L ) A 1 = t 0 A 2 + ( μ L R + t 0 ) A 1 , ( ω μ R R ) A 1 = t 0 A 2 + ( μ R L + t 0 ) A 1 ,
A n = { e i k n + r e i k n , ( n 0 ) τ ~ e i k n , ( n > 0 ) .
[ ( ω μ L L t 0 e i k ) e i k ( μ L R + t 0 ) e i k ( μ R L + t 0 ) e i k ( ω μ R R ) e i k t 0 e 2 i k ] [ r τ ~ ] = [ ( ω μ L L t 0 e i k ) e i k ( μ R L + t 0 ) e i k ] .
τ ~ = t 0 ( μ R L + t 0 ) D μ ( e i k e i k ) e 2 i k
T ( ω ) = ( μ R L + t 0 ) 2 | D μ | 2 ( 4 t 0 2 ω 2 ) .

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