Abstract
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 nonHermitian 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 PTsymmetric 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 noiseimmune signal processing.
1 Introduction
Topological degrees of freedom (DOF) have provided a new phase of matter, including quantized bulk conductance and topological insulators [1]. The similarity between the Schrödinger equation and Maxwell’s equations has also stimulated the birth of topological photonics [2]. One of the major goals in this field is to realize a photonic analogy of the quantum Hall family and topological insulators, which enables defectimmune wave propagations. While topological photonics has been extended to synthetic dimensions [3], nonHermitian photonics [4], atom–photon interactions [5], and second and thirdorder optical nonlinearities [6], [7], [8], [9], [10], 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 welldefined parameter spaces [11]. 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 [14], which enabled the discovery of hyperbolic materials [12] and pure transverse spins [13]. 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 [17] or timevarying media [18], 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 [19]. 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 [19]. 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 nonHermitian photonics [20] to nonlinear or timevarying platforms. So far, numerous studies on nonlinear dynamics have been conducted for specific nonHermitian systems. For example, a laser is a traditional nonHermitian platform with nonlinearity, which enables chaos, injection locking, and synchronization in amplification [21], [22], [23], [24]. The recent developments in parity–time (PT) symmetry [25, 26] and topological physics [2, 27] have established new design freedom in nonlinear and nonHermitian optical dynamics: suppressed time reversals [28], optical isolation [29], amplified Fano resonances [30], singlemode lasing [31], quenching behaviors [32] in nonlinear PTsymmetric systems, and topological zero modes in Su–Schrieffer–Heeger chains [33]. 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 PTsymmetric dynamics.
In this paper, we show that the interpretation of PTsymmetric dynamics with the topological picture provides new design freedom for noiseimmune signal processing. We introduce the topological classification of nonlinear optical dynamics in PTsymmetric systems by employing dynamical systems theory [19]. We show that the topological class defined by optical state trajectories is classified by PT symmetry [20], revealing PTsymmetryprotected oscillation quenching mechanisms: amplitude death (AD) [34] and oscillation death (OD) [35]. 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: noiseimmune laser modulation and rectification. Our results can be readily implemented with electric circuits and acoustics.
2 Results
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 intensitydependent gain or loss. The photonic molecule is then described by temporal coupledmode theory (TCMT) [36] as
where a _{ m } and N _{ m } represent the field amplitude and realvalued nonlinearity function of the mth resonator (m = 1, 2), respectively. The abstract forms of N _{1,2} represent intensitydependent nonlinearities (Supplementary Note A): multiphoton processes [37], saturable responses [29], and their combinations, increasing the possible design freedom. Although Eq. (1) and its simplified form have been studied [22], [23], [24, 28], [29], [30], [31], [32], [33], 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 [38]
where I _{ m } and φ _{ m } are realvalued intensity and phase functions, respectively, and θ = φ _{1} − φ _{2} is the timevarying phase difference between each resonator field. Although φ _{1,2}(t) and θ(t) are determined by coupled timederivative equations (Supplementary Note B), at this stage, we consider the steadystate solution with static I _{1,2} and θ(t, I _{1}, I _{2}) = θ _{s}(I _{1}, I _{2}) near equilibria [19]. This synchronization around equilibria will be discussed later.
With the synchronization, we assign da _{ m }/dt = iωa _{ m } with static N _{ m }(I _{ m }). The static function θ _{s}(I _{1}, I _{2}) for all eigenmodes of Eq. (1) then satisfies (Supplementary Note C)
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 [20], 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 [19]. The equilibrium (I _{1E}, I _{2E}) in the twodimensional (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 [20].
The stability of the equilibrium (I _{1E}, I _{2E}) is examined by the first Lyapunov criterion [19] 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) [34] 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) [35] 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 PTsymmetric systems. In the static PTsymmetric system, the system dimensionality (or Hilbert space) is different only at the “point” of the parameter space, which is called the exceptional point [25]. However, the dynamical PTsymmetric 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 twophoton absorption (TPA) or twophoton 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 [19]). 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 [39], 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 _{−}) [19], 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 [11] 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 [19], analogous to negative, zero, and positive electric charges. Although the S and U phases are divided into the node (S^{N},U^{N}) and focus (S^{F},U^{F}) phases according to their detailed trajectories, the phases with the same (n _{+}, n _{−}) are topologically equivalent [19]. 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 saddlenode 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).
Figure 1:
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}) [34] in unbroken PT symmetry and OD in the S(0,2) phase with the IHSS (I _{1E} ≠ I _{2E}) [35] 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 [20]. However, because optical randomness does not affect the topology, the trajectories of each topological phase are eventually recovered.
Figure 2:
2.4 Topological protection against system perturbations
In this section, we investigate topological protection against system perturbations. In topological photonics, topological protections enable backscatteringfree 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 timevarying 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 backscatteringfree 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).
Figure 3:
Notably, the timevarying 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 Applications
3.1 Noiseimmune 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 equilibriumbased, noiseimmune signal processing in the temporal domain, such as noiseimmune laser modulation (Figure 4) and rectification.
Figure 4:
Figure 5:
Firstly, we show noiseimmune 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 [42] or graphene layers [43]. 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 noiseimmune 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 defectinduced local field perturbation in conventional topological structures [2, 27, 40, 41].
In Supplementary Note Q, we examine the effect of noise rates on noiseimmune 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 Noiseimmune optical rectification
We now show another example with a higherlevel functionality using a dynamical transition between AD and OD: a noiseimmune halfwave rectifier utilizing the different nature of each oscillation quenching mechanism. The platform for this application is shown in Figure 5a. The coupledresonator 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 halfwave 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.
4 Conclusion
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 intensitydependent 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 noiseimmune 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 coupledresonator 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 [21], 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 [21]. 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 [44]. 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 intensitydependent 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 [47] or threephoton absorption (3PA [48]) and TPE [49] or threephoton emission (3PE [50]) provide multifaceted DOF for achieving distinctive topologically protected state trajectories in nonlinear optical dynamics. The existing highqualityfactor waveguide [47, 48] or resonator [29, 45] elements with intensitydependent 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 datadriven method such as the deep learning approach [51] 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 backscatteringfree 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 PTsymmetryprotected oscillation quenching mechanisms, we expect that our platform will have significant implications across dynamical optical devices, such as noiseimmune optical memory, electrooptical logic gates, and the dynamical control of chiral eigenmodes at the exceptional point [52]. The construction and utilization of oscillation quenching phenomena in manybody systems [53] 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.
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