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
1. INTRODUCTION
Eigenvalues (wavenumber) in open quantum (wave) systems described in terms of non-Hermitian Hamiltonian are complex, and their corresponding eigenvectors (modes) are not completely orthogonal. One fascinating issue arising in an open system is the unique point of non-Hermitian degeneracy, the so-called “exceptional point” (EP), where the eigenvalues as well as the corresponding eigenvectors coalesce simultaneously [1–3]. So far, much effort has been devoted to figuring out this issue, and EPs have been found in such extensive systems as atomic systems [4], lasers [5,6], microwave and optical systems [7–10], acoustics [11,12], and electronic circuits [13,14].
In an earlier stage of studying EPs, the focus was on a two-level system , which helps to understand the physical mechanism of a second-order EP (EP2). Several types of EP2s have been found in various systems, depending on their intrinsically different mechanism of mode coupling: a parity-time symmetry [15–17], asymmetric back scattering [18–21], supermode coupling [22–26], an internal-external mode pair [27], a concentric layered microdisk [28], and resonance-assisted tunneling [29].
After the reports of applicability of EPs, e.g., sensors [30] for nanoparticle detection [31–35] or gyroscopes [36–39], a growing interest has been concentrated on higher-order EPs . This is because, as splitting of eigenvalues against a small perturbation is proportional to in the vicinity of EPs, sensitivity can be enhanced by as much as the increased order . Hence, higher-order EPs have become a key agenda today. Considerable attempts have been made theoretically to obtain higher-order EPs for several systems [28,40–50]. However, because of their fragile configurations, only a restricted number of experiments to obtain higher-order EPs were successful [45,49,50].
Each of those higher-order EPs mentioned above uses its own single coupling mechanism, and yet, no attempt has been made to hybridize two different types of EP2s for a higher-order EP. In this paper, we show that different types of EP2s can be hybridized for a higher-order EP. As a demonstration, we hybridize a chiral EP and a supermode EP, where the former and the latter are the results of intra-disk and inter-disk mode coupling in an optical microdisk system [51] comprised of two Mie-scale microdisks and one Rayleigh-scale scatterer as shown in Fig. 1.
When an even and an odd parity mode in a single microdisk coalesce due to asymmetric backscattering, we call this type of EP2 a “chiral EP” [18–21,36,52]. On the other hand, optical modes in separate microdisks can be coupled to form a “supermodes” pair. When they coalesce [22], we call this type of EP2 a “supermode EP.” These intra- and inter-disk mode couplings can contribute simultaneously in our system to generation of a chiral EP and a supermode EP, respectively, as well as their hybridization for third-order EPs (EP3s). The setup in Fig. 1 is experimentally feasible since the system parameters are easily controllable as in Refs. [35,52].
2. NON-HYBRIDIZED EXCEPTIONAL POINTS
Optical modes in our system are obtained by solving Maxwell’s equations reduced to a two-dimensional wave equation , where , , and are the component of electric field (i.e., TM polarization), the refractive index, and the vacuum wavenumber, respectively. We set inside of the microdisks and the scatterer and outside of them. In numerical computations, we use the boundary element method [53], and the modes obtained are expressed in terms of dimensionless wavenumbers .
Figure 2 shows the of modes as a function of in coupled microdisks without a scatterer for . The horizontal and oblique curves represent modes localized on the left and the right microdisk, respectively. The highlighted thick curves originate from the and the (1, 5) mode, which are to be used as four basis modes in the following discussions. Here and are the radial and the azimuthal mode number, respectively, in each microdisk. The four basis modes are defined as follows: inset I () is the “left-even” mode, II () the “left-odd”, III () the “right-even”, and IV () the “right-odd.” Here, “right” and “left” indicate the location of modes on the microdisks, and “even” and “odd” indicate the parity with respect to the axis of mirror-reflection symmetry ( axis). We focus on the region marked by the blue dashed circle, where the four modes interact with each other. The pairs and are in the weak and the strong coupling, respectively [29]. This is because the real and the imaginary parts (not shown) of the wavenumbers of the pair cross and repulse, respectively, while those of the pair exhibit an avoided crossing and a crossing, respectively. Note that the even (solid curves) and odd (dashed curves) parity modes in Fig. 2 are orthogonal because of the mirror-reflection symmetry of the system, even though the system is non-Hermitian. It is known that the orthogonality of eigenvectors in a non-Hermitian system can be characterized by the definition of the phase rigidity [54,55].
In the marked region, we add a scatterer and observe the wavenumber variations of the four modes depending on at , , , and . Figures 3(a) and 3(b) show and , respectively. By introducing the scatterer, the even and odd parity modes are now highly nonorthogonal because of the asymmetric backscattering induced by the scatterer. This additional non-Hermiticity, caused by the scatterer, can induce couplings among all the basis modes, I, II, III, and IV, in Fig. 2, no matter what their parity symmetries were before. Therefore, the wave functions corresponding to the curves in Figs. 3(a) and 3(b) exhibit the superpositions of the basis modes. To quantify how close the modes are to the EPs, we obtain wavenumber differences for all combinations of the four modes as shown in Fig. 3(c). The figure exhibits three sharp local minima at (A) , (B) 0.909, and (C) 1.969. According to the locations and parities of the modes, we can infer that A and B are the proximity of a chiral EP and a supermode EP because the modes there originate from and , respectively. C is an EP3 of three coalescent modes originating from . In this case, since the combination implies a chiral EP and the one a supermode EP, this EP3 is a hybridization of the two different types of EP2. Hence, accessing points A, B, and C, we can switch the three different EPs by controlling the scatterer in our system.
We first analyze the supermode EP at B. By means of “downhill simplex method” [56], we obtain more accurate parameters of the EP2. Figures 4(a) and 4(b) show Riemann surfaces of and around the supermode EP at and . The coalescent modes shown in Fig. 4(c) are not completely symmetric to the axis, unlike the ones in Ref. [22], which are obtained by controlling the size of one microdisk and the distance between two microdisks. In our system, just by controlling the size and position of the scatterer, we can achieve a supermode EP. Note that the other modes in Figs. 4(d) and 4(e) are not symmetric as well.
In the case of the chiral EP at A, Riemann surfaces of and around the EP at and are shown in Figs. 5(a) and 5(b). The coalescent mode at this EP is given in Fig. 5(c), and the other two are in Figs. 5(d) and 5(e). Although we tune the scatterer belonging to the left microdisk, the chiral EP mode in the right one becomes strongly chiral [i.e., traveling wave; see Fig. 5(c)]. This implies that we can remotely control the traveling waves in the target microdisk (i.e., the right one) without a direct perturbation.
The chirality of modes can be quantified by employing the Husimi function [57] of where , which encodes a probability of propagation directions in the phase space as for the counter-clockwise direction and for the clockwise direction, respectively. Here, and represent the position of the impact point and the incident angle of the ray with respect to the outward normal direction on the microdisk boundary. The chirality is given as
The chirality of the modes in the left and right microdisk in Fig. 5(c) is . These values that are slightly less than 1 are counterintuitive when we consider the previous works on a chiral EP in a different system [58]. Note that the chiralities of all the remaining modes in Figs. 5(d) and 5(e) as well as of the supermode EPs in Figs. 4(c)–4(e) are found to have non-zero values less than 1. It is also noted that the mode properties of our supermode EP2s and chiral EP2s are not the same as the ones found in typical systems consisting of only two microdisks or a single microdisk with two scatterers.3. HYBRIDIZATION OF EXCEPTIONAL POINTS
Eventually, we turn our attention to the EP3 at C in Fig. 3. After carrying out tedious but definite procedures of numerical scanning in a multi-valued parameter space, EP3s are obtained around and . Riemann surfaces for and , shown in Figs. 6(a) and 6(b), clearly exhibit the typical structure of an EP3 (see, e.g., Ref. [59]). We find that three nearly identical of the coalescent modes of , , and are , , and , respectively. The wavenumber of the remaining one, , is . The chirality of EP3 modes in each microdisk is as shown in Fig. 7(a), and that of the remaining mode is (0.35,0.10). Again, we can see that the chirality is neither 1 nor 0, in spite of the fact that the chirality of the mode in the left microdisk should be if our EP3 is a pure combination of three basis modes excluding .
Though all three coalescent modes at the EP3 seem to be identical like the modes in Fig. 6(c), it is necessary to verify whether they are really identical or not. To this end, as pointed out in Ref. [60], it is useful to compute the overlaps of the modes at the EPs by using the following equation:
where and imply two different modes. The overlaps of all the combinations of the three coalescent modes at the EP3 are above 0.9999 as shown by the red symbols (square, circle, and downward triangle) in Fig. 7(b), while the overlaps between the three coalescent modes () and are about 0.065 as shown by the black symbols (upward triangle, diamond, and star) in Fig. 7(b). The overlap of 0.065 implies that and are not orthogonal and that all four basis modes contribute to the EP3.To estimate the contribution of the four basis modes to the EP3, we obtain overlaps between and the four basis modes by using the definition [40,55,61,62]. Then the ratio of the overlaps among the four basis modes are . Although the EP3 is a coalescence of three modes originating from , the non-negligible contribution of is observed as the value of . Hence, we describe our EP3 by using a Hamiltonian as follows:
where the subscripts and denote the left (right) microdisk and the even (odd) parity, respectively. In the Hamiltonian, is the intra-disk coupling of in the left microdisk due to a scatterer, and is the inter-disk coupling of even (odd) parity modes between the left and the right microdisks. The left-upper and the right-lower dashed boxes represent the left and right microdisk, respectively. For the Hamiltonian, we conjecture: (i) all off-diagonal elements for a supermode (not EP) without a scatterer are 0, except and , because a supermode can be formed by the same parity modes; and (ii) when a scatterer is added, and are smaller than , , and , but are not exactly zero because the scatterer globally perturbs all basis modes in the formation of an EP3.Now, we numerically obtain all elements by the expansion and the definition . For , 2, 3, and 4, we can identify all elements in Eq. (3) by setting 16 equations of , where is the unit matrix. Conjecture (i) is confirmed, as a supermode (not EP) has and , while all the other off-diagonal elements are zero. In the case of a heterogeneous EP3, the Hamiltonian is obtained as follows:
We can see that the elements , , and are much bigger than non-zero and as are conjectured in (ii). Moreover, it turns out that in the EP3 is similar to the one in the supermode. Based on this observation, we can conclude that a contribution of the remaining mode cannot be neglected in our EP3 described in a Hamiltonian.Now, we test the sensitivity of our EP3 against external perturbations in comparison with that of a chiral EP and a supermode EP. We exemplify a deviation as a function of refractive index variation outside of the microdisks. This could be interpreted as the impurity concentration in a solvent, like in [63]. In Fig. 8, we can see that the slopes of the chiral and the supermode EPs are ruled by , whereas our EP3 exhibits . This is further evidence of an EP3. This impressive sensitivity was already predicted in previous literature, e.g., in Refs. [28,49]. To emphasize our point, our proposed system is obviously feasible since we can take advantage of the system implemented in Ref. [35].
4. CONCLUSION
To summarize, we have demonstrated hybridization of two different types of EP2. This hybridization results in a third-order EP in a system comprised of two microdisks and a scatterer. The high-order sensitivity against an external perturbation can function to resolve the imperative issue concerning real applications of EP3s to sensors. We believe that the unveiled hybridization mechanism can shed light on both improving the performance of applications and advancing the physics related to EPs. Most importantly, since this hybridization mechanism can extend to various cases of EP2s in extensive physical systems, we believe that our findings will initiate new directions of studying EPs.
Acknowledgment
This research was supported by Ministry of Health and Welfare, Republic of Korea (Government-wide R&D Fund project for infectious disease research, HG18C0069).
REFERENCES
1. T. Kato, Perturbation Theory for Linear Operators (Springer, 1966).
2. W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000). [CrossRef]
3. M. Berry, “Physics of nonhermitian degeneracies,” Czech. J. Phys. 54, 1039–1047 (2004). [CrossRef]
4. H. Cartarius, J. Main, and G. Wunner, “Exceptional points in atomic spectra,” Phys. Rev. Lett. 99, 173003 (2007). [CrossRef]
5. 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]
6. 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]
7. 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]
8. 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]
9. 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]
10. 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]
11. 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]
12. 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]
13. T. Stehmann, W. D. Heiss, and F. G. Scholtz, “Observation of exceptional points in electronic circuits,” J. Phys. A 37, 7813–7819 (2004). [CrossRef]
14. 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]
15. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998). [CrossRef]
16. 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]
17. 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]
18. 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]
19. J. Wiersig, “Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles,” Phys. Rev. A 84, 063828 (2011). [CrossRef]
20. 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]
21. 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]
22. 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]
23. S. V. Boriskina, “Coupling of whispering-gallery modes in size-mismatched microdisk photonic molecules,” Opt. Lett. 32, 1557–1559 (2007). [CrossRef]
24. 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]
25. J.-B. Shim and J. Wiersig, “Semiclassical evaluation of frequency splittings in coupled optical microdisks,” Opt. Express 21, 24240–24253 (2013). [CrossRef]
26. D. W. Schönleber, A. Eisfeld, and R. El-Ganainy, “Optomechanical interactions in non-Hermitian photonic molecules,” New J. Phys. 18, 045014 (2016). [CrossRef]
27. 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]
28. 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]
29. 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]
30. J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016). [CrossRef]
31. 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]
32. 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]
33. 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]
34. 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]
35. 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]
36. R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015). [CrossRef]
37. S. Sunada, “Large Sagnac frequency splitting in a ring resonator operating at an exceptional point,” Phys. Rev. A 96, 033842 (2017). [CrossRef]
38. 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]
39. Y.-H. Lai, Y.-K. Lu, M.-G. Suh, and K. Vahala, “Enhanced sensitivity operation of an optical gyroscope near an exceptional point,” arXiv:1901.08217 (2019).
40. H. Eleuch and I. Rotter, “Clustering of exceptional points and dynamical phase transitions,” Phys. Rev. A 93, 042116 (2016). [CrossRef]
41. 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]
42. 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]
43. H. Jing, S. K. Özdemir, H. Lü, and F. Nori, “High-order exceptional points in optomechanics,” Sci. Rep. 7, 3386 (2017). [CrossRef]
44. W. D. Heiss and G. Wunner, “Resonance scattering at third-order exceptional points,” J. Phys. A 48, 345203 (2015). [CrossRef]
45. 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]
46. 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]
47. 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]
48. R. El-Ganainy, M. Khajavikhan, and L. Ge, “Exceptional points and lasing self-termination in photonic molecules,” Phys. Rev. A 90, 013802 (2014). [CrossRef]
49. 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]
50. 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]
51. H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015). [CrossRef]
52. 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]
53. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A 5, 53–60 (2002). [CrossRef]
54. H. Eleuch and I. Rotter, “Resonances in open quantum systems,” Phys. Rev. A 95, 022117 (2017). [CrossRef]
55. I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A 42, 153001 (2009). [CrossRef]
56. J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965). [CrossRef]
57. 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]
58. 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]
59. G. Demange and E.-M. Graefe, “Signatures of three coalescing eigenfunctions,” J. Phys. A 45, 025303 (2011). [CrossRef]
60. J. T. Chalker and B. Mehlig, “Eigenvector statistics in non-Hermitian random matrix ensembles,” Phys. Rev. Lett. 81, 3367–3370 (1998). [CrossRef]
61. M. Müller, F.-M. Dittes, W. Iskra, and I. Rotter, “Level repulsion in the complex plane,” Phys. Rev. E 52, 5961–5973 (1995). [CrossRef]
62. 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]
63. A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high-Q microcavities,” Opt. Lett. 31, 1896–1898 (2006). [CrossRef]