High-order exceptional points in non-Hermitian Moiré lattices

Abstract

This study proposes an approach to generate high-order exceptional points (EPs) in non-Hermitian systems. A system comprising a homogenous waveguide is considered wherein the imaginary part of the refractive index is modulated using a one-dimensional Moiré profile. This gain-loss modulation couples different lossless waveguide modes, and these hybrid modes can be modeled using a non-Hermitian matrix with complex off-diagonal elements. Results indicate that third-order EPs can be produced by the coalescence of two second-order EPs. Then, the necessary requirements are analyzed using coupled-wave equations and the physical effects of the singularities are discussed.

Keywords

exceptional points / non-Hermitian quantum physics / quantum optics / phase transitions

References

[1] T. Kato, Perturbation Theory of Linear Operators, Berlin: Springer-Verlag Berlin Heidelberg, 1966

[2] W. D. Heiss and H. L. Harney, The chirality of exceptional points, Eur. Phys. J. D 17(2), 149 (2001)

[3] C. Dembowski, B. Dietz, H. D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, Observation of a chiral state in a microwave cavity, Phys. Rev. Lett. 90(3), 034101 (2003)

[4] J. Wiersig, Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities, Phys. Rev. Lett. 97(25), 253901 (2006)

[5] B. Dietz, T. Friedrich, J. Metz, M. Miski-Oglu, A. Richter, F. Schafer, and C. A. Stafford, Rabi oscillations at exceptional points in microwave billiards, Phys. Rev. E 75(2), 027201 (2007)

[6] S. B. Lee, J. Yang, S. Moon, S. Y. Lee, J. B. Shim, S. W. Kim, J. H. Lee, and K. An, Observation of an exceptional point in a chaotic optical microcavity, Phys. Rev. Lett. 103(13), 134101 (2009)

[7] Q. H. Song and H. Cao, Improving optical confinement in nanostructures via external mode coupling, Phys. Rev. Lett. 105(5), 053902 (2010)

[8] S. Bittner, B. Dietz, U. Gunther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schafer, PTsymmetry and spontaneous symmetry breaking in a microwave billiard, Phys. Rev. Lett. 108(2), 024101 (2012)

[9] M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Tureci, and S. Rotter, Pump-induced exceptional points in lasers, Phys. Rev. Lett. 108(17), 173901 (2012)

[10] X. Yin and X. Zhang, Unidirectional light propagation at exceptional points, Nat. Mater. 12(3), 175 (2013)

[11] M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schoberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, Reversing the pump dependence of a laser at an exceptional point, Nat. Commun. 5(1), 4034 (2014)

[12] M. Kang, H. X. Cui, T. F. Li, J. Chen, W. Zhu, and M. Premaratne, Unidirectional phase singularity in ultrathin metamaterials at exceptional points, Phys. Rev. A 89(6), 065801 (2014)

[13] S. Longhi and G. Della Valle, Optical lattices with exceptional points in the continuum, Phys. Rev. A 89(5), 052132 (2014)

[14] H. Cao and J. Wiersig, Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics, Rev. Mod. Phys. 87(1), 61 (2015)

[15] H. Hodaei, M. A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, Parity–time-symmetric coupled microring lasers operating around an exceptional point, Opt. Lett. 40(21), 4955 (2015)

[16] B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S. Chua, J. D. Joannopoulos, and M. Soljacic, Spawning rings of exceptional points out of Dirac cones, Nature 525(7569), 354 (2015)

[17] A. Cerjan, A. Raman, and S. Fan, Exceptional contours and band structure design in parity–time symmetric photonic crystals, Phys. Rev. Lett. 116(20), 203902 (2016)

[18] 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(7618), 76 (2016)

[19] L. Ge, Anomalous parity–time-symmetry transition away from an exceptional point, Phys. Rev. A 94(1), 013837 (2016)

[20] M. Kang, J. Chen, and Y. D. Chong, Chiral exceptional points in metasurfaces, Phys. Rev. A 94(3), 033834 (2016)

[21] K. H. Kim, M. S. Hwang, H. R. Kim, J. H. Choi, Y. S. No, and H. G. Park, Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains, Nat. Commun. 7(1), 13893 (2016)

[22] M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, Manifestation of PTsymmetry breaking in polarization space with terahertz metasurfaces, Phys. Rev. Lett. 113(9), 093901 (2014)

[23] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Observation of PT-symmetry breaking in complex optical potentials, Phys. Rev. Lett. 103(9), 093902 (2009)

[24] S. Longhi, PT-symmetric laser absorber, Phys. Rev. A 82(3), 031801 (2010) (R)

[25] L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, Experimental demonstration of a unidirectional reflectionless parity–time metamaterial at optical frequencies, Nat. Mater. 12(2), 108 (2013)

[26] W. Wang, L. Q. Wang, R. D. Xue, H. L. Chen, R. P. Guo, Y. Liu, and J. Chen, Unidirectional excitation of radiative-loss-free surface plasmon polaritons in PT-symmetric systems, Phys. Rev. Lett. 119(7), 077401 (2017)

[27] T. Goldzak, A. A. Mailybaev, and N. Moiseyev, Light stops at exceptional points, Phys. Rev. Lett. 120(1), 013901 (2018)

[28] H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, Enhanced sensitivity at higher-order exceptional points, Nature 548(7666), 187 (2017)

[29] W. D. Heiss, Chirality of wavefunctions for three coalescing levels, J. Phys. A 41(24), 244010 (2008)

[30] 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(23), 235310 (2015)

[31] W. D. Heiss and G. Wunner, Resonance scattering at third-order exceptional points, J. Phys. A 48(34), 345203 (2015)

[32] Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals, Phys. Rev. Lett. 117(10), 107402 (2016)

[33] H. Jing, K. Ozdemir, H. Lu, and F. Nori, High-order exceptional points in optomechanics, Sci. Rep. 7(1), 3386 (2017)

[34] M. Y. Nada, M. A. K. Othman, and F. Capolino, Theory of coupled resonator optical waveguides exhibiting high-order exceptional points of degeneracy, Phys. Rev. B 96(18), 184304 (2017)

[35] J. Schnabel, H. Cartarius, J. Main, G. Wunner, and W. D. Heiss, PT-symmetric waveguide system with evidence of a third-order exceptional point, Phys. Rev. A 95(5), 053868 (2017)

[36] J. Kullig, C. H. Yi, M. Hentschel, and J. Wiersig, Exceptional points of third-order in a layered optical microdisk cavity, New J. Phys. 20(8), 083016 (2018)

[37] Q. Zhong, D. N. Christodoulides, M. Khajavikhan, K. G. Makris, and R. El-Ganainy, Power-law scaling of extreme dynamics near higher-order exceptional points, Phys. Rev. A 97(2), 020105 (2018) (R)

[38] X. Zhou, S. K. Gupta, Z. Huang, Z. Yan, P. Zhan, Z. Chen, M. Lu, and Z. Wang, Optical lattices with higher-order exceptional points by non-Hermitian coupling, Appl. Phys. Lett. 113(10), 101108 (2018)

[39] K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Beam dynamics in PT symmetric optical lattices, Phys. Rev. Lett. 100(10), 103904 (2008)

[40] J. B. Khurgin, Light slowing down in Moiré fiber gratings and its implications for nonlinear optics, Phys. Rev. A 62(1), 013821 (2000)

[41] R. D. Xue, W. Wang, L. Q. Wang, H. L. Chen, R. P. Guo, and J. Chen, Localization and oscillation of optical beams in Moiré lattices,Opt. Express 25(5), 5788 (2017)

[42] L. Feng, R. El-Ganainy, and L. Ge, Non-Hermitian photonics based on parity–time symmetry, Nat. Photonics 11(12), 752 (2017)

[43] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non- Hermitian physics and PTsymmetry, Nat. Phys. 14(1), 11 (2018)

[44] I. Rotter, A non-Hermitian Hamilton operator and the physics of open quantum systems, J. Phys. A 42(15), 153001 (2009)

[45] W. Zhang, A. Hu, X. Lei, N. Xu, and N. Ming, Photonic band structures of a two-dimensional ionic dielectric medium, Phys. Rev. B 54(15), 10280 (1996)

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