The concept of topology is universally observed in various physical objects when the objects can be described by geometric structures. Although a representative example is the knotted geometry of wavefunctions in reciprocal space for quantum Hall family and topological insulators, topological states have also been defined for other physical quantities, such as topologically distinct Fermi surfaces and enhanced lattice degrees of freedom in hyperbolic geometry. Here, we investigate a different class of topological states – topological geometry of dynamical state trajectories – in non-Hermitian and nonlinear optical dynamics, revealing topologically protected oscillation quenching mechanisms determined by parity–time (PT) symmetry. For coupled systems composed of nonlinear gain and loss elements, we classify the topology of equilibria separately for unbroken and broken PT symmetry, which result in distinct oscillation quenching mechanisms: amplitude death and oscillation death. We then show that these PT-symmetric quenching mechanisms lead to immunity against temporal perturbations, enabling the applications of topologically protected laser modulation and rectification. The observed connection between the topological geometry of dynamical states, oscillation quenching phenomena in dynamical systems theory, and PT symmetry provides a powerful toolkit for noise-immune signal processing.
Topological degrees of freedom (DOF) have provided a new phase of matter, including quantized bulk conductance and topological insulators . The similarity between the Schrödinger equation and Maxwell’s equations has also stimulated the birth of topological photonics . One of the major goals in this field is to realize a photonic analogy of the quantum Hall family and topological insulators, which enables defect-immune wave propagations. While topological photonics has been extended to synthetic dimensions , non-Hermitian photonics , atom–photon interactions , and second- and third-order optical nonlinearities , , , , , most of these efforts have focused on the topological nature of optical wavefunctions in dispersion bands.
Topological properties are ubiquitous in any physical objects that can be expressed as geometric structures in well-defined parameter spaces . Therefore, there have been efforts to exploit other classes of topological invariants, which are defined for optical quantities other than band structures. An important example is the topology of an isofrequency surface and its topological transition [12, 13] as the optical equivalent of the Lifshitz transition , which enabled the discovery of hyperbolic materials  and pure transverse spins . Another example can be found in unique flat bands in hyperbolic geometry [15, 16], which originate from the lattices that are topologically distinct from Euclidean ones. Considering recent interest in photonic systems using nonlinearities  or time-varying media , we expect the utilization of a certain type of topological invariants defined for the dynamics of light.
Notably, the study of the topological equivalence between different dynamical systems has been a traditional focus of dynamical systems theory . The topologically equivalent dynamical systems can be defined by the existence of an invertible map between the state trajectories of each system, i.e. the homeomorphism of phase portraits . Because the dynamical evolution of the system states requires the energy exchange between the system and its environment, topological phases in optical dynamics also have a connection with the extension of static non-Hermitian photonics  to nonlinear or time-varying platforms. So far, numerous studies on nonlinear dynamics have been conducted for specific non-Hermitian systems. For example, a laser is a traditional non-Hermitian platform with nonlinearity, which enables chaos, injection locking, and synchronization in amplification , , , . The recent developments in parity–time (PT) symmetry [25, 26] and topological physics [2, 27] have established new design freedom in nonlinear and non-Hermitian optical dynamics: suppressed time reversals , optical isolation , amplified Fano resonances , single-mode lasing , quenching behaviors  in nonlinear PT-symmetric systems, and topological zero modes in Su–Schrieffer–Heeger chains . However, despite these various achievements, an essential analysis for dynamical systems – topological classification and protection of dynamical state trajectories [19, 34, 35] – is still absent in nonlinear PT-symmetric dynamics.
In this paper, we show that the interpretation of PT-symmetric dynamics with the topological picture provides new design freedom for noise-immune signal processing. We introduce the topological classification of nonlinear optical dynamics in PT-symmetric systems by employing dynamical systems theory . We show that the topological class defined by optical state trajectories is classified by PT symmetry , revealing PT-symmetry-protected oscillation quenching mechanisms: amplitude death (AD)  and oscillation death (OD) . Using the immunity of the quenching mechanisms against temporal perturbations, which originates from the topologically conserved state trajectories under the system deformation, two application examples are also presented: noise-immune laser modulation and rectification. Our results can be readily implemented with electric circuits and acoustics.
2.1 Model definition and theoretical analysis
Let us consider a system coupled to an external reservoir, such as a photonic molecule consisting of two coupled nonlinear resonators each with the same resonance frequency ω 0 and coupling coefficient κ. The nonlinearity of each resonator is assumed to be any form of an intensity-dependent gain or loss. The photonic molecule is then described by temporal coupled-mode theory (TCMT)  as
where a m and N m represent the field amplitude and real-valued nonlinearity function of the mth resonator (m = 1, 2), respectively. The abstract forms of N 1,2 represent intensity-dependent nonlinearities (Supplementary Note A): multiphoton processes , saturable responses , and their combinations, increasing the possible design freedom. Although Eq. (1) and its simplified form have been studied , , [24, 28], , , , , , the topological classification of optical state trajectories, their protections, and the connection to PT symmetry have not been considered.
We derive an equation for the intensity with a m = I m 1/2 exp (iφ m ) as 
where I m and φ m are real-valued intensity and phase functions, respectively, and θ = φ 1 − φ 2 is the time-varying phase difference between each resonator field. Although φ 1,2(t) and θ(t) are determined by coupled time-derivative equations (Supplementary Note B), at this stage, we consider the steady-state solution with static I 1,2 and θ(t, I 1, I 2) = θ s(I 1, I 2) near equilibria . This synchronization around equilibria will be discussed later.
where γ = [N 2(I 2) − N 1(I 1)]/(2κ) and sgn(x) is the sign function: sgn(x ≥ 0) = +1 and sgn(x < 0) = −1. The upper and lower conditions in Eq. (3) correspond to unbroken and broken PT symmetry , respectively, which implies that topological phases will be classified by the phase of PT symmetry as discussed later. To define the topology of our system, we explore the equilibria in Eq. (2) and examine their stability . The equilibrium (I 1E, I 2E) in the two-dimensional (2D) state space I 1–I 2 is obtained with dI 1,2/dt = 0 in Eq. (2), which results in
with the static function θ s(I 1E, I 2E) at equilibrium. Although we initially assign arbitrary nonlinearities, the first equality in Eq. (4) shows that N 1(I 1E) and N 2(I 2E) should have different signs for a nontrivial equilibrium I 1E,2E > 0, which exhibits the connection between nonlinear optical dynamics and PT symmetry with the gain–loss configuration .
The stability of the equilibrium (I 1E, I 2E) is examined by the first Lyapunov criterion  in which the eigenvalues of the Jacobian matrix of Eq. (2) are used (Supplementary Note D for the Jacobian matrix A). We note that Eq. (3) provides a separate analysis of each phase of PT symmetry. For unbroken PT symmetry (|γ| ≤ 1, Supplementary Note E), the system is in the homogeneous steady state (HSS)  at equilibrium as I 1E = I 2E = I HE, where I HE is obtained from N 1(I HE) = −N 2(I HE). This equilibrium reduces the system dimensionality from 2D to one dimension (1D) due to the degeneracy of A, resulting in a single Jacobian eigenvalue λ HE = [N 1′(I HE) + N 2′(I HE)]I HE, where N m ′ = dN m /dI m . In contrast, for broken PT symmetry (|γ| > 1, Supplementary Note F), the equilibrium is determined by the relation N 1(I 1E)N 2(I 2E) = −κ 2 with the intensity ratio I 2E/I 1E = −N 1(I 1E)/N 2(I 2E), forming an inhomogeneous steady state (IHSS)  with I 1E ≠ I 2E. The eigenvalues of A are then achieved as λ ±E = [N 1(I 1E) + N 2(I 2E)]/2 + N 1′(I 1E)I 1E + N 2′(I 2E)I 2E ± ρ 1/2/2 (Supplementary Note F for ρ). With Eq. (3) and the equilibrium, the static phase difference condition θ(t, I 1, I 2) = θ s(I 1, I 2) near equilibria is proved in all phases of PT symmetry regardless of the value of |γ| (Supplementary Note G).
The stability analysis shows the critical difference between static and dynamical PT-symmetric systems. In the static PT-symmetric system, the system dimensionality (or Hilbert space) is different only at the “point” of the parameter space, which is called the exceptional point . However, the dynamical PT-symmetric system has “continuous” phases with different system dimensionalities: unbroken and broken phases, each with the eigenvalues λ HE and λ ±E.
2.2 Topological classification
As an example, we investigate the simple functions of N 1(I 1) = η 11 I 1 + η 10 and N 2(I 2) = η 20, which give analytical solutions (Supplementary Note H). In practical systems, η 11 describes two-photon absorption (TPA) or two-photon emission (TPE) and η 10,20 represents linear gain or loss. We assume κ ≥ 0 and a fixed value of η 11. We then classify the topological phases of the nonlinear photonic molecule by the topological equivalence of the dynamical trajectories in the 2D state space I 1–I 2 (i.e. the homeomorphism of phase portraits ). We focus on the system near a hyperbolic equilibrium, which does not have Jacobian eigenvalues on the imaginary axis. According to the Grobman–Hartman theorem , the phase portraits of the system near a hyperbolic equilibrium are locally topologically equivalent to those of its linearized system. The topology of this linearized system is quantized by the pair (n +, n −) , where n ± denotes the numbers of Jacobian eigenvalues with positive and negative real parts. Therefore, the pair (n +, n −) corresponds to the quantized “topological charge of dynamical systems”, analogous to the genus number of differential geometry  or Chern number of dispersion bands [1, 2, 27].
For general 2D systems near hyperbolic equilibria, three topological phases exist according to the phase portrait (Figure 1a), with their topologies characterized by (n +, n −) (Figure 1b): the S(0,2) stable phase, D(1,1) saddle phase, and U(2,0) unstable phase , analogous to negative, zero, and positive electric charges. Although the S and U phases are divided into the node (SN,UN) and focus (SF,UF) phases according to their detailed trajectories, the phases with the same (n +, n −) are topologically equivalent . Figure 1c and d show the topological classification in the parameter space η 10–η 20 for the example of the TPA (η 11 < 0) nonlinearity. The Jacobian eigenvalues are obtained for unbroken (Figure 1c) and broken (Figure 1d) PT symmetry, except for forbidden regions (gray color, I 1E,2E < 0, Supplementary Note I). For unbroken PT symmetry with |γ| ≤ 1 at equilibria, a single topological phase exists, the S(0,1) stable phase. In contrast, for broken PT symmetry with |γ| > 1 at equilibria, three topological phases of 2D systems exist, the S(0,2), U(2,0), and D(1,1) phases (Supplementary Note J). Although the Andronov–Hopf (AH) bifurcation occurs between the S and U phases (red dashed line in Figure 1d), the saddle-node bifurcation does not occur in this example because of the forbidden region (gray color). For the complete realization of the topological phases in Figure 1a and b, the use of the TPE (η 11 > 0) nonlinearity is also required (Supplementary Note K). The goal of the later sections is then the understanding of optical behaviors in each topological phase and their applications to optical signal processing (Figure 1e).
2.3 Topological protection against optical randomness
Similar to topological protections of the dispersion band [3, 27, 40], the phase portraits of optical states (I 1, I 2) in our nonlinear system in which the topology is characterized as (n +, n −) are also robust against perturbations, here, in the temporal domain. We verify the topologically protected dynamics by testing the robustness of the phase portraits to random light incidences and system perturbations. Of the various topological phases in Figure 1, we focus on two “stable” phases for practical applications: the S(0,1) and S(0,2) phases, each with unbroken and broken PT symmetry (Supplementary Note L for other topological phases).
Figure 2 shows the calculated state trajectories (I 1, I 2) with random initial conditions around the equilibrium (I 1E, I 2E). The initial fields are completely random in their phases and amplitudes. Although the detailed trajectories of the S(0,1) phase and S(0,2) phases are different, the phase portraits of both topological phases converge to their equilibrium (I 1E, I 2E) (red circles) regardless of the initial conditions. Furthermore, the stabilization of each topological phase S(0,1) and S(0,2) involves a different type of oscillation quenching phenomenon protected by PT symmetry: AD in the S(0,1) phase with the HSS (I 1E = I 2E = I HE)  in unbroken PT symmetry and OD in the S(0,2) phase with the IHSS (I 1E ≠ I 2E)  in broken PT symmetry. The slower and oscillatory AD convergence (Figure 2a) and faster and monotonic OD convergence (Figure 2b) originate from the inherent properties of PT symmetry: the different and identical real parts of the eigenvalues in unbroken and broken PT symmetry, respectively . However, because optical randomness does not affect the topology, the trajectories of each topological phase are eventually recovered.
2.4 Topological protection against system perturbations
In this section, we investigate topological protection against system perturbations. In topological photonics, topological protections enable backscattering-free wave transport despite a local deformation of the field profiles for spatial perturbations: a deformation in the lattice constants or rod radii in photonic crystals [40, 41]. As a temporal equivalent, topological protection in optical dynamics leads to the phase portrait that is immune to temporal perturbations, e.g. a time-varying linear gain or loss η 10,20(t), despite a local deformation of the phase portraits.
Figure 3 presents the phase portraits of the stable phases S(0,1) and S(0,2) with an example of random perturbations in η 10(t) and η 20(t) in the temporal domain (black solid lines, for over 30% of maximum errors in η 10(t) and over 70% of maximum errors in η 20(t) in this example). The temporal variations in the system parameters result in a deformation of the local phase portraits of the optical intensities (I 1, I 2), similar to the deformed field profiles near the spatial defects in backscattering-free transport examples [40, 41]. However, the convergences to the equilibria (red circles) of the stable phases are topologically protected and eventually lead to AD and OD, as long as the S(0,1) and S(0,2) phases (Figure 1c and d) are maintained with the randomly perturbed η 10(t) and η 20(t), respectively. This result suggests that the systematic laser stabilization based on oscillation quenching mechanisms can be achieved with the coupling of a parasitic nonlinear resonator (here, resonator 2) to a lasing gain resonator (here, resonator 1).
Notably, the time-varying system perturbation with η 10(t) and η 20(t) results in the dynamical evolution of the equilibrium point (I 1E, I 2E), while preserving topological protection to each altered equilibrium. Considering the convergence time to equilibria (Figure 2), the deviation from the initial equilibrium depends on the time scale of the system perturbations, though the convergence to the original optical state is eventually achieved when the initial system parameter is recovered (see Supplementary Notes M and N for detailed analysis). In terms of noise suppression, this result shows that better performance will be achieved for faster system perturbations as discussed later.
3.1 Noise-immune signal modulation
Analogous to the effect of a negative charge on an electric field, the topology of the stable phases leads to the convergence of the optical states (I 1, I 2) to the equilibrium (I 1E, I 2E). This topologically protected convergence against random light incidences and system perturbations allows equilibrium-based, noise-immune signal processing in the temporal domain, such as noise-immune laser modulation (Figure 4) and rectification.
Firstly, we show noise-immune laser modulation by exploiting the S(0,2) phase that produces OD. The OD modulation is obtained by controlling a linear gain or loss parameter η 20 (Figure 4a and yellow dashed arrows in Figure 1c and d), which can be achieved with optical or electrical pumping to quantum dot films  or graphene layers . The variation in η 20 leads to a gradual variation in (I 1E, I 2E) (Figure S3b and c in Supplementary Note I), which allows noise-immune modulation of the laser outputs |S 1−|2 and |S 2−|2, each determined by I 1E(η 20(t)) and I 2E(η 20(t)) (Supplementary Note O for the TCMT model). Figure 4b–g presents the results of the modulation. When we set the spectral noise component in η 20(t) (Figure 4b–d, Supplementary Note P for details), evident suppression of the noise component in η 20(t) is observed in Figure 4f and g for both |S 1−|2 and |S 2−|2 from the topologically protected convergence to equilibrium and the η 20-dependent gradual variation in (I 1E, I 2E). We note that the noise suppression in the output is a temporal equivalent of the suppression of the defect-induced local field perturbation in conventional topological structures [2, 27, 40, 41].
In Supplementary Note Q, we examine the effect of noise rates on noise-immune signal processing performance, which was previously discussed in Section 2.4. The results agree with the prediction based on the results of Supplementary Note N, demonstrating better noise suppression for faster system perturbations.
3.2 Noise-immune optical rectification
We now show another example with a higher-level functionality using a dynamical transition between AD and OD: a noise-immune half-wave rectifier utilizing the different nature of each oscillation quenching mechanism. The platform for this application is shown in Figure 5a. The coupled-resonator laser consists of a nonlinear photonic molecule in which each resonator is coupled to a waveguide with a lifetime τ W. The nonlinearity functions are thus transformed into N 1(I 1) = η 11 I 1 + η 10 − 1/τ W and N 2(I 2) = η 20 − 1/τ W due to radiation loss (Supplementary Note O for the TCMT model). The input signal is set to be the control of a linear gain or loss parameter η 20, and the output signal is defined as |S 1−|2 − |S 2−|2, the difference between the powers of the outgoing waves through waveguides 1 and 2.
The input signal η 20(t) changes the topological phases between two regimes (black dashed arrows in Figure 1c and d): the AD regime (orange parts) protected by unbroken PT symmetry for the S(0,1) phase and the OD regime (blue parts) protected by broken PT symmetry for the S(0,2) phase. AD and OD then lead to distinct output signals; the output signal |S 1−|2 − |S 2−|2 becomes zero for all regimes of AD and achieves a continuous variation in the OD regime due to the gradual change (I 1E, I 2E). From these contrasting modulations of |S 1−|2 − |S 2−|2, a dynamical transition between the digital AD operation and the analog OD operation, which also accompanies the switching between 1D and 2D dynamics, constitutes the rectification of η 20(t). We note that the half-wave rectification in Figure 5a is accompanied by noise immunity according to the topologically protected convergence to the equilibrium (I 1E, I 2E). For the spectral noise component in η 20(t) (Figure 5b–d, Supplementary Note P), the operation of this application is presented in Figure 5b–g for different noise levels. As expected for faster OD convergence (Figure 2b), the analog OD modulation exhibits superior noise suppression.
In conclusion, we theoretically studied the topological nature of nonlinear optical dynamics with PT symmetry, which is manifested by the topological invariance of the trajectory in the optical state space. For generic intensity-dependent nonlinearities in coupled photonic systems, we revealed the crucial link between the topological phases of nonlinear optical dynamics and PT symmetry. Along with the topological protection of each phase in 1D and 2D dynamics, we demonstrated two representative oscillation quenching mechanisms, AD and OD, which are protected by unbroken and broken PT symmetry, respectively. We also showed that topologically stable AD and OD phases allow noise-immune signal processing, such as the combination of digital AD and analog OD operations for rectification. We note that the AD and OD quenching mechanisms in our coupled-resonator platform are achieved without the assistance of phase delay lines, in sharp contrast to previous realizations of AD and OD in general dynamical systems [34, 35].
When compared with traditional laser stabilizations, such as the injection locking method , AD and OD phases show unique mechanisms and phenomena originating from PT symmetry. In conventional injection locking, the slave laser is stabilized by the unidirectional effect from the injection by the master laser . The influence from the slave laser to the master laser should be prohibited through optical isolators because this effect may lead to an unstable injection. However, the observed quenching mechanisms in our work are necessarily based on the mutual interactions between gain and loss nonlinear resonators. This mutual interaction enables the simultaneous and synchronized stabilization of both resonators with the perfectly equal intensity (AD phase) or designed different intensities (OD phase). Therefore, our example, demonstrating the emergence of oscillation quenching mechanisms [34, 35] in coupled optical systems, will provide new design freedom distinct from traditional injection locking methods.
The stabilization based on AD and OD phenomena is also distinct from another example of stabilization in optics: temporal solitons . Although there are some similarities between these phenomena in terms of achieving the balance between two different physical origins – the balance between nonlinear gain/loss and evanescent coupling in AD and OD phenomena, and the balance between nonlinearity and dispersion in temporal solitons – the topological nature imposes uniqueness on AD and OD phenomena. As shown in topological phases in 2D dynamical systems, AD and OD phenomena originate from the quantized states of system dynamical trajectories, in sharp contrast to the spectrally continuous compensation of the dispersive properties in temporal solitons. The quantization in AD and OD phenomena leads to the robustness against continuous deformation as intensively studied in topological photonics.
The dynamical system studied here is based on the intensity-dependent nonlinearity, which is described by an abstract nonlinearity function N(I). A variety of saturable responses [29, 45] and multiphoton processes [37, 46] such as TPA  or three-photon absorption (3PA ) and TPE  or three-photon emission (3PE ) provide multifaceted DOF for achieving distinctive topologically protected state trajectories in nonlinear optical dynamics. The existing high-quality-factor waveguide [47, 48] or resonator [29, 45] elements with intensity-dependent nonlinearities and their coupled systems can be applied to observe our result in the spatial or temporal domain. We can also envisage the realization of composite materials that can provide a more complex form of nonlinearity function N(I). In this case, applying a numerical approach rather than theoretical analysis will be necessary to examine the stable regime in dynamical systems. To handle such highly complex and nonlinear problems, the data-driven method such as the deep learning approach  may be useful by exploiting the dataset generated by numerical simulations.
In terms of the extension of the topological charge into general dynamical systems, the achieved noise immunity against temporal perturbations originates from the topological nature of state trajectories, which are robust to the deformation of the system as similar to backscattering-free propagation in topological band theory or other topological phenomena. This result also suggests a new systematic laser stabilization methodology: the coupling of a parasitic nonlinear resonator to a lasing gain resonator, which will lead to oscillation quenching mechanisms. With the newly found PT-symmetry-protected oscillation quenching mechanisms, we expect that our platform will have significant implications across dynamical optical devices, such as noise-immune optical memory, electro-optical logic gates, and the dynamical control of chiral eigenmodes at the exceptional point . The construction and utilization of oscillation quenching phenomena in many-body systems  will also be expected in terms of realizing active functionalities for photonic neural networks.
Funding source: National Research Foundation of Korea
Award Identifier / Grant number: 2014M3A6B3063708
Award Identifier / Grant number: 2021R1C1C1005031
Award Identifier / Grant number: 2021R1A4A3032027
Author contribution: S. Y. and N. P. conceived the presented idea. S. Y. and X. P. developed the theory and performed the computations. N. P. encouraged S. Y. to investigate nonlinear dynamics for topological photonics while supervising the findings of this work. All authors discussed the results and contributed to the final manuscript.
Research funding: This work was supported by the National Research Foundation of Korea (NRF) through the Global Frontier Program (GFP, 2014M3A6B3063708) funded by the Korean government. S. Yu was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2021R1C1C1005031 and No. 2021R1A4A3032027).
Conflict of interest statement: The authors declare no competing financial interest.
 T. Ozawa, H. M. Price, A. Amo, et al.., “Topological photonics,” Rev. Mod. Phys., vol. 91, 2018, Art no. 015006. Search in Google Scholar
 E. Lustig, S. Weimann, Y. Plotnik, et al.., “Photonic topological insulator in synthetic dimensions,” Nature, vol. 567, pp. 356–360, 2019, https://doi.org/10.1038/s41586-019-0943-7. Search in Google Scholar
 A. V. Poshakinskiy, J. Zhong, Y. Ke, et al.., “Quantum Hall phases emerging from atom–photon interactions,” npj Quantum Inf., vol. 7, pp. 1–8, 2021, https://doi.org/10.1038/s41534-021-00372-8. Search in Google Scholar
 D. Leykam and Y. D. Chong, “Edge solitons in nonlinear-photonic topological insulators,” Phys. Rev. Lett., vol. 117, p. 143901, 2016, https://doi.org/10.1103/physrevlett.117.143901. Search in Google Scholar
 L. He, Z. Addison, J. Jin, E. J. Mele, S. G. Johnson, and B. Zhen, “Floquet Chern insulators of light,” Nat. Commun., vol. 10, pp. 1–6, 2019, https://doi.org/10.1038/s41467-019-12231-4. Search in Google Scholar
 J. Lu, L. He, Z. Addison, E. J. Mele, and B. Zhen, “Floquet topological phases in one-dimensional nonlinear photonic crystals,” Phys. Rev. Lett., vol. 126, p. 113901, 2021, https://doi.org/10.1103/physrevlett.126.113901. Search in Google Scholar
 L. J. Maczewsky, M. Heinrich, M. Kremer, et al.., “Nonlinearity-induced photonic topological insulator,” Science, vol. 370, pp. 701–704, 2020, https://doi.org/10.1126/science.abd2033. Search in Google Scholar
 S. Mukherjee and M. C. Rechtsman, “Observation of Floquet solitons in a topological bandgap,” Science, vol. 368, pp. 856–859, 2020, https://doi.org/10.1126/science.aba8725. Search in Google Scholar
 J. R. Munkres, Topology: A First Course, vol. 23, Englewood Cliffs, NJ, Prentice-Hall, 1975. Search in Google Scholar
 H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science, vol. 336, pp. 205–209, 2012, https://doi.org/10.1126/science.1219171. Search in Google Scholar
 X. Piao, S. Yu, and N. Park, “Design of transverse spinning of light with globally unique handedness,” Phys. Rev. Lett., vol. 120, p. 203901, 2018, https://doi.org/10.1103/physrevlett.120.203901. Search in Google Scholar
 I. Lifshitz, “Anomalies of electron characteristics of a metal in the high pressure region,” Sov. Phys. JETP, vol. 11, pp. 1130–1135, 1960. Search in Google Scholar
 A. J. Kollár, M. Fitzpatrick, and A. A. Houck, “Hyperbolic lattices in circuit quantum electrodynamics,” Nature, vol. 571, pp. 45–50, 2019, https://doi.org/10.1038/s41586-019-1348-3. Search in Google Scholar
 K. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-kerr nonlinearity,” Phys. Rev. Lett., vol. 121, p. 203602, 2018, https://doi.org/10.1103/physrevlett.121.203602. Search in Google Scholar
 Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, vol. 112, Berlin, Germany, Springer Science & Business Media, 2013. Search in Google Scholar
 R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys., vol. 14, pp. 11–19, 2018, https://doi.org/10.1038/nphys4323. Search in Google Scholar
 J. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos, vol. 111, 4th ed. International Publishing AG, Springer, 2012. Search in Google Scholar
 T. Simpson, J. Liu, A. Gavrielides, V. Kovanis, and P. Alsing, “Period‐doubling route to chaos in a semiconductor laser subject to optical injection,” Appl. Phys. Lett., vol. 64, pp. 3539–3541, 1994, https://doi.org/10.1063/1.111218. Search in Google Scholar
 V. Kovanis, A. Gavrielides, T. Simpson, and J.-M. Liu, “Instabilities and chaos in optically injected semiconductor lasers,” Appl. Phys. Lett., vol. 67, pp. 2780–2782, 1995, https://doi.org/10.1063/1.114591. Search in Google Scholar
 A. Hohl, A. Gavrielides, T. Erneux, and V. Kovanis, “Localized synchronization in two coupled nonidentical semiconductor lasers,” Phys. Rev. Lett., vol. 78, p. 4745, 1997, https://doi.org/10.1103/physrevlett.78.4745. Search in Google Scholar
 L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photonics, vol. 11, pp. 752–762, 2017, https://doi.org/10.1038/s41566-017-0031-1. Search in Google Scholar
 A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A, vol. 82, 2010, Art no. 043818, https://doi.org/10.1103/physreva.82.043818. Search in Google Scholar
 L. Chang, X. Jiang, S. Hua, et al.., “Parity–time symmetry and variable optical isolation in active–passive-coupled microresonators,” Nat. Photonics, vol. 8, pp. 524–529, 2014, https://doi.org/10.1038/nphoton.2014.133. Search in Google Scholar
 A. E. Miroshnichenko, B. A. Malomed, and Y. S. Kivshar, “Nonlinearly PT-symmetric systems: spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A, vol. 84, 2011, Art no. 012123, https://doi.org/10.1103/physreva.84.012123. Search in Google Scholar
 M. H. Teimourpour, L. Ge, D. N. Christodoulides, and R. El-Ganainy, “Non-Hermitian engineering of single mode two dimensional laser arrays,” Sci. Rep., vol. 6, 2016, https://doi.org/10.1038/srep33253. Search in Google Scholar
 S. Yu, X. Piao, and N. Park, “Neuromorphic functions of light in parity‐time‐symmetric systems,” Adv. Sci., vol. 6, p. 1900771, 2019, https://doi.org/10.1002/advs.201900771. Search in Google Scholar
 S. Malzard, E. Cancellieri, and H. Schomerus, “Topological dynamics and excitations in lasers and condensates with saturable gain or loss,” Opt. Express, vol. 26, pp. 22506–22518, 2018, https://doi.org/10.1364/oe.26.022506. Search in Google Scholar
 G. Saxena, A. Prasad, and R. Ramaswamy, “Amplitude death: the emergence of stationarity in coupled nonlinear systems,” Phys. Rep., vol. 521, pp. 205–228, 2012, https://doi.org/10.1016/j.physrep.2012.09.003. Search in Google Scholar
 A. Koseska, E. Volkov, and J. Kurths, “Oscillation quenching mechanisms: amplitude vs. oscillation death,” Phys. Rep., vol. 531, pp. 173–199, 2013, https://doi.org/10.1016/j.physrep.2013.06.001. Search in Google Scholar
 H. A. Haus, Waves and Fields in Optoelectronics, vol. 464, Englewood Cliffs, NJ, Prentice-Hall, 1984. Search in Google Scholar
 S. Yu, X. Piao, and N. Park, “Bohmian photonics for independent control of the phase and amplitude of waves,” Phys. Rev. Lett., vol. 120, p. 193902, 2018, https://doi.org/10.1103/physrevlett.120.193902. Search in Google Scholar
 F. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett., vol. 100, 2008, Art no. 013904, https://doi.org/10.1103/PhysRevLett.100.013904. Search in Google Scholar
 D. Pacifici, H. J. Lezec, and H. A. Atwater, “All-optical modulation by plasmonic excitation of CdSe quantum dots,” Nat. Photonics, vol. 1, pp. 402–406, 2007, https://doi.org/10.1038/nphoton.2007.95. Search in Google Scholar
 Y. Zhang, Y. Feng, B. Zhu, J. Zhao, and T. Jiang, “Graphene based tunable metamaterial absorber and polarization modulation in terahertz frequency,” Opt. Express, vol. 22, pp. 22743–22752, 2014, https://doi.org/10.1364/oe.22.022743. Search in Google Scholar
 T. Herr, V. Brasch, J. D. Jost, et al.., “Temporal solitons in optical microresonators,” Nat. Photonics, vol. 8, pp. 145–152, 2014, https://doi.org/10.1038/nphoton.2013.343. Search in Google Scholar
 H. K. Tsang, C. Wong, T. Liang, et al.., “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett., vol. 80, pp. 416–418, 2002, https://doi.org/10.1063/1.1435801. Search in Google Scholar
 Z. Wang, H. Liu, N. Huang, Q. Sun, J. Wen, and X. Li, “Influence of three-photon absorption on mid-infrared cross-phase modulation in silicon-on-sapphire waveguides,” Opt. Express, vol. 21, pp. 1840–1848, 2013, https://doi.org/10.1364/oe.21.001840. Search in Google Scholar
 A. Nevet, N. Berkovitch, A. Hayat, et al.., “Plasmonic nanoantennas for broad-band enhancement of two-photon emission from semiconductors,” Nano Lett., vol. 10, pp. 1848–1852, 2010, https://doi.org/10.1021/nl1005806. Search in Google Scholar
 S. Yu, X. Piao, and N. Park, “Machine learning identifies scale-free properties in disordered materials,” Nat. Commun., vol. 11, p. 4842, 2020, https://doi.org/10.1038/s41467-020-18653-9. Search in Google Scholar
 S. Yu, H. S. Park, X. Piao, B. Min, and N. Park, “Low-dimensional optical chirality in complex potentials,” Optica, vol. 3, p. 1025, 2016, https://doi.org/10.1364/optica.3.001025. Search in Google Scholar
 S. Yu, C.-W. Qiu, Y. Chong, S. Torquato, and N. Park, “Engineered disorder in photonics,” Nat. Rev. Mater., vol. 6, pp. 226–243, 2020, https://doi.org/10.1038/s41578-020-00263-y. Search in Google Scholar
The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph-2021-0215).
© 2021 Sunkyu Yu et al., published by De Gruyter, Berlin/Boston
This work is licensed under the Creative Commons Attribution 4.0 International License.