Abstract

We report a specially configured open optical microcavity, imposing a spatially imbalanced gain–loss profile, to host an exclusively proposed next-nearest-neighbor resonance coupling scheme. Adopting the scattering matrix (S-matrix) formalism, the effect of interplay between such proposed resonance interactions and the incorporated non-Hermiticity in the microcavity is analyzed drawing a special attention to the existence of hidden singularities, namely exceptional points (EPs), where at least two coupled resonances coalesce. We establish adiabatic flip-of-state phenomenon of the coupled resonances in the complex frequency plane (k-plane), which is essentially an outcome of the fact that the respective EP is being encircled in the system parameter plane. Encountering such multiple EPs, the robustness of flip-of-states phenomena has been analyzed via continuous tuning of coupling parameters along a special hidden singular line which connects all the EPs in the cavity. Such a numerically devised cavity, incorporating the exclusive next neighbor coupling scheme, has been designed for the first time to study the unconventional optical phenomena in the vicinity of EPs.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  3. K. Fang, G. W. Fernando, and A. N. Kocharian, “Pairing enhancement in Betts lattices with next nearest neighbor couplings: exact results,” Phys. Lett. A 376, 538–543 (2012).
    [Crossref]
  4. S.-J. Gu, H. Li, Y.-Q. Li, and H.-Q. Lin, “Entanglement of the Heisenberg chain with the next-nearest-neighbor interaction,” Phys. Rev. A 70, 052302 (2004).
    [Crossref]
  5. G. Santhosh, D. Kumar, and R. Ramaswamy, “Thermal transport in low-dimensional lattices with nearest- and next-nearest-neighbor coupling,” J. Stat. Mech. 7, P07005 (2005).
  6. V. Kadirko, K. Ziegler, and E. Kogan, “Next-nearest-neighbor tight-binding model of plasmons in graphene,” Graphene 2, 97–101 (2013).
    [Crossref]
  7. G. Csaba, À. Csurgay, and W. Porod, “Computing architecture composed of next-neighbor-coupled optically pumped nanodevices,” Int. J. Circuit Theory Appl. 29, 73–91 (2001).
    [Crossref]
  8. T. A. Zaleski and T. K. Kopeć, “Effect of next-nearest-neighbor hopping on Bose–Einstein condensation in optical lattices,” J. Phys. B 43, 085303 (2010).
    [Crossref]
  9. R. Keil, J. M. Zeuner, F. Dreisow, M. Heinrich, A. Tünnermann, S. Nolte, and A. Szameit, “The random mass Dirac model and long-range correlations on an integrated optical platform,” Nat. Commun. 4, 1368 (2013).
    [Crossref]
  10. L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
    [Crossref]
  11. A. Laha and S. N. Ghosh, “Suppression of excess noise of longer-lived high-quality states in nonuniformly pumped optical microcavities,” Opt. Lett. 41, 942–945 (2016).
    [Crossref]
  12. A. Laha and S. N. Ghosh, “Connected hidden singularities and toward successive state flipping in degenerate optical microcavities,” J. Opt. Soc. Am. B 34, 238–244 (2017).
    [Crossref]
  13. W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000).
    [Crossref]
  14. H. Cartarius, J. Main, and G. Wunner, “Exceptional points in the spectra of atoms in external fields,” Phys. Rev. A 79, 053408 (2009).
    [Crossref]
  15. S. N. Ghosh and Y. D. Chong, “Hidden singularities and asymmetric mode conversion in quasi-guided dual-mode optical waveguides,” Sci. Rep. 6, 19837 (2016).
    [Crossref]
  16. H. Menke, M. Klett, H. Cartarius, J. Main, and G. Wunner, “State flip at exceptional points in atomic spectra,” Phys. Rev. A 93, 013401 (2016).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  22. W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17, 149–151 (2001).
    [Crossref]
  23. S. Bittner, B. Dietz, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “Scattering experiments with microwave billiards at an exceptional point under broken time-reversal invariance,” Phys. Rev. E 89, 032909 (2014).
    [Crossref]
  24. R. Gutöhrlein, J. Main, H. Cartarius, and G. Wunner, “Bifurcations and exceptional points in dipolar Bose–Einstein condensates,” J. Phys. A 46, 305001 (2013).
    [Crossref]
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    [Crossref]
  26. K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals,” Phys. Rev. B 92, 235310 (2015).
    [Crossref]
  27. J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016).
    [Crossref]
  28. J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
    [Crossref]
  29. H. Hodaei, A. U. Hassan, W. E. Hayenga, M. A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Dark-state lasers: mode management using exceptional points,” Opt. Lett. 41, 3049–3052 (2016).
    [Crossref]
  30. G. Chen, R. Zhang, and J. Sun, “On-chip optical mode conversion based on dynamic grating in photonic-phononic hybrid waveguide,” Sci. Rep. 5, 10346 (2015).
    [Crossref]
  31. 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]
  32. K. Igarashi, D. Souma, T. Tsuritani, and I. Morita, “Performance evaluation of selective mode conversion based on phase plates for a 10-mode fiber,” Opt. Express 22, 20881–20893 (2014).
    [Crossref]
  33. G. Demange and E.-M. Graefe, “Signatures of three coalescing eigenfunctions,” J. Phys. A 45, 025303 (2012).
    [Crossref]
  34. A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, and A. A. Pukhov, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
    [Crossref]
  35. S. Phang, A. Vukovic, S. C. Creagh, T. M. Benson, P. D. Sewell, and G. Gradoni, “Parity-time symmetric coupled microresonators with a dispersive gain/loss,” Opt. Express 23, 11493–11507 (2015).
    [Crossref]

2017 (1)

2016 (6)

S. N. Ghosh and Y. D. Chong, “Hidden singularities and asymmetric mode conversion in quasi-guided dual-mode optical waveguides,” Sci. Rep. 6, 19837 (2016).
[Crossref]

H. Menke, M. Klett, H. Cartarius, J. Main, and G. Wunner, “State flip at exceptional points in atomic spectra,” Phys. Rev. A 93, 013401 (2016).
[Crossref]

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016).
[Crossref]

A. Laha and S. N. Ghosh, “Suppression of excess noise of longer-lived high-quality states in nonuniformly pumped optical microcavities,” Opt. Lett. 41, 942–945 (2016).
[Crossref]

H. Hodaei, A. U. Hassan, W. E. Hayenga, M. A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Dark-state lasers: mode management using exceptional points,” Opt. Lett. 41, 3049–3052 (2016).
[Crossref]

2015 (3)

S. Phang, A. Vukovic, S. C. Creagh, T. M. Benson, P. D. Sewell, and G. Gradoni, “Parity-time symmetric coupled microresonators with a dispersive gain/loss,” Opt. Express 23, 11493–11507 (2015).
[Crossref]

G. Chen, R. Zhang, and J. Sun, “On-chip optical mode conversion based on dynamic grating in photonic-phononic hybrid waveguide,” Sci. Rep. 5, 10346 (2015).
[Crossref]

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals,” Phys. Rev. B 92, 235310 (2015).
[Crossref]

2014 (4)

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
[Crossref]

S. Bittner, B. Dietz, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “Scattering experiments with microwave billiards at an exceptional point under broken time-reversal invariance,” Phys. Rev. E 89, 032909 (2014).
[Crossref]

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, and A. A. Pukhov, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[Crossref]

K. Igarashi, D. Souma, T. Tsuritani, and I. Morita, “Performance evaluation of selective mode conversion based on phase plates for a 10-mode fiber,” Opt. Express 22, 20881–20893 (2014).
[Crossref]

2013 (3)

V. Kadirko, K. Ziegler, and E. Kogan, “Next-nearest-neighbor tight-binding model of plasmons in graphene,” Graphene 2, 97–101 (2013).
[Crossref]

R. Keil, J. M. Zeuner, F. Dreisow, M. Heinrich, A. Tünnermann, S. Nolte, and A. Szameit, “The random mass Dirac model and long-range correlations on an integrated optical platform,” Nat. Commun. 4, 1368 (2013).
[Crossref]

R. Gutöhrlein, J. Main, H. Cartarius, and G. Wunner, “Bifurcations and exceptional points in dipolar Bose–Einstein condensates,” J. Phys. A 46, 305001 (2013).
[Crossref]

2012 (2)

K. Fang, G. W. Fernando, and A. N. Kocharian, “Pairing enhancement in Betts lattices with next nearest neighbor couplings: exact results,” Phys. Lett. A 376, 538–543 (2012).
[Crossref]

G. Demange and E.-M. Graefe, “Signatures of three coalescing eigenfunctions,” J. Phys. A 45, 025303 (2012).
[Crossref]

2011 (1)

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[Crossref]

2010 (1)

T. A. Zaleski and T. K. Kopeć, “Effect of next-nearest-neighbor hopping on Bose–Einstein condensation in optical lattices,” J. Phys. B 43, 085303 (2010).
[Crossref]

2009 (2)

H. Cartarius, J. Main, and G. Wunner, “Exceptional points in the spectra of atoms in external fields,” Phys. Rev. A 79, 053408 (2009).
[Crossref]

R. Lefebvre, O. Atabek, M. Šindelka, and N. Moiseyev, “Resonance coalescence in molecular photodissociation,” Phys. Rev. Lett. 103, 123003 (2009).
[Crossref]

2008 (1)

E. M. Graefe, U. Günther, H. J. Korsch, and A. E. Niederle, “A non-Hermitian PT-symmetric Bose–Hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A 41, 255206 (2008).
[Crossref]

2005 (2)

G. Santhosh, D. Kumar, and R. Ramaswamy, “Thermal transport in low-dimensional lattices with nearest- and next-nearest-neighbor coupling,” J. Stat. Mech. 7, P07005 (2005).

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]

2004 (1)

S.-J. Gu, H. Li, Y.-Q. Li, and H.-Q. Lin, “Entanglement of the Heisenberg chain with the next-nearest-neighbor interaction,” Phys. Rev. A 70, 052302 (2004).
[Crossref]

2003 (2)

C. Dembowski, B. Dietz, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Observation of a chiral state in a microwave cavity,” Phys. Rev. Lett. 90, 034101 (2003).
[Crossref]

M. V. Berry, “Mode degeneracies and the Petermann excess-noise factor for unstable lasers,” J. Mod. Opt. 50, 63–81 (2003).
[Crossref]

2001 (3)

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17, 149–151 (2001).
[Crossref]

G. Csaba, À. Csurgay, and W. Porod, “Computing architecture composed of next-neighbor-coupled optically pumped nanodevices,” Int. J. Circuit Theory Appl. 29, 73–91 (2001).
[Crossref]

2000 (1)

W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000).
[Crossref]

1989 (1)

M. Bernaschi, L. Biferale, L. A. Fernandez, U. M. B. Marconi, R. Petronzio, and A. Tarancon, “Renormalization group study of the three state three dimensional Potts model,” Phys. Lett. B 231, 157–160 (1989).
[Crossref]

1970 (1)

L. J. Kijewski and M. P. Kawatra, “One-dimensional Ising model with long-range interaction,” Phys. Lett. A 31, 479–480 (1970).
[Crossref]

Atabek, O.

R. Lefebvre, O. Atabek, M. Šindelka, and N. Moiseyev, “Resonance coalescence in molecular photodissociation,” Phys. Rev. Lett. 103, 123003 (2009).
[Crossref]

Benson, T. M.

Bernaschi, M.

M. Bernaschi, L. Biferale, L. A. Fernandez, U. M. B. Marconi, R. Petronzio, and A. Tarancon, “Renormalization group study of the three state three dimensional Potts model,” Phys. Lett. B 231, 157–160 (1989).
[Crossref]

Berry, M. V.

M. V. Berry, “Mode degeneracies and the Petermann excess-noise factor for unstable lasers,” J. Mod. Opt. 50, 63–81 (2003).
[Crossref]

Biferale, L.

M. Bernaschi, L. Biferale, L. A. Fernandez, U. M. B. Marconi, R. Petronzio, and A. Tarancon, “Renormalization group study of the three state three dimensional Potts model,” Phys. Lett. B 231, 157–160 (1989).
[Crossref]

Bittner, S.

S. Bittner, B. Dietz, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “Scattering experiments with microwave billiards at an exceptional point under broken time-reversal invariance,” Phys. Rev. E 89, 032909 (2014).
[Crossref]

Böhm, J.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Cartarius, H.

H. Menke, M. Klett, H. Cartarius, J. Main, and G. Wunner, “State flip at exceptional points in atomic spectra,” Phys. Rev. A 93, 013401 (2016).
[Crossref]

R. Gutöhrlein, J. Main, H. Cartarius, and G. Wunner, “Bifurcations and exceptional points in dipolar Bose–Einstein condensates,” J. Phys. A 46, 305001 (2013).
[Crossref]

H. Cartarius, J. Main, and G. Wunner, “Exceptional points in the spectra of atoms in external fields,” Phys. Rev. A 79, 053408 (2009).
[Crossref]

Chan, C. T.

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals,” Phys. Rev. B 92, 235310 (2015).
[Crossref]

Chen, G.

G. Chen, R. Zhang, and J. Sun, “On-chip optical mode conversion based on dynamic grating in photonic-phononic hybrid waveguide,” Sci. Rep. 5, 10346 (2015).
[Crossref]

Chong, Y. D.

S. N. Ghosh and Y. D. Chong, “Hidden singularities and asymmetric mode conversion in quasi-guided dual-mode optical waveguides,” Sci. Rep. 6, 19837 (2016).
[Crossref]

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[Crossref]

Christodoulides, D. N.

Creagh, S. C.

Csaba, G.

G. Csaba, À. Csurgay, and W. Porod, “Computing architecture composed of next-neighbor-coupled optically pumped nanodevices,” Int. J. Circuit Theory Appl. 29, 73–91 (2001).
[Crossref]

Csurgay, À.

G. Csaba, À. Csurgay, and W. Porod, “Computing architecture composed of next-neighbor-coupled optically pumped nanodevices,” Int. J. Circuit Theory Appl. 29, 73–91 (2001).
[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]

Demange, G.

G. Demange and E.-M. Graefe, “Signatures of three coalescing eigenfunctions,” J. Phys. A 45, 025303 (2012).
[Crossref]

Dembowski, C.

C. Dembowski, B. Dietz, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Observation of a chiral state in a microwave cavity,” Phys. Rev. Lett. 90, 034101 (2003).
[Crossref]

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

Dietz, B.

S. Bittner, B. Dietz, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “Scattering experiments with microwave billiards at an exceptional point under broken time-reversal invariance,” Phys. Rev. E 89, 032909 (2014).
[Crossref]

C. Dembowski, B. Dietz, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Observation of a chiral state in a microwave cavity,” Phys. Rev. Lett. 90, 034101 (2003).
[Crossref]

Ding, K.

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals,” Phys. Rev. B 92, 235310 (2015).
[Crossref]

Doppler, J.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Dorofeenko, A. V.

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, and A. A. Pukhov, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[Crossref]

Dreisow, F.

R. Keil, J. M. Zeuner, F. Dreisow, M. Heinrich, A. Tünnermann, S. Nolte, and A. Szameit, “The random mass Dirac model and long-range correlations on an integrated optical platform,” Nat. Commun. 4, 1368 (2013).
[Crossref]

Fang, K.

K. Fang, G. W. Fernando, and A. N. Kocharian, “Pairing enhancement in Betts lattices with next nearest neighbor couplings: exact results,” Phys. Lett. A 376, 538–543 (2012).
[Crossref]

Fernandez, L. A.

M. Bernaschi, L. Biferale, L. A. Fernandez, U. M. B. Marconi, R. Petronzio, and A. Tarancon, “Renormalization group study of the three state three dimensional Potts model,” Phys. Lett. B 231, 157–160 (1989).
[Crossref]

Fernando, G. W.

K. Fang, G. W. Fernando, and A. N. Kocharian, “Pairing enhancement in Betts lattices with next nearest neighbor couplings: exact results,” Phys. Lett. A 376, 538–543 (2012).
[Crossref]

Ge, L.

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[Crossref]

Ghosh, S. N.

Girschik, A.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Gradoni, G.

Graefe, E. M.

E. M. Graefe, U. Günther, H. J. Korsch, and A. E. Niederle, “A non-Hermitian PT-symmetric Bose–Hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A 41, 255206 (2008).
[Crossref]

Graefe, E.-M.

G. Demange and E.-M. Graefe, “Signatures of three coalescing eigenfunctions,” J. Phys. A 45, 025303 (2012).
[Crossref]

Gräf, H. D.

C. Dembowski, B. Dietz, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Observation of a chiral state in a microwave cavity,” Phys. Rev. Lett. 90, 034101 (2003).
[Crossref]

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

Gu, S.-J.

S.-J. Gu, H. Li, Y.-Q. Li, and H.-Q. Lin, “Entanglement of the Heisenberg chain with the next-nearest-neighbor interaction,” Phys. Rev. A 70, 052302 (2004).
[Crossref]

Günther, U.

E. M. Graefe, U. Günther, H. J. Korsch, and A. E. Niederle, “A non-Hermitian PT-symmetric Bose–Hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A 41, 255206 (2008).
[Crossref]

Gutöhrlein, R.

R. Gutöhrlein, J. Main, H. Cartarius, and G. Wunner, “Bifurcations and exceptional points in dipolar Bose–Einstein condensates,” J. Phys. A 46, 305001 (2013).
[Crossref]

Harney, H. L.

S. Bittner, B. Dietz, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “Scattering experiments with microwave billiards at an exceptional point under broken time-reversal invariance,” Phys. Rev. E 89, 032909 (2014).
[Crossref]

C. Dembowski, B. Dietz, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Observation of a chiral state in a microwave cavity,” Phys. Rev. Lett. 90, 034101 (2003).
[Crossref]

W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17, 149–151 (2001).
[Crossref]

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

Hassan, A. U.

Hayenga, W. E.

Heine, A.

C. Dembowski, B. Dietz, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Observation of a chiral state in a microwave cavity,” Phys. Rev. Lett. 90, 034101 (2003).
[Crossref]

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

Heinrich, M.

R. Keil, J. M. Zeuner, F. Dreisow, M. Heinrich, A. Tünnermann, S. Nolte, and A. Szameit, “The random mass Dirac model and long-range correlations on an integrated optical platform,” Nat. Commun. 4, 1368 (2013).
[Crossref]

Heiss, W. D.

C. Dembowski, B. Dietz, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Observation of a chiral state in a microwave cavity,” Phys. Rev. Lett. 90, 034101 (2003).
[Crossref]

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17, 149–151 (2001).
[Crossref]

W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000).
[Crossref]

Hodaei, H.

Igarashi, K.

Kadirko, V.

V. Kadirko, K. Ziegler, and E. Kogan, “Next-nearest-neighbor tight-binding model of plasmons in graphene,” Graphene 2, 97–101 (2013).
[Crossref]

Kawatra, M. P.

L. J. Kijewski and M. P. Kawatra, “One-dimensional Ising model with long-range interaction,” Phys. Lett. A 31, 479–480 (1970).
[Crossref]

Keil, R.

R. Keil, J. M. Zeuner, F. Dreisow, M. Heinrich, A. Tünnermann, S. Nolte, and A. Szameit, “The random mass Dirac model and long-range correlations on an integrated optical platform,” Nat. Commun. 4, 1368 (2013).
[Crossref]

Khajavikhan, M.

Kijewski, L. J.

L. J. Kijewski and M. P. Kawatra, “One-dimensional Ising model with long-range interaction,” Phys. Lett. A 31, 479–480 (1970).
[Crossref]

Klett, M.

H. Menke, M. Klett, H. Cartarius, J. Main, and G. Wunner, “State flip at exceptional points in atomic spectra,” Phys. Rev. A 93, 013401 (2016).
[Crossref]

Kocharian, A. N.

K. Fang, G. W. Fernando, and A. N. Kocharian, “Pairing enhancement in Betts lattices with next nearest neighbor couplings: exact results,” Phys. Lett. A 376, 538–543 (2012).
[Crossref]

Kogan, E.

V. Kadirko, K. Ziegler, and E. Kogan, “Next-nearest-neighbor tight-binding model of plasmons in graphene,” Graphene 2, 97–101 (2013).
[Crossref]

Kopec, T. K.

T. A. Zaleski and T. K. Kopeć, “Effect of next-nearest-neighbor hopping on Bose–Einstein condensation in optical lattices,” J. Phys. B 43, 085303 (2010).
[Crossref]

Korsch, H. J.

E. M. Graefe, U. Günther, H. J. Korsch, and A. E. Niederle, “A non-Hermitian PT-symmetric Bose–Hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A 41, 255206 (2008).
[Crossref]

Kuhl, U.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Kumar, D.

G. Santhosh, D. Kumar, and R. Ramaswamy, “Thermal transport in low-dimensional lattices with nearest- and next-nearest-neighbor coupling,” J. Stat. Mech. 7, P07005 (2005).

Laha, A.

Lefebvre, R.

R. Lefebvre, O. Atabek, M. Šindelka, and N. Moiseyev, “Resonance coalescence in molecular photodissociation,” Phys. Rev. Lett. 103, 123003 (2009).
[Crossref]

Li, H.

S.-J. Gu, H. Li, Y.-Q. Li, and H.-Q. Lin, “Entanglement of the Heisenberg chain with the next-nearest-neighbor interaction,” Phys. Rev. A 70, 052302 (2004).
[Crossref]

Li, Y.-Q.

S.-J. Gu, H. Li, Y.-Q. Li, and H.-Q. Lin, “Entanglement of the Heisenberg chain with the next-nearest-neighbor interaction,” Phys. Rev. A 70, 052302 (2004).
[Crossref]

Libisch, F.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Lin, H.-Q.

S.-J. Gu, H. Li, Y.-Q. Li, and H.-Q. Lin, “Entanglement of the Heisenberg chain with the next-nearest-neighbor interaction,” Phys. Rev. A 70, 052302 (2004).
[Crossref]

Mailybaev, A. A.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Main, J.

H. Menke, M. Klett, H. Cartarius, J. Main, and G. Wunner, “State flip at exceptional points in atomic spectra,” Phys. Rev. A 93, 013401 (2016).
[Crossref]

R. Gutöhrlein, J. Main, H. Cartarius, and G. Wunner, “Bifurcations and exceptional points in dipolar Bose–Einstein condensates,” J. Phys. A 46, 305001 (2013).
[Crossref]

H. Cartarius, J. Main, and G. Wunner, “Exceptional points in the spectra of atoms in external fields,” Phys. Rev. A 79, 053408 (2009).
[Crossref]

Marconi, U. M. B.

M. Bernaschi, L. Biferale, L. A. Fernandez, U. M. B. Marconi, R. Petronzio, and A. Tarancon, “Renormalization group study of the three state three dimensional Potts model,” Phys. Lett. B 231, 157–160 (1989).
[Crossref]

Menke, H.

H. Menke, M. Klett, H. Cartarius, J. Main, and G. Wunner, “State flip at exceptional points in atomic spectra,” Phys. Rev. A 93, 013401 (2016).
[Crossref]

Milburn, T. J.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Miri, M. A.

Miski-Oglu, M.

S. Bittner, B. Dietz, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “Scattering experiments with microwave billiards at an exceptional point under broken time-reversal invariance,” Phys. Rev. E 89, 032909 (2014).
[Crossref]

Moiseyev, N.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

R. Lefebvre, O. Atabek, M. Šindelka, and N. Moiseyev, “Resonance coalescence in molecular photodissociation,” Phys. Rev. Lett. 103, 123003 (2009).
[Crossref]

Morita, I.

Muga, J. G.

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]

Niederle, A. E.

E. M. Graefe, U. Günther, H. J. Korsch, and A. E. Niederle, “A non-Hermitian PT-symmetric Bose–Hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A 41, 255206 (2008).
[Crossref]

Nolte, S.

R. Keil, J. M. Zeuner, F. Dreisow, M. Heinrich, A. Tünnermann, S. Nolte, and A. Szameit, “The random mass Dirac model and long-range correlations on an integrated optical platform,” Nat. Commun. 4, 1368 (2013).
[Crossref]

Petronzio, R.

M. Bernaschi, L. Biferale, L. A. Fernandez, U. M. B. Marconi, R. Petronzio, and A. Tarancon, “Renormalization group study of the three state three dimensional Potts model,” Phys. Lett. B 231, 157–160 (1989).
[Crossref]

Phang, S.

Porod, W.

G. Csaba, À. Csurgay, and W. Porod, “Computing architecture composed of next-neighbor-coupled optically pumped nanodevices,” Int. J. Circuit Theory Appl. 29, 73–91 (2001).
[Crossref]

Pukhov, A. A.

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, and A. A. Pukhov, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[Crossref]

Rabl, P.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Ramaswamy, R.

G. Santhosh, D. Kumar, and R. Ramaswamy, “Thermal transport in low-dimensional lattices with nearest- and next-nearest-neighbor coupling,” J. Stat. Mech. 7, P07005 (2005).

Rehfeld, H.

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

Richter, A.

S. Bittner, B. Dietz, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “Scattering experiments with microwave billiards at an exceptional point under broken time-reversal invariance,” Phys. Rev. E 89, 032909 (2014).
[Crossref]

C. Dembowski, B. Dietz, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Observation of a chiral state in a microwave cavity,” Phys. Rev. Lett. 90, 034101 (2003).
[Crossref]

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

Rotter, S.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[Crossref]

Ruschhaupt, A.

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]

Santhosh, G.

G. Santhosh, D. Kumar, and R. Ramaswamy, “Thermal transport in low-dimensional lattices with nearest- and next-nearest-neighbor coupling,” J. Stat. Mech. 7, P07005 (2005).

Schäfer, F.

S. Bittner, B. Dietz, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “Scattering experiments with microwave billiards at an exceptional point under broken time-reversal invariance,” Phys. Rev. E 89, 032909 (2014).
[Crossref]

Sewell, P. D.

Šindelka, M.

R. Lefebvre, O. Atabek, M. Šindelka, and N. Moiseyev, “Resonance coalescence in molecular photodissociation,” Phys. Rev. Lett. 103, 123003 (2009).
[Crossref]

Souma, D.

Stone, A. D.

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[Crossref]

Sun, J.

G. Chen, R. Zhang, and J. Sun, “On-chip optical mode conversion based on dynamic grating in photonic-phononic hybrid waveguide,” Sci. Rep. 5, 10346 (2015).
[Crossref]

Szameit, A.

R. Keil, J. M. Zeuner, F. Dreisow, M. Heinrich, A. Tünnermann, S. Nolte, and A. Szameit, “The random mass Dirac model and long-range correlations on an integrated optical platform,” Nat. Commun. 4, 1368 (2013).
[Crossref]

Tarancon, A.

M. Bernaschi, L. Biferale, L. A. Fernandez, U. M. B. Marconi, R. Petronzio, and A. Tarancon, “Renormalization group study of the three state three dimensional Potts model,” Phys. Lett. B 231, 157–160 (1989).
[Crossref]

Tsuritani, T.

Tünnermann, A.

R. Keil, J. M. Zeuner, F. Dreisow, M. Heinrich, A. Tünnermann, S. Nolte, and A. Szameit, “The random mass Dirac model and long-range correlations on an integrated optical platform,” Nat. Commun. 4, 1368 (2013).
[Crossref]

Türeci, H. E.

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[Crossref]

Vinogradov, A. P.

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, and A. A. Pukhov, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[Crossref]

Vukovic, A.

Wiersig, J.

J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016).
[Crossref]

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
[Crossref]

Wunner, G.

H. Menke, M. Klett, H. Cartarius, J. Main, and G. Wunner, “State flip at exceptional points in atomic spectra,” Phys. Rev. A 93, 013401 (2016).
[Crossref]

R. Gutöhrlein, J. Main, H. Cartarius, and G. Wunner, “Bifurcations and exceptional points in dipolar Bose–Einstein condensates,” J. Phys. A 46, 305001 (2013).
[Crossref]

H. Cartarius, J. Main, and G. Wunner, “Exceptional points in the spectra of atoms in external fields,” Phys. Rev. A 79, 053408 (2009).
[Crossref]

Zaleski, T. A.

T. A. Zaleski and T. K. Kopeć, “Effect of next-nearest-neighbor hopping on Bose–Einstein condensation in optical lattices,” J. Phys. B 43, 085303 (2010).
[Crossref]

Zeuner, J. M.

R. Keil, J. M. Zeuner, F. Dreisow, M. Heinrich, A. Tünnermann, S. Nolte, and A. Szameit, “The random mass Dirac model and long-range correlations on an integrated optical platform,” Nat. Commun. 4, 1368 (2013).
[Crossref]

Zhang, R.

G. Chen, R. Zhang, and J. Sun, “On-chip optical mode conversion based on dynamic grating in photonic-phononic hybrid waveguide,” Sci. Rep. 5, 10346 (2015).
[Crossref]

Zhang, Z. Q.

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals,” Phys. Rev. B 92, 235310 (2015).
[Crossref]

Ziegler, K.

V. Kadirko, K. Ziegler, and E. Kogan, “Next-nearest-neighbor tight-binding model of plasmons in graphene,” Graphene 2, 97–101 (2013).
[Crossref]

Zyablovsky, A. A.

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, and A. A. Pukhov, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[Crossref]

Eur. Phys. J. D (1)

W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17, 149–151 (2001).
[Crossref]

Graphene (1)

V. Kadirko, K. Ziegler, and E. Kogan, “Next-nearest-neighbor tight-binding model of plasmons in graphene,” Graphene 2, 97–101 (2013).
[Crossref]

Int. J. Circuit Theory Appl. (1)

G. Csaba, À. Csurgay, and W. Porod, “Computing architecture composed of next-neighbor-coupled optically pumped nanodevices,” Int. J. Circuit Theory Appl. 29, 73–91 (2001).
[Crossref]

J. Mod. Opt. (1)

M. V. Berry, “Mode degeneracies and the Petermann excess-noise factor for unstable lasers,” J. Mod. Opt. 50, 63–81 (2003).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. A (4)

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]

G. Demange and E.-M. Graefe, “Signatures of three coalescing eigenfunctions,” J. Phys. A 45, 025303 (2012).
[Crossref]

R. Gutöhrlein, J. Main, H. Cartarius, and G. Wunner, “Bifurcations and exceptional points in dipolar Bose–Einstein condensates,” J. Phys. A 46, 305001 (2013).
[Crossref]

E. M. Graefe, U. Günther, H. J. Korsch, and A. E. Niederle, “A non-Hermitian PT-symmetric Bose–Hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A 41, 255206 (2008).
[Crossref]

J. Phys. B (1)

T. A. Zaleski and T. K. Kopeć, “Effect of next-nearest-neighbor hopping on Bose–Einstein condensation in optical lattices,” J. Phys. B 43, 085303 (2010).
[Crossref]

J. Stat. Mech. (1)

G. Santhosh, D. Kumar, and R. Ramaswamy, “Thermal transport in low-dimensional lattices with nearest- and next-nearest-neighbor coupling,” J. Stat. Mech. 7, P07005 (2005).

Nat. Commun. (1)

R. Keil, J. M. Zeuner, F. Dreisow, M. Heinrich, A. Tünnermann, S. Nolte, and A. Szameit, “The random mass Dirac model and long-range correlations on an integrated optical platform,” Nat. Commun. 4, 1368 (2013).
[Crossref]

Nature (1)

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Lett. A (2)

K. Fang, G. W. Fernando, and A. N. Kocharian, “Pairing enhancement in Betts lattices with next nearest neighbor couplings: exact results,” Phys. Lett. A 376, 538–543 (2012).
[Crossref]

L. J. Kijewski and M. P. Kawatra, “One-dimensional Ising model with long-range interaction,” Phys. Lett. A 31, 479–480 (1970).
[Crossref]

Phys. Lett. B (1)

M. Bernaschi, L. Biferale, L. A. Fernandez, U. M. B. Marconi, R. Petronzio, and A. Tarancon, “Renormalization group study of the three state three dimensional Potts model,” Phys. Lett. B 231, 157–160 (1989).
[Crossref]

Phys. Rev. A (6)

J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016).
[Crossref]

H. Cartarius, J. Main, and G. Wunner, “Exceptional points in the spectra of atoms in external fields,” Phys. Rev. A 79, 053408 (2009).
[Crossref]

H. Menke, M. Klett, H. Cartarius, J. Main, and G. Wunner, “State flip at exceptional points in atomic spectra,” Phys. Rev. A 93, 013401 (2016).
[Crossref]

S.-J. Gu, H. Li, Y.-Q. Li, and H.-Q. Lin, “Entanglement of the Heisenberg chain with the next-nearest-neighbor interaction,” Phys. Rev. A 70, 052302 (2004).
[Crossref]

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[Crossref]

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, and A. A. Pukhov, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[Crossref]

Phys. Rev. B (1)

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals,” Phys. Rev. B 92, 235310 (2015).
[Crossref]

Phys. Rev. E (2)

S. Bittner, B. Dietz, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “Scattering experiments with microwave billiards at an exceptional point under broken time-reversal invariance,” Phys. Rev. E 89, 032909 (2014).
[Crossref]

W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000).
[Crossref]

Phys. Rev. Lett. (4)

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
[Crossref]

R. Lefebvre, O. Atabek, M. Šindelka, and N. Moiseyev, “Resonance coalescence in molecular photodissociation,” Phys. Rev. Lett. 103, 123003 (2009).
[Crossref]

C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86, 787–790 (2001).
[Crossref]

C. Dembowski, B. Dietz, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Observation of a chiral state in a microwave cavity,” Phys. Rev. Lett. 90, 034101 (2003).
[Crossref]

Sci. Rep. (2)

S. N. Ghosh and Y. D. Chong, “Hidden singularities and asymmetric mode conversion in quasi-guided dual-mode optical waveguides,” Sci. Rep. 6, 19837 (2016).
[Crossref]

G. Chen, R. Zhang, and J. Sun, “On-chip optical mode conversion based on dynamic grating in photonic-phononic hybrid waveguide,” Sci. Rep. 5, 10346 (2015).
[Crossref]

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Figures (7)

Fig. 1.
Fig. 1.

Schematic diagram of the proposed coupling scheme between the resonances appeared in the complex energy plane. The resonances labeled by green colors interact, whereas the resonances labeled by red colors remain isolated.

Fig. 2.
Fig. 2.

Level repulsion phenomena with variation of λ showing crossing/anticrossing of real and imaginary parts of the energy eigenvalues E 1 and E 3 , keeping E 2 as an isolated state. For λ I = 0.1 (top panel), R ( E ) of two interacting levels anticross and I ( E ) cross, whereas for λ I = 0.2 (bottom panel), vice versa. The parameters are chosen as ϵ 1 = 1 , ϵ 2 = 1.5 , ϵ 3 = 2 , ω 1 = 1 , ω 2 = 0.2 , ω 3 = 1 , and ξ = 0.2 .

Fig. 3.
Fig. 3.

(a)  3 D schematic diagram of the designed Fabry–Perot-type optical microcavity with nonuniform background refractive index. (b)  2 D cross-sectional view of the same microcavity occupying the region 0 x L with L = 10    μm . Here L G = 3    μm and L R = 7    μm . The real background refractive indices are n R 1 = 1.5 and n R 2 = 4.5 . The eigenstates ψ L + and ψ R indicate the incident waves with complex amplitudes A and D , whereas the eigenstates ψ L and ψ R + indicate the scattered waves with complex amplitudes B and C , respectively. (c) Schematic nonlinear distribution of S -matrix poles in the complex k -plane of the microcavity under the operating condition. The poles indicated by green circles represent the pair of interacted states, whereas the poles indicated by red circles represent the isolated states.

Fig. 4.
Fig. 4.

Trajectories of three chosen poles (dotted blue, red, and black lines) exhibiting A R C s [clearly shown in the upper panels for both (a) and (b)] followed by the partial pumping in terms of unbalanced spatial gain–loss distribution in the cavity. In the passive cavity two green circles indicate the position of two poles which are interacting and the red circle denotes the position of an intermediate isolated pole. The loss-to-gain ratio is set at τ = 5.32 in (a) and τ = 5.33 in (b), respectively. The crossing/anticrossing behaviors of R ( k ) and I ( k ) are separately depicted as a function of γ for both the τ values in the lower panels. The red crosses in the respective top right panels represent the approximate position of the branch point singularity.

Fig. 5.
Fig. 5.

Trajectories of the three poles in the complex k -plane (initial positions are marked by the brown circles) associated with the identified singularity denoted by the red cross in the ( γ , τ ) -plane in the inset for an encircling process (blue circular path in the inset) centering it with a = 0.04 a.u. In the k -plane, the dotted red and blue lines represent the trajectories of the coupled poles, whereas the dotted black line represents the trajectories of the intermediate isolated poles after one round encirclement around the singularity in the ( γ , τ ) -plane. The dynamics of coupled poles are depicted with respect to the I ( k ) axis where the ticks labels are shown in the right sides, while the left side distribution in ticks labels correspond to the I ( k ) axis to depict the dynamics of the isolated pole. Such two different distributions in ticks labels corresponding to the I ( k ) axis are considered for clear visibility. A zoomed view around the initial position of the intermediate pole is also shown for clear visibility in loop formation.

Fig. 6.
Fig. 6.

Approximate locations of three embedded second-order EPs denoted by three red crosses in the ( γ , τ )-plane. At suffixes the numbers are used only to distinguish them. The blue dashed line with negative tangent represents the exceptional line, which connects the discrete locations of the identified EPs.

Fig. 7.
Fig. 7.

Trajectories of the coupled poles (initial positions are marked as the brown circles) corresponding to all three consecutive EPs in the complex k -plane (denoted as three red crosses in the inset) for a common encircling process in the ( γ , τ ) -plane around the center at ( 0.056 , 5.325 ) (marked as the blue dot in the inset) with a = 0.075    a.u. in the presence of modest random fluctuations on the enclosing contour (described as blue fluctuated circular path in the inset). Here the described parametric path encloses both E P 1 and E P 2 , except E P 3 .

Tables (1)

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Table 1. Complex Resonances of the Microcavity for Both Passive and Initial Pumped Conditions a

Equations (16)

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H = p i , j i = j C 2 i 1,2 j + 1 Λ 2 i 1 Λ 2 j + 1 + q k , l k = l C 2 k 1,2 l + 1 Λ 2 k 1 Λ 2 l + 1 + r n h n Λ n .
H = p i , j i = j C 2 i 1,2 j + 1 Λ 2 i 1 Λ 2 j + 1 ; i , j = 1 , 3 , 5
H = ( ϵ 1 0 0 0 ϵ 2 0 0 0 ϵ 3 ) + λ U ( ω 1 0 0 0 ω 2 0 0 0 ω 3 ) U ,
U ( ξ ) = ( cos ξ 0 sin ξ 0 1 0 sin ξ 0 cos ξ ) .
E 1,3 ( λ ) = ϵ 1 + ϵ 3 + λ ( ω 1 + ω 3 ) 2 ± C ,
E 2 ( λ ) = ϵ 2 + λ ω 2 ,
C = [ ( ϵ 1 ϵ 3 2 ) 2 + ( λ ( ω 1 ω 3 ) 2 ) 2 + λ 2 ( ϵ 1 ϵ 3 ) ( ω 1 ω 3 ) cos ( 2 ξ ) ] 1 / 2 .
λ E P = ϵ 1 ϵ 3 ω 1 ω 3 exp ( ± 2 i ξ ) .
E 1,3 ( λ ) = E E P ± c 1 λ λ E P .
n R ( x ) = { n R 1 , 0 x L G n R 2 , L G x L R n R 1 , L R x L .
n G = n R 1 i γ , 0 x L G ,
n L = n R 1 + i τ γ , L R x L .
( B C ) = S ( n ( x ) , ω ) ( A D ) .
1 max [ eig S ( ω ) ] = 0 .
γ ( ϕ ) = γ 0 [ 1 + a cos ( ϕ ) ] ,
τ ( ϕ ) = τ 0 [ 1 + a sin ( ϕ ) ] ,

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