Topological protection in nonlinear optical dynamics with parity-time symmetry

Topological phases exhibit properties that are conserved for continuous deformations, as demonstrated in topological protections in condensed-matter physics and electromagnetic waves. Despite its ubiquitous nature and recent extensions to synthetic dimensions, non-Hermitian Hamiltonians, and nonlinear dynamics, topological protection has generally been described in spatial lattices with the Chern number in the Brillouin zone, focusing on the realization of backscattering-free wave transport. Here, we investigate a different class of topological protection in parity-time-symmetric nonlinear optical dynamics, exploiting the topological invariance of optical state trajectories. For coupled nonlinear photonic systems composed of gain and loss atoms, we classify the topology of equilibria separately for unbroken and broken parity-time symmetry. Utilizing the immunity of topological phases against temporal perturbations, we develop noise-immune laser modulation and rectification with a parasitic nonlinear resonator based on oscillation quenching mechanisms that are protected by parity-time symmetry. The connection between topological photonics and parity-time symmetry through nonlinear dynamics provides a powerful platform for noise-immune signal processing.


Introduction
The topological degree of freedom in band theory has provided a new phase of matter, such as quantized bulk conductance and topologically protected edge states 1 .The similarity between the Schrödinger equation and Maxwell's wave equations has also stimulated the birth of topological photonics 2,3 .One of the major goals in this field is to realize a photonic analogy of topological phenomena in condensed-matter physics, which enables backscattering-free light propagation 4,5 .Topological photonics has recently been extended to synthetic dimensions 6 , non-Hermitian photonics 7 , and optical nonlinearities 8,9 .All of these efforts share a common definition of topology: the topological nature of optical wavefunctions in the dispersion band 1,2 .
However, given the ubiquitous property of topology 10 , we need to explore other classes of topological invariants in photonics, which are defined for optical quantities or phenomena rather than band structures.An important example is the topology of an isofrequency surface and its phase transition 11,12 as the optical equivalent of the Lifshitz transition for a Fermi surface 13 .This type of topology enabled the discovery of hyperbolic materials that have provided new design freedom in metamaterials 14 .When considering recent interest in optical dynamics using nonlinearities 15 or time-varying media 16 , we also expect the utilization of a certain type of topological invariants in optical dynamics, which will offer further design freedom for wave devices in the temporal domain.
It is well known that the topological equivalence between different dynamical systems is defined by the existence of an invertible map between the state trajectories of each system, i.e., the homeomorphism of phase portraits 17 .Because the topology of the state trajectory of a dynamical system is closely related to the energy exchange between the system and its environment, topological phases in optical dynamics should have a natural connection with non-Hermitian photonics 18 .So far, numerous studies on nonlinear optical dynamics have been conducted for specific types of optical nonlinearities in non-Hermitian systems.For example, a lasing platform, which is a traditional non-Hermitian system with inherent nonlinearity in amplification, has been extensively studied in terms of nonlinear optical dynamics in order to examine chaos, instability, and synchronization in lasing dynamics [19][20][21] .The recent developments in parity-time (PT) symmetry and topological physics have established new design freedom in nonlinear optical dynamics: the suppression of time reversals 22 , optical isolation 23 , amplified Fano resonances 24 , single-mode lasing [25][26][27] , and quenching behaviors 28 in nonlinear PTsymmetric systems, and topological zero modes in Su-Schrieffer-Heeger chains 29 .However, despite these various achievements, we emphasize that a common and essential analysis for general dynamical systems-topological classification and protection of state trajectories 17 -are still absent in nonlinear PT-symmetric dynamics.
In this paper, we study nonlinear optical dynamics in PT-symmetric systems in terms of topological protection by employing dynamical system theory 17 and thus achieve noise-immune signal processing using topologically protected phases.From the topological class defined by the topological equivalence of optical state trajectories, we show that the topological classification of nonlinear optical dynamics is determined by PT symmetry 18 .This classification also reveals PTsymmetry-protected oscillation quenching mechanisms: amplitude death (AD) 30 and oscillation death (OD) 31 without the assistance of phase delay lines, in sharp contrast to previous approaches 30,31 .Using topological immunity against temporal perturbations, two application examples are also presented: noise-immune laser modulation and rectification.The extension of topological protection into non-Hermitian photonics and nonlinear dynamics can be readily implemented with other platforms, such as electric circuits and acoustics.

Results
Model definition.Let us consider a generic photonic system coupled to an external reservoir, such as a photonic molecule consisting of two coupled nonlinear resonators each with the same resonance frequency ω0 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 described by temporal coupled mode theory (TCMT) 32 as ( ) ( ) where am and Nm represent the field amplitude and real-valued nonlinearity function of the m th resonator (m = 1, 2), respectively.The abstract forms of N1,2 represent universal intensitydependent nonlinearities (Supplementary Note S1): multiphoton processes 33 , saturable responses 23 , and their arbitrary combinations, dramatically increasing the possible design freedom.We note that although Eq. ( 1) and its similar form has been widely studied [19][20][21][22][23][24][25][26][27][28][29] , the topological classification of optical state trajectories and their protections have not been considered.
To analyze the evolution of the optical energy, we derive an equation for the intensity with am = Im 1/2 exp(iφm) as 34 where Im and φm are real-valued intensity and phase functions, respectively, and θ = φ1 -φ2 is the time-varying phase difference between each resonator field.
The upper and lower conditions in Eq. ( 3) correspond to unbroken and broken PT symmetry 18 , respectively.

Stability analysis.
To define the topology of our system, we explore the equilibria in Eq. ( 2) and examine their stability 17 .The equilibrium (I1E,I2E) in the two-dimensional (2D) state space I1-I2 is obtained with dI1,2/dt = 0 in Eq. ( 2), which results in with the static function θs(I1E,I2E) at equilibrium.Although we initially assign arbitrary nonlinearities, the first equality in Eq. ( 4) shows that N1(I1E) and N2(I2E) should have different signs for a nontrivial equilibrium I1E,2E > 0, which exhibits the connection between nonlinear optical dynamics and PT symmetry with the gain-loss configuration 18 .
The stability of the equilibrium (I1E,I2E) is examined by the first Lyapunov criterion 17 in which the eigenvalues of the Jacobian matrix of Eq. ( 2) are used (Supplementary Note S4 for the Jacobian matrix A).We note that Eq. (3) provides a separate analysis of each phase of PT symmetry.For the case of unbroken PT symmetry (|γ| ≤ 1, Supplementary Note S5), the system is in the homogeneous steady state (HSS) 30

Topological classification.
As an example, we investigate the simple functions of N1(I1) = η11I1 + η10 and N2(I2) = η20, which gives analytical solutions (Supplementary Note S8).In practical systems, η11 describes two-photon absorption (TPA) or emission (TPE), and η10,20 represents linear gain or loss.We also assume κ ≥ 0 and a fixed value of η11.We classify the topological phases of the nonlinear photonic molecule by the topological equivalence of the dynamical trajectories in the 2D state space I1-I2 (i.e., the homeomorphism of phase portraits 17 ).According to the Grobman-Hartman theorem 35 , the phase portraits of the system near a hyperbolic equilibrium, which does not have Jacobian eigenvalues on the imaginary axis, are locally topologically equivalent to those of its linearized system.Because the topology of this linearized system is quantized by the pair (n+,n-) 17 , where n± denote the numbers of Jacobian eigenvalues with positive and negative real parts, we use (n+,n-) as the quantized "topological charge of dynamical systems", analogous to the genus number 10 or Chern number [1][2][3] .For general 2D systems near hyperbolic equilibria, three topological phases exist according to the phase portrait (Fig. 1a), with their topologies characterized by (n+,n-) (Fig. 1b): the S(0,2) stable phase, D(1,1) saddle phase, and U(2,0) unstable phase 17 , analogous to negative, zero, and positive electric charges.Although the S and U phases are divided into 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 17 .
Figure 1c and d shows the topological classification in the parameter space η10-η20 for the example of the TPA (η11 < 0) nonlinearity.The Jacobian eigenvalues are obtained for unbroken (Fig. 1c) and broken (Fig. 1d) PT symmetry, except for forbidden regions (gray color, I1E,2E < 0, Supplementary Note S9).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 S10).Although the Andronov-Hopf (AH) bifurcation occurs between the S and U phases (red dashed line in Fig. 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 Fig. 1a and b, the use of the TPE (η11 > 0) nonlinearity is also required (Supplementary Note S11).
Topological protection against optical randomness.Similar to topological protections of the dispersion band 2,5,6 , the phase portraits of optical states (I1,I2) in our nonlinear system in which the topology is characterized as (n+,n-) are also robust against temporal perturbations.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 Fig. 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 S12 for other topological phases).
Figure 2 shows the calculated state trajectories (I1,I2) with random initial conditions around the equilibrium (I1E,I2E).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 (I1E,I2E) (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: amplitude death (AD) in the S(0,1) phase with the HSS (I1E = I2E = IHE) 30 in unbroken PT symmetry, and oscillation death (OD) in the S(0,2) phase with the IHSS (I1E ≠ I2E) 31 in broken PT symmetry.We note that the slower and oscillatory AD convergence (Fig. 2a) and faster and monotonic OD convergence (Fig. 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 18 .
Topological protection against system perturbations.Next, we investigate topological protection against system perturbations.In conventional topological photonics, topological protections of the dispersion band enable backscattering-free wave transport despite a local deformation of the field profiles 4,5 for spatial perturbations in the system parameters such as a deformation in the lattice constants or rod radii in photonic crystals.As a temporal equivalent, topological protection in optical dynamics leads to a topology of phase portraits that is immune to temporal perturbations in the system parameters, e.g., a linear gain or loss in η10(t) and η20(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 random perturbations in η10(t) and η20(t) in the temporal domain (black solid lines, for over 30% maximum errors in η10(t) and over 70% maximum errors in η20(t) in this example).The temporal variations in the system parameters result in a deformation of the local phase portraits of the optical intensities (I1,I2), similar to the deformed field profiles near the spatial defects in backscattering-free transport examples 4,5 .However, the convergences to the equilibria (red circles) of the stable phases are topologically protected and eventually lead to AD and OD in the S(0,1) and S(0,2) phases, respectively.This result suggests that the systematic laser stabilization can be achieved with the coupling of a parasitic nonlinear resonator (here, resonator 2) to a lasing gain resonator (here, resonator 1), which leads to oscillation quenching mechanisms.
Noise-immune signal processing.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 (I1,I2) to the equilibrium (I1E,I2E).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 (Fig. 4) and rectification (Fig. 5).
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 η20 (Fig. 4a and yellow dashed arrows in Fig. 1c and d), which can be achieved with optical or electrical pumping to quantum dot films 36 or graphene layers 37 .The variation in η20 leads to a gradual variation in (I1E,I2E) (Fig. S3b and c in Supplementary Note S9), which allows noiseimmune modulation of the laser outputs |S1-| 2 and |S2-| 2, each determined by I1E(η20(t)) and 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][3][4][5] .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 Fig. 5a.

I2E(η20(t))
The coupled-resonator laser consists of a nonlinear photonic molecule in which each resonator is coupled to a waveguide with a lifetime τW.The nonlinearity functions are thus transformed into Such 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 η20(t).We note that the half-wave rectification in Fig. 5a is accompanied with noise immunity according to the topologically protected convergence to the equilibrium (I1E,I2E).For the spectral noise component in η20(t) (Fig. 5b-d, Supplementary Note S14), the operation of this application is presented in Fig. 5b-g for different noise levels.As expected for faster OD convergence (Fig. 2b), the analog OD modulation exhibits superior noise suppression.

Discussion
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 stable topological phases and the transition between them allow noise-immune signal processing achieved by AD and OD protected by PT symmetry, such as the combination of digital AD and analog OD operations for rectification.
In terms of the extension of the topological charge into general dynamical systems in an analogy to electrical charges, the achieved noise immunity against temporal perturbations is the dynamical equivalent of backscattering-free propagation.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 a noise-immune optical memory and electro-optical logic gates.

Temporal coupled mode theory model for a laser platform
The TCMT formulation for the platform of Figs. 4 and 5 in the main text is 32 ( ) ( ) 12 , where S1+,2+ denote the incident waves through each waveguide.The emitted waves S1-,2-from the nonlinear photonic molecule are obtained as The incident waves S1+,2+ are used for the initial excitation of the nonlinear photonic molecule.

Note S2. Phase equations
By replacing am with Im 1/2 exp(iφm) in Eq. ( 1) and employing Eq. ( 2) in the main text, we achieve sin , sin .Because the imaginary parts of Eq. (S1) are zero, an equation for the field phase φm inside each resonator is derived, as The time derivatives of the phase functions dφ1,2(t)/dt represent the instantaneous frequencies of the fields in resonators 1 and 2, correspondingly.The static condition of the phase difference θ(t) = φ1(t) -φ2(t) = θs is found in Eq. ( S2), which will be discussed later in Note S7.
The intensity value IHE of the equilibrium is then obtained from the specific mathematical forms

Note S7. Proof of static phase difference
In the analysis of the equilibrium and its stability, we employed the static phase difference condition θ(t,I1,I2) = θs(I1,I2) for separate analysis of the phases of PT symmetry.This condition requires dθ/dt = d(φ1 -φ2)/dt = 0. From Eq. (S2), we achieve ( ) Therefore, dθ/dt = 0 for the equilibria in all phases of PT symmetry, due to the conditions of (i) .
The Jacobian eigenvalues for the equilibrium are then achieved as

Note S9. Equilibria in the TPA example
Figure S3 shows the equilibria of unbroken (Fig. S3a, IHE) and broken (Fig. S3b and c, each for I1E and I2E) PT symmetry.In the parameter space η10-η20, the nontrivial equilibria exist except for the gray areas, which represent the "forbidden regions" of nontrivial equilibria.First, due to the necessary condition κ ≥ |η20| for unbroken PT symmetry, the equilibrium in Fig. S3a

Note S10. Jacobian eigenvalues of broken PT symmetry in the TPA example
Figure S4 shows the imaginary part of λ+E (Fig. S4a) and complex-valued λ-E (Fig. S4b and c) to provide all the information of λ±E for broken PT symmetry with Fig. 1d   The S and U phases in the region of η20 / κ < 0 can be further classified by the types of phase portraits (Fig. S5): node (S N and U N ) and focus (S F and U F ) phases, which are topologically equivalent [16] if the phases have the same (n+,n-).While node phases have

Note S12. Topologically protected dynamics in the saddle and unstable phases
The phase portraits in the saddle and unstable phases are presented in Fig. S8.While Fig. S8a shows the saddle phase dynamics obtained with the TPA resonator (η11 < 0), Fig. S8b and c represent the unstable dynamics obtained with TPE resonators (η11 > 0).Although both phases are unstable with respect to the nontrivial equilibrium, some of the initial states can converge to (I1E,I2E) = (0,0), depending on the topological phase of the trivial equilibrium (0,0).

Note S14. Noise in time-varying loss
In the time domain analysis in Figs. 4 and 5 in the main text, sinusoidal and square pulses including noise components are applied to test noise-immune laser rectification and modulation, respectively.For the target signal ftarget(t) (square pulse in Fig. 4b and sinusoidal pulse in Fig. 5b in the main text), we set the modulation input η20(t) by adding a random perturbation δ20(t) to ftarget(t) as η20(t) = ftarget(t) + δ20(t), where: Figure 4b-g presents the results of the modulation.When we set the spectral noise component in η20(t) (Fig. 4b-d, Note S15 for details), from the topologically protected convergence to equilibrium and the η20-dependent gradual variation in (I1E,I2E), evident suppression of the noise component in η20(t) is observed in Fig. 4f and g for both |S1-| 2 and |S2-| 2 .
τW and N2(I2) = η20 -1/τW due to radiation loss (Materials and methods for the TCMT model).The input signal is set to be the control of a linear gain or loss parameter η20, and the output signal is defined as |S1-| 2 -|S2-| 2 , the difference between the powers of the outgoing waves through waveguides 1 and 2.The input signal η20(t) changes the topological phases between two regimes (black dashed arrows in Fig.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: |S1-| 2 -|S2-| 2 = 0 for all regimes of AD, and the continuous change in (I1E,I2E) in the OD regime results in a modulation of |S1-| 2 -|S2-| 2 .

Fig. 1 .
Fig. 1.Topological classification of nonlinear photonic molecules.(a) Topological phases in general 2D dynamical systems determined by state trajectories.(b) Jacobian eigenvalues of each phase.An orange boundary between the S F and U F phases represents the AH bifurcation, while the green boundaries around the D(1,1) phase denote the saddle-node bifurcation.The black dashed line divides node and focus phases.(c,d) PT-symmetry-dependent phase diagrams.(c) λHE for unbroken PT symmetry.(d) Re[λ+E] for broken PT symmetry.The red dashed line denotes the AH bifurcation, and the black dashed lines represent the boundaries between the node and focus phases.The yellow and black dashed arrows in (c,d) represent the transitions for the devices in Figs. 4 and 5, respectively.A schematic of the nonlinear photonic molecules is shown in the third quadrant of (c,d).η11/κ = -0.5.

Figure 5 Supplementary
Figure 5 These responses are described by the nonlinearity function N(I) = η / [1 + (I/Is)], where η > 0 for saturable gain, η < 0 for saturable loss, and Is denotes the saturation intensity.On the other hand, multiphoton absorption and emission processes exhibit nonlinearity functions with polynomial expressions, as shown in the TPA and TPE with N(I) = ηI [5,7] and the 3PA and 3PE with N(I) = ηI 2 [6,8], where η > 0 for emission and η < 0 for absorption.Figure S1 shows the N(I) functions for different types of intensitydependent optical nonlinearities.

Fig. S1 .
Fig. S1.Nonlinearity functions N(I) of intensity-dependent optical nonlinearities: red lines for TPA and TPE with N(I) = ±η0I and blue lines for saturable gain and loss with N(I) = ±η0 / [1 + (I/Is)].The green dashed lines represent linear gain and loss with N(I) = ±η0.We set η0 > 0 for all cases.

Fig. S2 .
Fig. S2.The ratio between each resonator field near equilibria.(a,b) The ratio a1/a2 of two steady-state eigenmodes with various values of γ.The blue curve is for γ < 0, and the red curve is for γ ≥ 0.

Fig. S3 .
Fig. S3.Equilibria in the system parameter space η10-η20.(a) IHE for unbroken PT symmetry.(b) I1E and (c) I2E for broken PT symmetry.The gray areas denote the forbidden regions, originating from unbroken PT symmetry and nonnegative intensity values.η11/κ = -0.5 for all cases.

W
= [ωL,ωH] is the spectral bandwidth of the noise component, and u[p,q] is the uniform random function.The strength of the noise is then determined by the magnitude of δ (Fig.S10afor Figs.4c and 5c, and Fig.S10bfor Figs.4d and 5d).In both examples, ωL = 0.1ω0 and ωH = 0.2ω0.