Abstract

We reveal a novel topological property of the exceptional points in a two-level parity-time symmetric system and then propose a scheme to detect the topological exceptional points in the system, which is embedded in a larger Hilbert space constructed by a four-level cold atomic system. We show that a tunable parameter in the presented system for simulating the non-Hermitian Hamiltonian can be tuned to sweep the eigenstates through the exceptional points in parameter space. The non-trivial Berry phases of the eigenstates obtained in this loop from the exceptional points can be measured by the atomic interferometry. Since the proposed operations and detection are experimentally feasible, our scheme may pave a promising way to explore the novel properties of non-Hermitian systems.

© 2017 Optical Society of America

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References

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  1. C. M. Bender and S. Boettcher, “Real spectra of non-hermitian Hamiltonian having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
    [Crossref]
  2. C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
    [Crossref]
  3. T. Kottos, “Broken symmetry makes light work,” Nat. Phys. 6, 166–167 (2010).
    [Crossref]
  4. J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84, 040101 (2011).
    [Crossref]
  5. Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
    [Crossref]
  6. C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [Crossref]
  7. X. Zhou and Y. D. Chong, “PT symmetry breaking and nonlinear optical isolation in coupled microcavities,” Opt. Express 24(7) 6916–6930 (2016).
    [Crossref] [PubMed]
  8. B. Wu, J.-M. Wang, M. Xiao, J. Xu And, and Y.-T. Chen, “Strong hybridization of edge and bulk states in dimerized PT-symmetric coupled waveguide chain,” Opt. Express 25(2) 1040–1049 (2017).
    [Crossref] [PubMed]
  9. A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “PT-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84, 021806 (2011).
    [Crossref]
  10. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature (London) 488, 167–171 (2012).
    [Crossref]
  11. S. Bittner and B. Dietz, U. Gü nther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schä fer, “PT symmetry and spontaneous symmetry breaking in a microwave billiard,” Phys. Rev. Lett. 108, 024101 (2012).
    [Crossref]
  12. Y. Sun, W. Tan, H.-Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
    [Crossref] [PubMed]
  13. L. Chang, X.-S. Jiang, S.-Y. Hua, C. Yang, J.-M. Wen, L. Jiang, G.-Y. Li, G.-Z. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
    [Crossref]
  14. B. Peng, S. K. Özdemir, F.-C. Lei, F. Monifi, M. Gianfreda, G.-L. Long, S.-H. Fan, F. Nori, Carl M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Physics 10, 394–398 (2014).
    [Crossref]
  15. L. Feng, Z.-J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
    [Crossref] [PubMed]
  16. H. Hodaei, M. Miri, M Heinrich, Demetrios N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
    [Crossref] [PubMed]
  17. C. Hang, G.-X. Huang, and Vladimir V. Konotop, “PT Symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
    [Crossref] [PubMed]
  18. J.-H. Wu, M. Artoni, and G.C. La Rocca, “Non-hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
    [Crossref] [PubMed]
  19. C. Hang and G.-X. Huang, “Weak-light solitons and their active control in a parity-time-symmetric atomic system,” Phys. Rev. A 91, 043833 (2015).
    [Crossref]
  20. Diana A. Antonosyan, Alexander S. Solntsev, and Andrey A. Sukhorukov, “Parity-tme anti-symmetric parametric amplifier,” Opt. Letters 40(20) 4575–4578 (2015).
    [Crossref]
  21. P. Peng, W.-x Cao, C. Shen, W.-z Qu, J.-m Wen, L. Jiang, and Y.-h Xiao, “Anti-parity-time symmetric optics via flying atoms,” Nat. Phys. 12, 1139–1145 (2016).
    [Crossref]
  22. T. Kato, Perturbation theory of linear operators (Springer, 1966).
  23. H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-hermitian physics,” Rev. Mod. Phys. 87, 61 (2015).
    [Crossref]
  24. W.D. Heiss and H.L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17, 149–151 (2001).
    [Crossref]
  25. 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] [PubMed]
  26. Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
    [Crossref] [PubMed]
  27. W. D. Heiss, “Phases of wave functions and level repulsion,” Eur. Phys. J. D 7, 1–4 (1999).
    [Crossref]
  28. W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929 (2000).
    [Crossref]
  29. 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 (2001).
    [Crossref] [PubMed]
  30. C. Dembowski, B. Dietz, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
    [Crossref]
  31. J. Doppler, A. A. Mailybaev, J. Bohm, 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] [PubMed]
  32. H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
    [Crossref] [PubMed]
  33. I. Rotter, “A non-hermitian hamilton operator and the physics of open quantum systems,” J. Phys. A 42, 153001 (2009).
    [Crossref]
  34. H. Eleuch and I. Rotter, “Quantum states talk via the environment,” https://arXiv:1511.01751 (2015).
  35. B. Wahlstrand, I. I. Yakimenko, and K.-F. Berggren, “Wave transport and statistical properties of an open non-hermitian quantum dot with parity-time symmetry,” Phy. Rev. E 89, 062910 (2014).
    [Crossref]
  36. Uwe Günther and Boris F. Samsonov, “Naimark-dilated PT-symmetric brachistochrone,” Phys. Rev. Lett. 101, 230404 (2008).
    [Crossref] [PubMed]
  37. X. Zhang, Yangchao Shen, J.-h. Zhang, J. Casanova, L. Lamata, E. Solano, M.-H. Yung, J.-N. Zhang, and K. Kim, “Time reversal and charge conjugation in an embedding quantum simulator,” Nat. Commun. 6, 7917 (2015).
    [Crossref] [PubMed]
  38. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).
  39. S. W. Kim, T. Cheon, and A. Tanaka, “Exotic quantum holonomy induced by degeneracy hidden in complex parameter space,” Phys. Lett. A 374, 1958–1961 (2010).
    [Crossref]
  40. Y.-X. Du, Z.-T. Liang, W. Huang, and S.-L. Zhu, “Experimental observation of double coherent stimulated raman adiabatic passages in three-level Λ systems in a cold atomic ensemble,” Phys. Rev. A 90, 023821 (2014).
    [Crossref]
  41. P. Kral, I. Thanopulos, and M. Shapiro, “Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79, 53 (2007).
    [Crossref]
  42. Y.-X. Du, Z.-T. Liang, Y.-C. Li, X.-X. Yue, Q.-X. Lv, W. Huang, X. Chen, H. Yan, and S.-L. Zhu, “Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms,” Nat. Commun. 7, 12479 (2016).
    [Crossref] [PubMed]
  43. V. A. Andreev, A. B. Klimov, and P. B. Lerner, “Berry phases in the atomic interferometer,” Euro. Phys. Lett. 12, 2 (1990).
    [Crossref]
  44. C. L. Webb, R. M. Godun, G. S. Summy, M. K. Oberthaler, P. D. Featonby, C. J. Foot, and K. Burnett, “Measurement of Berry’s phase using an atom interferometer,” Phys. Rev. A 60, 1783 (1999).
    [Crossref]

2017 (1)

2016 (5)

X. Zhou and Y. D. Chong, “PT symmetry breaking and nonlinear optical isolation in coupled microcavities,” Opt. Express 24(7) 6916–6930 (2016).
[Crossref] [PubMed]

P. Peng, W.-x Cao, C. Shen, W.-z Qu, J.-m Wen, L. Jiang, and Y.-h Xiao, “Anti-parity-time symmetric optics via flying atoms,” Nat. Phys. 12, 1139–1145 (2016).
[Crossref]

J. Doppler, A. A. Mailybaev, J. Bohm, 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] [PubMed]

H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
[Crossref] [PubMed]

Y.-X. Du, Z.-T. Liang, Y.-C. Li, X.-X. Yue, Q.-X. Lv, W. Huang, X. Chen, H. Yan, and S.-L. Zhu, “Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms,” Nat. Commun. 7, 12479 (2016).
[Crossref] [PubMed]

2015 (4)

X. Zhang, Yangchao Shen, J.-h. Zhang, J. Casanova, L. Lamata, E. Solano, M.-H. Yung, J.-N. Zhang, and K. Kim, “Time reversal and charge conjugation in an embedding quantum simulator,” Nat. Commun. 6, 7917 (2015).
[Crossref] [PubMed]

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-hermitian physics,” Rev. Mod. Phys. 87, 61 (2015).
[Crossref]

C. Hang and G.-X. Huang, “Weak-light solitons and their active control in a parity-time-symmetric atomic system,” Phys. Rev. A 91, 043833 (2015).
[Crossref]

Diana A. Antonosyan, Alexander S. Solntsev, and Andrey A. Sukhorukov, “Parity-tme anti-symmetric parametric amplifier,” Opt. Letters 40(20) 4575–4578 (2015).
[Crossref]

2014 (8)

B. Wahlstrand, I. I. Yakimenko, and K.-F. Berggren, “Wave transport and statistical properties of an open non-hermitian quantum dot with parity-time symmetry,” Phy. Rev. E 89, 062910 (2014).
[Crossref]

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

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

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

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

L. Feng, Z.-J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref] [PubMed]

H. Hodaei, M. Miri, M Heinrich, Demetrios N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Y.-X. Du, Z.-T. Liang, W. Huang, and S.-L. Zhu, “Experimental observation of double coherent stimulated raman adiabatic passages in three-level Λ systems in a cold atomic ensemble,” Phys. Rev. A 90, 023821 (2014).
[Crossref]

2013 (1)

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

2012 (3)

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

S. Bittner and B. Dietz, U. Gü nther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schä fer, “PT symmetry and spontaneous symmetry breaking in a microwave billiard,” Phys. Rev. Lett. 108, 024101 (2012).
[Crossref]

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[Crossref]

2011 (2)

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

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

2010 (4)

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

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

S. W. Kim, T. Cheon, and A. Tanaka, “Exotic quantum holonomy induced by degeneracy hidden in complex parameter space,” Phys. Lett. A 374, 1958–1961 (2010).
[Crossref]

Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

2009 (1)

I. Rotter, “A non-hermitian hamilton operator and the physics of open quantum systems,” J. Phys. A 42, 153001 (2009).
[Crossref]

2008 (1)

Uwe Günther and Boris F. Samsonov, “Naimark-dilated PT-symmetric brachistochrone,” Phys. Rev. Lett. 101, 230404 (2008).
[Crossref] [PubMed]

2007 (1)

P. Kral, I. Thanopulos, and M. Shapiro, “Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79, 53 (2007).
[Crossref]

2004 (1)

C. Dembowski, B. Dietz, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
[Crossref]

2003 (1)

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] [PubMed]

2002 (1)

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

2001 (2)

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 (2001).
[Crossref] [PubMed]

2000 (1)

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

1999 (2)

W. D. Heiss, “Phases of wave functions and level repulsion,” Eur. Phys. J. D 7, 1–4 (1999).
[Crossref]

C. L. Webb, R. M. Godun, G. S. Summy, M. K. Oberthaler, P. D. Featonby, C. J. Foot, and K. Burnett, “Measurement of Berry’s phase using an atom interferometer,” Phys. Rev. A 60, 1783 (1999).
[Crossref]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra of non-hermitian Hamiltonian having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

1990 (1)

V. A. Andreev, A. B. Klimov, and P. B. Lerner, “Berry phases in the atomic interferometer,” Euro. Phys. Lett. 12, 2 (1990).
[Crossref]

An, K.

Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

And, J. Xu

Andreev, V. A.

V. A. Andreev, A. B. Klimov, and P. B. Lerner, “Berry phases in the atomic interferometer,” Euro. Phys. Lett. 12, 2 (1990).
[Crossref]

Antonosyan, Diana A.

Diana A. Antonosyan, Alexander S. Solntsev, and Andrey A. Sukhorukov, “Parity-tme anti-symmetric parametric amplifier,” Opt. Letters 40(20) 4575–4578 (2015).
[Crossref]

Artoni, M.

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

Bahat-Treidel, O.

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

Bender, C. M.

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

C. M. Bender and S. Boettcher, “Real spectra of non-hermitian Hamiltonian having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

Bender, Carl M.

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

Berggren, K.-F.

B. Wahlstrand, I. I. Yakimenko, and K.-F. Berggren, “Wave transport and statistical properties of an open non-hermitian quantum dot with parity-time symmetry,” Phy. Rev. E 89, 062910 (2014).
[Crossref]

Bersch, C.

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

Bittner, S.

S. Bittner and B. Dietz, U. Gü nther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schä fer, “PT symmetry and spontaneous symmetry breaking in a microwave billiard,” Phys. Rev. Lett. 108, 024101 (2012).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra of non-hermitian Hamiltonian having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

Bohm, J.

J. Doppler, A. A. Mailybaev, J. Bohm, 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] [PubMed]

Brody, D. C.

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

Burnett, K.

C. L. Webb, R. M. Godun, G. S. Summy, M. K. Oberthaler, P. D. Featonby, C. J. Foot, and K. Burnett, “Measurement of Berry’s phase using an atom interferometer,” Phys. Rev. A 60, 1783 (1999).
[Crossref]

Cao, H.

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-hermitian physics,” Rev. Mod. Phys. 87, 61 (2015).
[Crossref]

Cao, W.-x

P. Peng, W.-x Cao, C. Shen, W.-z Qu, J.-m Wen, L. Jiang, and Y.-h Xiao, “Anti-parity-time symmetric optics via flying atoms,” Nat. Phys. 12, 1139–1145 (2016).
[Crossref]

Casanova, J.

X. Zhang, Yangchao Shen, J.-h. Zhang, J. Casanova, L. Lamata, E. Solano, M.-H. Yung, J.-N. Zhang, and K. Kim, “Time reversal and charge conjugation in an embedding quantum simulator,” Nat. Commun. 6, 7917 (2015).
[Crossref] [PubMed]

Chang, L.

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

Chen, H.

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

Chen, X.

Y.-X. Du, Z.-T. Liang, Y.-C. Li, X.-X. Yue, Q.-X. Lv, W. Huang, X. Chen, H. Yan, and S.-L. Zhu, “Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms,” Nat. Commun. 7, 12479 (2016).
[Crossref] [PubMed]

Chen, Y.-T.

Cheon, T.

S. W. Kim, T. Cheon, and A. Tanaka, “Exotic quantum holonomy induced by degeneracy hidden in complex parameter space,” Phys. Lett. A 374, 1958–1961 (2010).
[Crossref]

Choi, Y.

Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

Chong, Y. D.

Christodoulides, D. N.

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

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

Christodoulides, Demetrios N.

H. Hodaei, M. Miri, M Heinrich, Demetrios N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Dembowski, C.

C. Dembowski, B. Dietz, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
[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] [PubMed]

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 (2001).
[Crossref] [PubMed]

Dietz, B.

S. Bittner and B. Dietz, U. Gü nther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schä fer, “PT symmetry and spontaneous symmetry breaking in a microwave billiard,” Phys. Rev. Lett. 108, 024101 (2012).
[Crossref]

C. Dembowski, B. Dietz, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
[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] [PubMed]

Doppler, J.

J. Doppler, A. A. Mailybaev, J. Bohm, 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] [PubMed]

Du, Y.-X.

Y.-X. Du, Z.-T. Liang, Y.-C. Li, X.-X. Yue, Q.-X. Lv, W. Huang, X. Chen, H. Yan, and S.-L. Zhu, “Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms,” Nat. Commun. 7, 12479 (2016).
[Crossref] [PubMed]

Y.-X. Du, Z.-T. Liang, W. Huang, and S.-L. Zhu, “Experimental observation of double coherent stimulated raman adiabatic passages in three-level Λ systems in a cold atomic ensemble,” Phys. Rev. A 90, 023821 (2014).
[Crossref]

El-Ganainy, R.

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

Ellis, F. M.

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[Crossref]

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

Fan, S.-H.

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

Featonby, P. D.

C. L. Webb, R. M. Godun, G. S. Summy, M. K. Oberthaler, P. D. Featonby, C. J. Foot, and K. Burnett, “Measurement of Berry’s phase using an atom interferometer,” Phys. Rev. A 60, 1783 (1999).
[Crossref]

Feng, L.

L. Feng, Z.-J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref] [PubMed]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

Foot, C. J.

C. L. Webb, R. M. Godun, G. S. Summy, M. K. Oberthaler, P. D. Featonby, C. J. Foot, and K. Burnett, “Measurement of Berry’s phase using an atom interferometer,” Phys. Rev. A 60, 1783 (1999).
[Crossref]

Gianfreda, M.

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

Girschik, A.

J. Doppler, A. A. Mailybaev, J. Bohm, 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] [PubMed]

Godun, R. M.

C. L. Webb, R. M. Godun, G. S. Summy, M. K. Oberthaler, P. D. Featonby, C. J. Foot, and K. Burnett, “Measurement of Berry’s phase using an atom interferometer,” Phys. Rev. A 60, 1783 (1999).
[Crossref]

Gräf, H.-D.

C. Dembowski, B. Dietz, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
[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] [PubMed]

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 (2001).
[Crossref] [PubMed]

Günther, Uwe

Uwe Günther and Boris F. Samsonov, “Naimark-dilated PT-symmetric brachistochrone,” Phys. Rev. Lett. 101, 230404 (2008).
[Crossref] [PubMed]

Hang, C.

C. Hang and G.-X. Huang, “Weak-light solitons and their active control in a parity-time-symmetric atomic system,” Phys. Rev. A 91, 043833 (2015).
[Crossref]

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

Harney, H. L.

C. Dembowski, B. Dietz, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
[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] [PubMed]

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 (2001).
[Crossref] [PubMed]

Harney, H.L.

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

Harris, J. G. E.

H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
[Crossref] [PubMed]

Heine, A.

C. Dembowski, B. Dietz, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
[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] [PubMed]

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 (2001).
[Crossref] [PubMed]

Heinrich, M

H. Hodaei, M. Miri, M Heinrich, Demetrios N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Heiss, W. D.

C. Dembowski, B. Dietz, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
[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] [PubMed]

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 (2001).
[Crossref] [PubMed]

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

W. D. Heiss, “Phases of wave functions and level repulsion,” Eur. Phys. J. D 7, 1–4 (1999).
[Crossref]

Heiss, W.D.

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

Hodaei, H.

H. Hodaei, M. Miri, M Heinrich, Demetrios N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Hua, S.-Y.

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

Huang, G.-X.

C. Hang and G.-X. Huang, “Weak-light solitons and their active control in a parity-time-symmetric atomic system,” Phys. Rev. A 91, 043833 (2015).
[Crossref]

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

Huang, W.

Y.-X. Du, Z.-T. Liang, Y.-C. Li, X.-X. Yue, Q.-X. Lv, W. Huang, X. Chen, H. Yan, and S.-L. Zhu, “Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms,” Nat. Commun. 7, 12479 (2016).
[Crossref] [PubMed]

Y.-X. Du, Z.-T. Liang, W. Huang, and S.-L. Zhu, “Experimental observation of double coherent stimulated raman adiabatic passages in three-level Λ systems in a cold atomic ensemble,” Phys. Rev. A 90, 023821 (2014).
[Crossref]

Jiang, L.

H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
[Crossref] [PubMed]

P. Peng, W.-x Cao, C. Shen, W.-z Qu, J.-m Wen, L. Jiang, and Y.-h Xiao, “Anti-parity-time symmetric optics via flying atoms,” Nat. Phys. 12, 1139–1145 (2016).
[Crossref]

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

Jiang, X.-S.

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

Jones, H. F.

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

Kang, S.

Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

Kato, T.

T. Kato, Perturbation theory of linear operators (Springer, 1966).

Khajavikhan, M.

H. Hodaei, M. Miri, M Heinrich, Demetrios N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Kim, J.-R.

Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

Kim, K.

X. Zhang, Yangchao Shen, J.-h. Zhang, J. Casanova, L. Lamata, E. Solano, M.-H. Yung, J.-N. Zhang, and K. Kim, “Time reversal and charge conjugation in an embedding quantum simulator,” Nat. Commun. 6, 7917 (2015).
[Crossref] [PubMed]

Kim, S. W.

S. W. Kim, T. Cheon, and A. Tanaka, “Exotic quantum holonomy induced by degeneracy hidden in complex parameter space,” Phys. Lett. A 374, 1958–1961 (2010).
[Crossref]

Kim, W.

Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

Kip, D.

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

Klimov, A. B.

V. A. Andreev, A. B. Klimov, and P. B. Lerner, “Berry phases in the atomic interferometer,” Euro. Phys. Lett. 12, 2 (1990).
[Crossref]

Konotop, Vladimir V.

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

Kottos, T.

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[Crossref]

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

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

Kral, P.

P. Kral, I. Thanopulos, and M. Shapiro, “Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79, 53 (2007).
[Crossref]

Kuhl, U.

J. Doppler, A. A. Mailybaev, J. Bohm, 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] [PubMed]

La Rocca, G.C.

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

Lamata, L.

X. Zhang, Yangchao Shen, J.-h. Zhang, J. Casanova, L. Lamata, E. Solano, M.-H. Yung, J.-N. Zhang, and K. Kim, “Time reversal and charge conjugation in an embedding quantum simulator,” Nat. Commun. 6, 7917 (2015).
[Crossref] [PubMed]

Lee, J.-H.

Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

Lei, F.-C.

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

Lerner, P. B.

V. A. Andreev, A. B. Klimov, and P. B. Lerner, “Berry phases in the atomic interferometer,” Euro. Phys. Lett. 12, 2 (1990).
[Crossref]

Li, A.

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

Li, G.-Y.

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

Li, H.-Q.

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

Li, J.

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

Li, Y.-C.

Y.-X. Du, Z.-T. Liang, Y.-C. Li, X.-X. Yue, Q.-X. Lv, W. Huang, X. Chen, H. Yan, and S.-L. Zhu, “Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms,” Nat. Commun. 7, 12479 (2016).
[Crossref] [PubMed]

Liang, Z.-T.

Y.-X. Du, Z.-T. Liang, Y.-C. Li, X.-X. Yue, Q.-X. Lv, W. Huang, X. Chen, H. Yan, and S.-L. Zhu, “Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms,” Nat. Commun. 7, 12479 (2016).
[Crossref] [PubMed]

Y.-X. Du, Z.-T. Liang, W. Huang, and S.-L. Zhu, “Experimental observation of double coherent stimulated raman adiabatic passages in three-level Λ systems in a cold atomic ensemble,” Phys. Rev. A 90, 023821 (2014).
[Crossref]

Libisch, F.

J. Doppler, A. A. Mailybaev, J. Bohm, 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] [PubMed]

Lim, S.

Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

Lin, Z.

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[Crossref]

Long, G.-L.

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

Lv, Q.-X.

Y.-X. Du, Z.-T. Liang, Y.-C. Li, X.-X. Yue, Q.-X. Lv, W. Huang, X. Chen, H. Yan, and S.-L. Zhu, “Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms,” Nat. Commun. 7, 12479 (2016).
[Crossref] [PubMed]

Ma, R.-M.

L. Feng, Z.-J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref] [PubMed]

Mailybaev, A. A.

J. Doppler, A. A. Mailybaev, J. Bohm, 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] [PubMed]

Makris, K. G.

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

Mason, D.

H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
[Crossref] [PubMed]

Milburn, T. J.

J. Doppler, A. A. Mailybaev, J. Bohm, 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] [PubMed]

Miri, M.

H. Hodaei, M. Miri, M Heinrich, Demetrios N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Miri, M.-A.

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

Moiseyev, N.

J. Doppler, A. A. Mailybaev, J. Bohm, 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] [PubMed]

Monifi, F.

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

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

Nori, F.

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

Oberthaler, M. K.

C. L. Webb, R. M. Godun, G. S. Summy, M. K. Oberthaler, P. D. Featonby, C. J. Foot, and K. Burnett, “Measurement of Berry’s phase using an atom interferometer,” Phys. Rev. A 60, 1783 (1999).
[Crossref]

Onishchukov, G.

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

Özdemir, S. K.

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

Peng, B.

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

Peng, P.

P. Peng, W.-x Cao, C. Shen, W.-z Qu, J.-m Wen, L. Jiang, and Y.-h Xiao, “Anti-parity-time symmetric optics via flying atoms,” Nat. Phys. 12, 1139–1145 (2016).
[Crossref]

Peschel, U.

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

Qu, W.-z

P. Peng, W.-x Cao, C. Shen, W.-z Qu, J.-m Wen, L. Jiang, and Y.-h Xiao, “Anti-parity-time symmetric optics via flying atoms,” Nat. Phys. 12, 1139–1145 (2016).
[Crossref]

Rabl, P.

J. Doppler, A. A. Mailybaev, J. Bohm, 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] [PubMed]

Rechtsman, M. C.

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

Regensburger, A.

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

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 (2001).
[Crossref] [PubMed]

Richter, A.

C. Dembowski, B. Dietz, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
[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] [PubMed]

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 (2001).
[Crossref] [PubMed]

Rotter, I.

I. Rotter, “A non-hermitian hamilton operator and the physics of open quantum systems,” J. Phys. A 42, 153001 (2009).
[Crossref]

Rotter, S.

J. Doppler, A. A. Mailybaev, J. Bohm, 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] [PubMed]

Ruter, C. E.

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

Samsonov, Boris F.

Uwe Günther and Boris F. Samsonov, “Naimark-dilated PT-symmetric brachistochrone,” Phys. Rev. Lett. 101, 230404 (2008).
[Crossref] [PubMed]

Schindler, J.

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[Crossref]

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

Segev, M.

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

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

Shapiro, M.

P. Kral, I. Thanopulos, and M. Shapiro, “Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79, 53 (2007).
[Crossref]

Shen, C.

P. Peng, W.-x Cao, C. Shen, W.-z Qu, J.-m Wen, L. Jiang, and Y.-h Xiao, “Anti-parity-time symmetric optics via flying atoms,” Nat. Phys. 12, 1139–1145 (2016).
[Crossref]

Shen, Yangchao

X. Zhang, Yangchao Shen, J.-h. Zhang, J. Casanova, L. Lamata, E. Solano, M.-H. Yung, J.-N. Zhang, and K. Kim, “Time reversal and charge conjugation in an embedding quantum simulator,” Nat. Commun. 6, 7917 (2015).
[Crossref] [PubMed]

Solano, E.

X. Zhang, Yangchao Shen, J.-h. Zhang, J. Casanova, L. Lamata, E. Solano, M.-H. Yung, J.-N. Zhang, and K. Kim, “Time reversal and charge conjugation in an embedding quantum simulator,” Nat. Commun. 6, 7917 (2015).
[Crossref] [PubMed]

Solntsev, Alexander S.

Diana A. Antonosyan, Alexander S. Solntsev, and Andrey A. Sukhorukov, “Parity-tme anti-symmetric parametric amplifier,” Opt. Letters 40(20) 4575–4578 (2015).
[Crossref]

Sukhorukov, Andrey A.

Diana A. Antonosyan, Alexander S. Solntsev, and Andrey A. Sukhorukov, “Parity-tme anti-symmetric parametric amplifier,” Opt. Letters 40(20) 4575–4578 (2015).
[Crossref]

Summy, G. S.

C. L. Webb, R. M. Godun, G. S. Summy, M. K. Oberthaler, P. D. Featonby, C. J. Foot, and K. Burnett, “Measurement of Berry’s phase using an atom interferometer,” Phys. Rev. A 60, 1783 (1999).
[Crossref]

Sun, Y.

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

Szameit, A.

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

Tan, W.

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

Tanaka, A.

S. W. Kim, T. Cheon, and A. Tanaka, “Exotic quantum holonomy induced by degeneracy hidden in complex parameter space,” Phys. Lett. A 374, 1958–1961 (2010).
[Crossref]

Thanopulos, I.

P. Kral, I. Thanopulos, and M. Shapiro, “Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79, 53 (2007).
[Crossref]

Wahlstrand, B.

B. Wahlstrand, I. I. Yakimenko, and K.-F. Berggren, “Wave transport and statistical properties of an open non-hermitian quantum dot with parity-time symmetry,” Phy. Rev. E 89, 062910 (2014).
[Crossref]

Wang, G.-Z.

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

Wang, J.-M.

Wang, Y.

L. Feng, Z.-J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref] [PubMed]

Webb, C. L.

C. L. Webb, R. M. Godun, G. S. Summy, M. K. Oberthaler, P. D. Featonby, C. J. Foot, and K. Burnett, “Measurement of Berry’s phase using an atom interferometer,” Phys. Rev. A 60, 1783 (1999).
[Crossref]

Wen, J.-m

P. Peng, W.-x Cao, C. Shen, W.-z Qu, J.-m Wen, L. Jiang, and Y.-h Xiao, “Anti-parity-time symmetric optics via flying atoms,” Nat. Phys. 12, 1139–1145 (2016).
[Crossref]

Wen, J.-M.

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

Wiersig, J.

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-hermitian physics,” Rev. Mod. Phys. 87, 61 (2015).
[Crossref]

Wong, Z.-J.

L. Feng, Z.-J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref] [PubMed]

Wu, B.

Wu, J.-H.

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

Xiao, M.

B. Wu, J.-M. Wang, M. Xiao, J. Xu And, and Y.-T. Chen, “Strong hybridization of edge and bulk states in dimerized PT-symmetric coupled waveguide chain,” Opt. Express 25(2) 1040–1049 (2017).
[Crossref] [PubMed]

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

Xiao, Y.-h

P. Peng, W.-x Cao, C. Shen, W.-z Qu, J.-m Wen, L. Jiang, and Y.-h Xiao, “Anti-parity-time symmetric optics via flying atoms,” Nat. Phys. 12, 1139–1145 (2016).
[Crossref]

Xu, H.

H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
[Crossref] [PubMed]

Yakimenko, I. I.

B. Wahlstrand, I. I. Yakimenko, and K.-F. Berggren, “Wave transport and statistical properties of an open non-hermitian quantum dot with parity-time symmetry,” Phy. Rev. E 89, 062910 (2014).
[Crossref]

Yan, H.

Y.-X. Du, Z.-T. Liang, Y.-C. Li, X.-X. Yue, Q.-X. Lv, W. Huang, X. Chen, H. Yan, and S.-L. Zhu, “Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms,” Nat. Commun. 7, 12479 (2016).
[Crossref] [PubMed]

Yang, C.

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

Yang, L.

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

Yue, X.-X.

Y.-X. Du, Z.-T. Liang, Y.-C. Li, X.-X. Yue, Q.-X. Lv, W. Huang, X. Chen, H. Yan, and S.-L. Zhu, “Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms,” Nat. Commun. 7, 12479 (2016).
[Crossref] [PubMed]

Yung, M.-H.

X. Zhang, Yangchao Shen, J.-h. Zhang, J. Casanova, L. Lamata, E. Solano, M.-H. Yung, J.-N. Zhang, and K. Kim, “Time reversal and charge conjugation in an embedding quantum simulator,” Nat. Commun. 6, 7917 (2015).
[Crossref] [PubMed]

Zhang, J.-h.

X. Zhang, Yangchao Shen, J.-h. Zhang, J. Casanova, L. Lamata, E. Solano, M.-H. Yung, J.-N. Zhang, and K. Kim, “Time reversal and charge conjugation in an embedding quantum simulator,” Nat. Commun. 6, 7917 (2015).
[Crossref] [PubMed]

Zhang, J.-N.

X. Zhang, Yangchao Shen, J.-h. Zhang, J. Casanova, L. Lamata, E. Solano, M.-H. Yung, J.-N. Zhang, and K. Kim, “Time reversal and charge conjugation in an embedding quantum simulator,” Nat. Commun. 6, 7917 (2015).
[Crossref] [PubMed]

Zhang, X.

X. Zhang, Yangchao Shen, J.-h. Zhang, J. Casanova, L. Lamata, E. Solano, M.-H. Yung, J.-N. Zhang, and K. Kim, “Time reversal and charge conjugation in an embedding quantum simulator,” Nat. Commun. 6, 7917 (2015).
[Crossref] [PubMed]

L. Feng, Z.-J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref] [PubMed]

Zheng, M. C.

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

Zhou, X.

Zhu, S.-L.

Y.-X. Du, Z.-T. Liang, Y.-C. Li, X.-X. Yue, Q.-X. Lv, W. Huang, X. Chen, H. Yan, and S.-L. Zhu, “Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms,” Nat. Commun. 7, 12479 (2016).
[Crossref] [PubMed]

Y.-X. Du, Z.-T. Liang, W. Huang, and S.-L. Zhu, “Experimental observation of double coherent stimulated raman adiabatic passages in three-level Λ systems in a cold atomic ensemble,” Phys. Rev. A 90, 023821 (2014).
[Crossref]

Eur. Phys. J. D (2)

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

W. D. Heiss, “Phases of wave functions and level repulsion,” Eur. Phys. J. D 7, 1–4 (1999).
[Crossref]

Euro. Phys. Lett. (1)

V. A. Andreev, A. B. Klimov, and P. B. Lerner, “Berry phases in the atomic interferometer,” Euro. Phys. Lett. 12, 2 (1990).
[Crossref]

J. Phys. A (1)

I. Rotter, “A non-hermitian hamilton operator and the physics of open quantum systems,” J. Phys. A 42, 153001 (2009).
[Crossref]

Nat. Commun. (2)

Y.-X. Du, Z.-T. Liang, Y.-C. Li, X.-X. Yue, Q.-X. Lv, W. Huang, X. Chen, H. Yan, and S.-L. Zhu, “Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms,” Nat. Commun. 7, 12479 (2016).
[Crossref] [PubMed]

X. Zhang, Yangchao Shen, J.-h. Zhang, J. Casanova, L. Lamata, E. Solano, M.-H. Yung, J.-N. Zhang, and K. Kim, “Time reversal and charge conjugation in an embedding quantum simulator,” Nat. Commun. 6, 7917 (2015).
[Crossref] [PubMed]

Nat. Photonics (1)

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

Nat. Phys. (3)

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

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

P. Peng, W.-x Cao, C. Shen, W.-z Qu, J.-m Wen, L. Jiang, and Y.-h Xiao, “Anti-parity-time symmetric optics via flying atoms,” Nat. Phys. 12, 1139–1145 (2016).
[Crossref]

Nat. Physics (1)

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

Nature (2)

J. Doppler, A. A. Mailybaev, J. Bohm, 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] [PubMed]

H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
[Crossref] [PubMed]

Nature (London) (1)

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

Opt. Express (2)

Opt. Letters (1)

Diana A. Antonosyan, Alexander S. Solntsev, and Andrey A. Sukhorukov, “Parity-tme anti-symmetric parametric amplifier,” Opt. Letters 40(20) 4575–4578 (2015).
[Crossref]

Phy. Rev. E (1)

B. Wahlstrand, I. I. Yakimenko, and K.-F. Berggren, “Wave transport and statistical properties of an open non-hermitian quantum dot with parity-time symmetry,” Phy. Rev. E 89, 062910 (2014).
[Crossref]

Phys. Lett. A (1)

S. W. Kim, T. Cheon, and A. Tanaka, “Exotic quantum holonomy induced by degeneracy hidden in complex parameter space,” Phys. Lett. A 374, 1958–1961 (2010).
[Crossref]

Phys. Rev. A (6)

Y.-X. Du, Z.-T. Liang, W. Huang, and S.-L. Zhu, “Experimental observation of double coherent stimulated raman adiabatic passages in three-level Λ systems in a cold atomic ensemble,” Phys. Rev. A 90, 023821 (2014).
[Crossref]

C. L. Webb, R. M. Godun, G. S. Summy, M. K. Oberthaler, P. D. Featonby, C. J. Foot, and K. Burnett, “Measurement of Berry’s phase using an atom interferometer,” Phys. Rev. A 60, 1783 (1999).
[Crossref]

C. Hang and G.-X. Huang, “Weak-light solitons and their active control in a parity-time-symmetric atomic system,” Phys. Rev. A 91, 043833 (2015).
[Crossref]

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

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

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[Crossref]

Phys. Rev. E (2)

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

C. Dembowski, B. Dietz, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Encircling an exceptional point,” Phys. Rev. E 69, 056216 (2004).
[Crossref]

Phys. Rev. Lett. (10)

Uwe Günther and Boris F. Samsonov, “Naimark-dilated PT-symmetric brachistochrone,” Phys. Rev. Lett. 101, 230404 (2008).
[Crossref] [PubMed]

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 (2001).
[Crossref] [PubMed]

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] [PubMed]

Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104, 153601 (2010).
[Crossref] [PubMed]

C. M. Bender and S. Boettcher, “Real spectra of non-hermitian Hamiltonian having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

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

S. Bittner and B. Dietz, U. Gü nther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schä fer, “PT symmetry and spontaneous symmetry breaking in a microwave billiard,” Phys. Rev. Lett. 108, 024101 (2012).
[Crossref]

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

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

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

Rev. Mod. Phys. (2)

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-hermitian physics,” Rev. Mod. Phys. 87, 61 (2015).
[Crossref]

P. Kral, I. Thanopulos, and M. Shapiro, “Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79, 53 (2007).
[Crossref]

Science (2)

L. Feng, Z.-J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref] [PubMed]

H. Hodaei, M. Miri, M Heinrich, Demetrios N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Other (3)

H. Eleuch and I. Rotter, “Quantum states talk via the environment,” https://arXiv:1511.01751 (2015).

T. Kato, Perturbation theory of linear operators (Springer, 1966).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

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Figures (2)

Fig. 1
Fig. 1 Schematic representation of the light-atom interaction configuration of the four-level Hamiltonian. The spin state can be mapped into the ground states of 87Rb, of which |1>=|F=1,mF=0>,|2>=|F=1,mF=−1>,|3>=|F=2,mF=0>,|4>=|F=2,mF=1>. The sublevels can be linked by microwave fields or radio frequency fields.
Fig. 2
Fig. 2 The phase difference and atomic population versus angle. The red, green, yellow and blue line denote the |E+(α)〉, |E(πα)〉, E(α)〉 and E+(πα)〉,respectively. (a,c) The dynamic γd = 0. (b,d) The dynamic phase γd = π/2.

Equations (23)

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

H P T = S ( i sin ( α ) 1 1 i sin ( α ) ) ,
| E + ( α ) = e i α / 2 2 cos ( α ) ( 1 e i α ) , | E ( α ) = e i α / 2 2 cos ( α ) ( 1 e i α ) .
H P T | E ± ( α ) = E ± ( α ) | E ± ( α ) , H P T * | Λ ± ( α ) = E ± * ( α ) | Λ ± ( α ) .
| E ± Λ ± | Λ ± | E ± = 1 , Λ ± | E = 0 .
| E ( π / 2 ) ( 1 i ) , | E ( π / 2 ) ( 1 i ) .
U ^ P T ( ± π / 2 , t ) = 1 cos ( α ) ( cos ( χ t α ) i sin ( χ t ) i sin ( χ t ) cos ( χ t + α ) ) ) .
U ^ P T ( ± π / 2 , t ) = ( 1 ± S t i S t i S t 1 S t ) .
γ = C A d α ,
A = i ( Λ + | d α | E + Λ + | d α | E Λ | d α | E + Λ | d α | E ) ,
γ α ± π α = ( 0 ± π 2 ± π 2 0 ) .
| E ± ( α ) e i π 2 | E ( π α ) | E ± ( α ) , | E ± ( α ) e i π 2 | E ( π α ) | E ± ( α ) .
U ^ F ( υ u ) = ( U ^ P T ( t ) 0 0 U ^ ) ( υ u ) ,
H F = S ( 0 cos ( α ) i sin ( α ) 0 cos ( α ) 0 0 i sin ( α ) i sin ( α ) 0 0 cos ( α ) 0 i sin ( α ) cos ( α ) 0 ) ,
U ^ F ( υ η υ ) = ( U ^ P T ( t ) 0 0 η U ^ P T ( t ) η 1 ) ( υ η υ ) .
H 0 = j ( ω e j ω e 1 ) | j j | , H int = Ω 1 e i ω 1 t | 3 1 | + Ω 2 e i ω 2 t | 3 4 | + Ω 3 e i ω 3 t | 4 2 | + Ω 4 e i ω 4 t | 2 1 | + H . c . ,
V = e i H 0 t = ( 1 0 0 0 0 e i ( ω e 2 ω e 1 ) t 0 0 0 0 e i ( ω e 3 ω e 1 ) t 0 0 0 0 e i ( ω e 4 ω e 1 ) t )
H ˜ = i d V d t V + V H V = ( 0 Ω 4 e i ( ω 4 + ω e 1 ω e 2 ) t Ω 1 e i ( ω 1 + ω e 1 ω e 3 ) t 0 Ω 4 * e i ( ω 4 + ω e 1 ω e 2 ) t 0 0 Ω 3 e i ( ω 3 + ω e 2 ω e 4 ) t Ω 1 * e i ( ω 1 + ω e 1 ω e 3 ) t 0 0 Ω 2 e i ( ω 2 + ω e 3 ω e 4 ) t 0 Ω 3 * e i ( ω 3 + ω e 2 ω e 4 ) t Ω 2 * e i ( ω 2 + ω e 3 ω e 4 ) t 0 ) .
H ˜ = ( 0 Ω 4 Ω 1 0 Ω 4 * 0 0 Ω 3 Ω 1 * 0 0 Ω 2 0 Ω 3 * Ω 2 * 0 ) = H F .
U π / 2 , ϕ = ( 1 i e i ϕ i e i ϕ 1 ) ,
N a , b = 1 2 [ 1 s i n ( ϕ + φ ) ] .
N 1 , 2 ( α ) = { [ 1 sin ( ϕ + α ) ] / 2 for | E + ( α ) [ 1 sin ( ϕ + π α ) ] / 2 for | E ( α ) .
N 0 , 1 ( α ) = { ( 1 sin φ + ) / 2 for | E + ( α ) ( 1 sin φ ) / 2 for | E ( α ) ,
d α d t Ω i , 2 χ .

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