Abstract

A large number of different types of second-order non-Hermitian degeneracies called exceptional points (EPs) were found in various physical systems depending on the mechanism of coupling between eigenstates. We show that these EPs can be hybridized to form higher-order EPs, which preserve the original properties of the initial EPs before hybridization. For a demonstration, we hybridize chiral and supermode second-order EPs, where the former and the latter are the results of intra-disk and inter-disk mode coupling in an optical system comprised of two Mie-scale microdisks and one Rayleigh-scale scatterer. The high sensitivity of the resulting third-order EP against external perturbations in our feasible system is emphasized.

© 2019 Chinese Laser Press

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References

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2019 (3)

W. R. Sweeney, C. W. Hsu, S. Rotter, and A. D. Stone, “Perfectly absorbing exceptional points and chiral absorbers,” Phys. Rev. Lett. 122, 093901 (2019).
[Crossref]

S. Wang, B. Hou, W. Lu, Y. Chen, Z. Q. Zhang, and C. T. Chan, “Arbitrary order exceptional point induced by photonic spin-orbit interaction in coupled resonators,” Nat. Commun. 10, 832 (2019).
[Crossref]

C.-H. Yi, J. Kullig, M. Hentschel, and J. Wiersig, “Non-Hermitian degeneracies of internal-external mode pairs in dielectric microdisks,” Photon. Res. 7, 464–472 (2019).
[Crossref]

2018 (7)

W. Chen, J. Zhang, B. Peng, Ş. K. Özdemir, X. Fan, and L. Yang, “Parity-time-symmetric whispering-gallery mode nanoparticle sensor,” Photon. Res. 6, A23–A30 (2018).
[Crossref]

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, 083016 (2018).
[Crossref]

C.-H. Yi, J. Kullig, and J. Wiersig, “Pair of exceptional points in a microdisk cavity under an extremely weak deformation,” Phys. Rev. Lett. 120, 093902 (2018).
[Crossref]

S. Liu, J. Wiersig, W. Sun, Y. Fan, L. Ge, J. Yang, S. Xiao, Q. Song, and H. Cao, “Transporting the optical chirality through the dynamical barriers in optical microcavities,” Laser Photon. Rev. 12, 1800027 (2018).
[Crossref]

J. Kullig, C.-H. Yi, and J. Wiersig, “Exceptional points by coupling of modes with different angular momenta in deformed microdisks: a perturbative analysis,” Phys. Rev. A 98, 023851 (2018).
[Crossref]

W. Zhu, X. Fang, D. Li, Y. Sun, Y. Li, Y. Jing, and H. Chen, “Simultaneous observation of a topological edge state and exceptional point in an open and non-Hermitian acoustic system,” Phys. Rev. Lett. 121, 124501 (2018).
[Crossref]

Y. Choi, C. Hahn, J. W. Yoon, and S. H. Song, “Observation of an anti-PT-symmetric exceptional point and energy-difference conserving dynamics in electrical circuit resonators,” Nat. Commun. 9, 2182 (2018).
[Crossref]

2017 (8)

H. Jing, S. K. Özdemir, H. Lü, and F. Nori, “High-order exceptional points in optomechanics,” Sci. Rep. 7, 3386 (2017).
[Crossref]

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, 053868 (2017).
[Crossref]

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

H. Eleuch and I. Rotter, “Resonances in open quantum systems,” Phys. Rev. A 95, 022117 (2017).
[Crossref]

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, 187–191 (2017).
[Crossref]

J. Ren, H. Hodaei, G. Harari, A. U. Hassan, W. Chow, M. Soltani, D. Christodoulides, and M. Khajavikhan, “Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope,” Opt. Lett. 42, 1556–1559 (2017).
[Crossref]

N. Zhang, Z. Gu, S. Liu, Y. Wang, S. Wang, Z. Duan, W. Sun, Y.-F. Xiao, S. Xiao, and Q. Song, “Far-field single nanoparticle detection and sizing,” Optica 4, 1151–1156 (2017).
[Crossref]

S. Sunada, “Large Sagnac frequency splitting in a ring resonator operating at an exceptional point,” Phys. Rev. A 96, 033842 (2017).
[Crossref]

2016 (10)

D. W. Schönleber, A. Eisfeld, and R. El-Ganainy, “Optomechanical interactions in non-Hermitian photonic molecules,” New J. Phys. 18, 045014 (2016).
[Crossref]

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

W. D. Heiss and G. Wunner, “A model of three coupled wave guides and third order exceptional points,” J. Phys. A 49, 495303 (2016).
[Crossref]

H. Eleuch and I. Rotter, “Clustering of exceptional points and dynamical phase transitions,” Phys. Rev. A 93, 042116 (2016).
[Crossref]

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
[Crossref]

B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Ylmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. USA 113, 6845–6850 (2016).
[Crossref]

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

C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, “Accessing the exceptional points of parity-time symmetric acoustics,” Nat. Commun. 7, 11110 (2016).
[Crossref]

Y. Shin, H. Kwak, S. Moon, S.-B. Lee, J. Yang, and K. An, “Observation of an exceptional point in a two-dimensional ultrasonic cavity of concentric circular shells,” Sci. Rep. 6, 38826 (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]

2015 (4)

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

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015).
[Crossref]

R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015).
[Crossref]

N. Zhang, S. Liu, K. Wang, Z. Gu, M. Li, N. Yi, S. Xiao, and Q. Song, “Single nanoparticle detection using far-field emission of photonic molecule around the exceptional point,” Sci. Rep. 5, 11912 (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]

J. Wiersig, “Chiral and nonorthogonal eigenstate pairs in open quantum systems with weak backscattering between counterpropagating traveling waves,” Phys. Rev. A 89, 012119 (2014).
[Crossref]

R. El-Ganainy, M. Khajavikhan, and L. Ge, “Exceptional points and lasing self-termination in photonic molecules,” Phys. Rev. A 90, 013802 (2014).
[Crossref]

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

2013 (1)

2012 (2)

Y. V. Fyodorov and D. V. Savin, “Statistics of resonance width shifts as a signature of eigenfunction nonorthogonality,” Phys. Rev. Lett. 108, 184101 (2012).
[Crossref]

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

2011 (4)

J. Wiersig, A. Eberspächer, J.-B. Shim, J.-W. Ryu, S. Shinohara, M. Hentschel, and H. Schomerus, “Nonorthogonal pairs of copropagating optical modes in deformed microdisk cavities,” Phys. Rev. A 84, 023845 (2011).
[Crossref]

J. Wiersig, “Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles,” Phys. Rev. A 84, 063828 (2011).
[Crossref]

G. Demange and E.-M. Graefe, “Signatures of three coalescing eigenfunctions,” J. Phys. A 45, 025303 (2011).
[Crossref]

M. Benyoucef, J.-B. Shim, J. Wiersig, and O. G. Schmidt, “Quality-factor enhancement of supermodes in coupled microdisks,” Opt. Lett. 36, 1317–1319 (2011).
[Crossref]

2010 (1)

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

2009 (3)

J.-W. Ryu, S.-Y. Lee, and S. W. Kim, “Coupled nonidentical microdisks: avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858 (2009).
[Crossref]

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, 134101 (2009).
[Crossref]

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

2007 (2)

S. V. Boriskina, “Coupling of whispering-gallery modes in size-mismatched microdisk photonic molecules,” Opt. Lett. 32, 1557–1559 (2007).
[Crossref]

H. Cartarius, J. Main, and G. Wunner, “Exceptional points in atomic spectra,” Phys. Rev. Lett. 99, 173003 (2007).
[Crossref]

2006 (1)

2004 (2)

M. Berry, “Physics of nonhermitian degeneracies,” Czech. J. Phys. 54, 1039–1047 (2004).
[Crossref]

T. Stehmann, W. D. Heiss, and F. G. Scholtz, “Observation of exceptional points in electronic circuits,” J. Phys. A 37, 7813–7819 (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. Hentschel, H. Schomerus, and R. Schubert, “Husimi functions at dielectric interfaces: inside-outside duality for optical systems and beyond,” Europhys. Lett. 62, 636–642 (2003).
[Crossref]

2002 (1)

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A 5, 53–60 (2002).
[Crossref]

2001 (1)

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]

2000 (1)

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

1998 (2)

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

J. T. Chalker and B. Mehlig, “Eigenvector statistics in non-Hermitian random matrix ensembles,” Phys. Rev. Lett. 81, 3367–3370 (1998).
[Crossref]

1995 (1)

M. Müller, F.-M. Dittes, W. Iskra, and I. Rotter, “Level repulsion in the complex plane,” Phys. Rev. E 52, 5961–5973 (1995).
[Crossref]

1965 (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
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Figures (8)

Fig. 1.
Fig. 1. System configuration comprised of two microdisks and one scatterer. The scatterer is magnified in the dotted box.
Fig. 2.
Fig. 2. Re(kR) of supermodes as a function of r1/R and their basis modes in coupled microdisks without a scatterer. The solid and the dashed curves are the even and odd parity modes, respectively. In the region marked by a blue dashed circle, the four modes originating from (l,m)=(1,6) and (1, 5) are highlighted by the thick solid and dotted curves. The insets I, II, III, and IV are the four basis modes defined as ϕ1, ϕ2, ϕ3, and ϕ4, respectively.
Fig. 3.
Fig. 3. (a) and (b) are the θ-dependent Re(kR) and Im(kR) of the four modes originating from the modes in the blue dashed circle in Fig. 2, respectively; (c) shows the absolute values of wavenumber differences for all combinations of the four modes. The black-solid curve in (c) is for the black-solid and black-dashed curves in (a) and (b). The orange-dashed curve in (c) is for the black-solid and orange-solid curves in (a) and (b). The blue-dotted curve in (c) is for the orange-solid and black-dashed curves in (a) and (b). Vertical lines at A, B, and C mark the local minima of these three curves in (c).
Fig. 4.
Fig. 4. (a) and (b) Riemann surfaces of Re(kR) and Im(kR), respectively, around the supermode EP kEPR4.219i0.0430. The dashed curves are the branch cuts, and the large dots are the EPs. Here Δr2=r2rEP and Δθ=θθEP. (c) Coalescent mode at the EP; (d) and (e) the two remaining modes at this EP parameter.
Fig. 5.
Fig. 5. (a) and (b) Riemann surfaces of Re(kR) and Im(kR), respectively, around the chiral EP kEPR4.232i0.0438. The dashed curves, the large dots, Δr2, and Δθ are the same as in Fig. 4. (c) Coalescent mode at the EP; (d) and (e) the two remaining modes at this EP parameter.
Fig. 6.
Fig. 6. Riemann surfaces and modes of the EP3 by hybridizing a supermode and a chiral EP; (a) and (b) are Re(kR) and Im(kR), respectively. The dashed curves are the branch cuts and the large dots are the EPs. Here Δr2=r2rEP and Δθ=θθEP. (c) and (d) The coalescent and the remaining modes at this EP parameter.
Fig. 7.
Fig. 7. (a) Accumulated Husimi function of the coalescent mode as a function of p. The black-solid and the red-dashed curves are for the left and right microdisk, respectively. (b) Overlaps of all combinations of the four modes at EP3. The subscripts 1, 2, and 3 are the modes forming EP3, and the subscript 4 is the remaining mode.
Fig. 8.
Fig. 8. Wavenumber splittings depending on the outside refractive index of the microdisk in double logarithmic scale. Thick magenta, blue, and green curves correspond to an EP3, a supermode EP, and a chiral EP, respectively. Thin black-solid and black-dashed curves are the fittings of |ΔkR|(Δn)1/3 and (Δn)1/2, respectively.

Equations (4)

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α1min{p<0H(γ)dqdp,p>0H(γ)dqdp}max{p<0H(γ)dqdp,p>0H(γ)dqdp}.
Sμν|dxdyΨμ*Ψν|dxdyΨμ*ΨμdxdyΨν*Ψν,
H=(kL,ERaakL,ORbϵ1ϵ2cbϵ2ϵ1ckR,ERδδkR,OR),
H(4.1990.041i0.013+0.007i0.012+0.008i4.2120.036i0.0310.006i0.0020.003i0.003+0.001i0.0060.007i0.0280.005i0.003+0.001i0.0020.002i0.0070.008i4.2090.052i0.001+0.001i0.001+0.001i4.2260.056i).