Abstract
We investigate experimentally and theoretically a system ruled by an intricate interplay between topology, nonlinearity, and spontaneous symmetry breaking. The experiment is based on a twomode coherentlydriven optical resonator where photons interact through the Kerr nonlinearity. In presence of a phase defect, the modal structure acquires a synthetic Möbius topology enabling the realization of spontaneous symmetry breaking in inherently biasfree conditions without fine tuning of parameters. Rigorous statistical tests confirm the robustness of the underlying symmetry protection, which manifests itself by a periodic alternation of the modes reminiscent of perioddoubling. This dynamic also confers long term stability to various localized structures including domain walls, solitons, and breathers. Our findings are supported by an effective Hamiltonian model and have relevance to other systems of interacting bosons and to the Floquet engineering of quantum matter. They could also be beneficial to the implementation of coherent Ising machines.
Similar content being viewed by others
Introduction
Topology, nonlinearity, and spontaneous symmetry breaking are key concepts of modern physics. Topology is concerned with the study of properties that are invariant under continuous deformations, hence inherently robust against fluctuations and disorder^{1,2}. Nonlinearity is the source of complexity, underpinning attractors, chaos, selforganization, and selflocalization (solitons)^{3,4}. Spontaneous symmetry breaking (SSB) underlies much of the diversity observed in Nature^{5,6,7}, from ferromagnetism^{8} to embryo development^{9}. Here, we report on an experiment in which these three concepts are fundamentally intertwined. Nonlinearity and SSB combine to endow our system with a topological invariant; in return, that invariance protects the underlying symmetry, enabling a robust realization of SSB that is immune to perturbations.
The discovery of topological states of matter in solidstate materials has revealed the central role of topology in the classification of quantum matter^{1,10}. Besides, recent progress in designing and controlling synthetic lattice systems has allowed for the exploration of topological properties in a broad class of physical contexts^{11,12}, including ultracold gases^{11}, photonic crystals^{12,13}, mechanics^{14}, and systems with synthetic dimensions^{13,15}. Topological robustness has potential wideranging applications, from faulttolerant quantum computers^{16,17} to lasers capable of singlemode operation at high power^{18}. While topological band structures concern the properties of singleparticle Bloch states, interesting behaviors were observed by combining these with classical nonlinearities^{19}. This includes, nonexhaustively, edge solitons, nonlinearityinduced topological phase transitions^{20,21}, novel symmetryprotected phases^{22}, or quantized Thouless pumping of solitons^{23}.
The results presented in this Article demonstrate a new facet of the potent bridging of topology and nonlinearity and how it can confer robustness to the realization of SSB. Upon SSB, a system bifurcates to states with a broken symmetry even though the equations of motion retain that symmetry^{5,6,7}. Intense interest has been paid to this process in recent years, especially in optical microresonators where it has been studied, e.g., for optical memories^{24}, nonreciprocal propagation in integrated photonic circuits^{25,26}, dynamical control of laser directionality, chirality, and polarization^{27,28}, or sensors with divergent sensitivities^{29,30}. Experiments on SSB have also been used for fundamental studies of domain walls^{31}, of the universal unfolding of the pitchfork bifurcation^{32,33}, or of the Bose–Hubbard dimer model^{34}. These experiments most often rely on an exchange symmetry (\({{\mathbb{Z}}}_{2}\)) between two states. Invariably, however, this symmetry is affected by manufacturing imperfections, experimental biases, or other nonidealities. As a consequence, instead of a spontaneous, random selection between the two states, one state is statistically favored over the other^{33,35}, hindering the experiments. Here, we show how the combination of nonlinearity and topology can solve that problem. Specifically, we report on the experimental realization of an optical resonator with a synthetic modal structure characterized by a Möbius topology protected by the Kerr nonlinearity. That protection leads to an exact exchange symmetry, which enables robust, biasfree SSB that does not require any finetuning of the system parameters. The generation of truly random binary sequences based on this platform confirms the robustness of the exchange symmetry. Additionally, we reveal a new class of topological symmetrybroken localized structures characterized by rapid periodic exchange of their symmetry. Our observations are supported by theory, which reveals, in particular, the key role played by degenerate parametric processes. This connects our work with some recent implementations of socalled coherent Ising machines^{36,37} that could potentially benefit from our symmetryprotection scheme. These benefits could extend to experiments on Bose–Einstein condensates. Indeed, the Kerr nonlinearity stems from interaction processes among photons, in direct analogy with the nonlinearity inherent to BoseEinstein condensates and described by the GrossPitaevskii equation^{38}. We also present an effective Hamiltonian approach, similar to that used for the Floquet engineering of quantum matter^{39,40,41}, to explain aspects of the emergence of symmetry in our system. Finally, our work could have relevance to the symmetry restoration techniques used in meanfield approaches of quantum manybody systems in nuclear, atomic, and molecular physics^{42}.
Results
Principle of symmetry protection
Our system consists of a passive nonlinear (Kerr) optical resonator coherently driven by a singlefrequency field that presents two distinct polarization mode families. Hereafter ψ_{x} and ψ_{y} denote the modes’ classical electric field (complex) amplitudes inside the resonator scaled such that ∣ψ_{x}∣^{2} and ∣ψ_{y}∣^{2} represent the number of photons in each mode. In this configuration, the nonlinearity is well known to induce a polarization SSB bifurcation contingent on the existence of an exchange symmetry between the two modes^{43,44}. That symmetry is typically realized by being as close as possible to conditions where the modes are equally driven and equally detuned from the driving field frequency^{25,26,33}. Here, we deviate from this paradigm. Specifically, we introduce a localized πphaseshift defect between the two mode families of the resonator. As a result, their resonance frequencies are separated by half a freespectral range (FSR). In these conditions, the exchange symmetry cannot be realized as described above: the singlefrequency driving field can only be resonant with one of the modes, assumed here to be the x mode; hence, only that mode is driven, and the roundtrip phase detunings differ by π. These different aspects are illustrated schematically in Fig. 1a.
Consider now an alternative description of this resonator in terms of the hybrid mode amplitudes ψ_{+} and ψ_{−} defined by the unitary transformation
Because of the πphase defect, the amplitude of the y mode flips its sign with respect to the driven mode at each roundtrip in the resonator, ψ_{y} → ψ_{y}e^{iπ} = − ψ_{y}. From the above relations, it is clear that this is associated with a roundtriptoroundtrip periodic swapping of the + and − hybrid mode amplitudes, ψ_{+} ⇌ ψ_{−} [right side in Fig. 1c]. The field must circulate two roundtrips in the resonator to be restored to its original state, revealing that the hybrid modes embody the topology of a Möbius strip^{45,46,47,48,49}. In terms of those hybrid modes, the resonator shown in Fig. 1a is thus equivalent to that represented in Fig. 1b. Note that the Möbius topology only exists in the synthetic modal dimension rather than in real physical space^{50}. A consequence of the singlesided, nonorientable nature of the Möbius topology is that the two hybrid mode amplitudes will necessarily accumulate the same changes over two resonator roundtrips. The resonator, therefore, exhibits an exchange symmetry, ψ_{+} ⇌ ψ_{−}. That symmetry, coupled with the nonlinearity, enables SSB between the hybrid modes.
Here, it is important to realize that SSB in our system is fundamentally intertwined with the generation of the y field. As can be seen from Eqs. (1), a symmetric state for the hybrid modes, ψ_{+} = ψ_{−}, is necessarily associated with a y field of zero amplitude. By contrast, a symmetrybroken hybrid state, ψ_{+} ≠ ψ_{−}, requires a nonzero y field. By symmetry, the mirror state, ψ_{+} ⇌ ψ_{−}, must also exist, corresponding to a y field with a phase differing by π, ψ_{y} ⇌ − ψ_{y}. Keeping in mind that the yfield is undriven, it must be generated internally through nonlinear interactions past a certain threshold of detuning and/or driving strength. What appears as an SSB bifurcation for the populations ∣ψ_{+}∣^{2}, ∣ψ_{−}∣^{2} of the + and − hybrid modes can, therefore, be seen as the threshold of parametric oscillation for the y mode^{51} [left side in Fig. 1c]. However, our system is not just a standard parametric oscillator. Above the bifurcation threshold, the underlying Möbius topology imparted by the π phase defect forces the system to alternate between the two symmetrybroken states one roundtrip to the next and to display an overall periodicity of two roundtrips. This is consistent with the spectral components of the ymode being generated near their corresponding resonance condition, i.e., shifted by ±FSR/2 from the driving field frequency. As a reminder of these specific characteristics, we will refer to this regime as P2 SSB. One can also note that, in the P2 SSB regime, the amplitudes of the two hybrid modes over one resonator roundtrip become indistinguishable from the amplitude of one of those modes over two subsequent roundtrips. An alternative interpretation is thus that SSB now occurs between subsequent resonator roundtrips, in essence breaking the time translation symmetry in a way reminiscent of period doubling instabilities^{48,52,53,54}.
Theoretical analysis
The conclusions of the previous Section are based on the premise that the phase defect seen by the y mode at each resonator roundtrip is exactly π. To understand how the Möbius topology and the associated symmetry can persist in the presence of deviations from this condition, we must examine some aspects in more detail. To that end, we introduce the Hamiltonian \({\hat{H}}_{{{{{{{{\rm{NL}}}}}}}}}\) describing the Kerrmediated interactions taking place along the resonator roundtrip,
Here g is the single photoninduced Kerr frequency shift of the resonator (g > 0)^{55} while \({\hat{a}}_{\sigma }^{{{{\dagger}}} }\) (resp. \({\hat{a}}_{\sigma }\)) represents creation (annihilation) operators for the mode σ = x, y. These operators satisfy bosonic commutation relations, \([{\hat{a}}_{\sigma },{\hat{a}}_{{\sigma }^{{\prime} }}^{{{{\dagger}}} }]={\delta }_{\sigma,{\sigma }^{{\prime} }}\). The three terms on the righthand side of Eq. (2) describe, respectively, onsite (Bose–Hubbard) interactions (selfphase modulation), intersite interactions (crossphase modulation), and pair hopping (parametric fourphoton mixing). The latter corresponds to two photons in the x mode being converted to two photons in the y mode (or vice versa); since the ymode is undriven, this term is required as the source of y photons. Without loss of generality, we will assume A = 1, B = 2/3, and C = 1/3. These are the values found in silica (which is the Kerr material used in our experimental demonstration presented below) for modes that are linearly polarized^{56,57}. Generalization to other polarization states will be discussed in the Methods.
To understand specifically the evolution of the phase of the y mode, we examine the equation of motion for \({\hat{a}}_{y}\), i.e., \(d{\hat{a}}_{y}/dt=i[{\hat{H}}_{{{{{{{{\rm{NL}}}}}}}}},{\hat{a}}_{y}]/\hslash\). In the classical limit, \({\hat{a}}_{y}^{{{{\dagger}}} }\to {\psi }_{y}\), we get
where the righthand side has the form of the interaction terms found in the GrossPitaevskii/nonlinear Schrödinger equation (* denotes complex conjugation). Separating amplitude and phase, \({\psi }_{\sigma }= {\psi }_{\sigma } {e}^{i{\phi }_{\sigma }}\), we find that the phase ϕ_{y} of the y mode obeys
The three terms in the above equation represent phase shifts induced by the Kerr nonlinearity. Without loss of generality and for simplicity, we can restrict ourselves to highQ resonators where all these terms can be assumed ≪ 1. Indeed, nonlinear effects will only have an important role when close to resonance. In these conditions, we can use a single Euler step to approximate the integration of Eq. (4) over one resonator roundtrip time t_{R}. To obtain the total change in the phase of the y mode from one roundtrip to the next, we still need to consider the role of the boundary conditions, which add a linear contribution due to the resonator detuning (see Methods for more detail), leading to
Here m is the roundtrip index, δ_{0} = δ_{x} is the roundtrip phase detuning of the driving field (modulo 2π), also assumed ≪ 1 (δ_{0} > 0 means reddetuned), and the corresponding detuning for the y mode is defined through δ_{x} − δ_{y} = π + δ_{π}, with δ_{π} representing deviations from an exact value of π. We can observe that all terms above are, in fact, ≪π, except π itself (a similar argument for the x mode would show that \({\phi }_{x}^{(m+1)}{\phi }_{x}^{(m)}\ll 1\)). Moreover, the pair hopping (parametric) contribution (∝C), which is the only phasesensitive term, is π periodic in ϕ_{y}. This is a wellknown characteristic of degenerate parametric oscillators^{58}. It follows from Eq. (5) that the only stable configuration is for the phase of the y mode to step by exactly π, one roundtrip to the next. That solution acts as an attractor for the system, irrespective of the value of δ_{π}. Effectively, the intracavity fields will selfadjust through the dissipative nonlinear dynamics to guarantee a roundtriptoroundtrip y mode phase step of π. In this way, the Möbius topology of the resonator is intrinsically robust and protected by nonlinearity.
As a second step, we examine how the Möbius topology induces an exact exchange symmetry ψ_{+} ⇌ ψ_{−}. To that end, we introduce an explicit asymmetry and show how it cancels out. Specifically, we consider a difference in detunings between the + and − hybrid modes. The Hamiltonian Eq. (2) can be expressed in terms of these modes with creation (annihilation) operators \({\hat{a}}_{\pm }^{{{{\dagger}}} }\) (\({\hat{a}}_{\pm }\)); these are related to the corresponding x, y operators by the same unitary transformations as Eqs. (1). We find
where \({A}^{{\prime} }=2/3\) and \({B}^{{\prime} }=4/3\). For convenience, let us define a set of angular momentum (Schwinger) operators as
These operators satisfy commutation relations \([\,{\hat{J}}_{\mu },\,{\hat{J}}_{\nu }]=i\hslash {\epsilon }_{\mu \nu \lambda }\,{\hat{J}}_{\lambda }\) similar to spins (ϵ_{μνλ} is the Levi–Civita symbol), while \(\hat{N}\) counts the total number of photons. In this part of our analysis, we assume a lossless resonator (conservative limit) so \(\hat{N}\) is constant. With these notations, a difference in angular frequency detuning δΩ between the two hybrid modes enters the Hamiltonian as \(\delta \Omega \,{\hat{J}}_{z}\), so that the total Hamiltonian of the system reads
where \(a={A}^{\prime}  {B}^{\prime}\), \(b=({A}^{\prime}+{B}^{\prime} )/4\), and \(c={A}^{\prime} /2\).
We now proceed by seeking an effective (Floquet) Hamiltonian \({\hat{H}}_{{{{{{{{\rm{eff}}}}}}}}}\)^{39,59}, which captures stroboscopically the dynamics of a full period of the Möbius configuration depicted in Fig. 1b, i.e., every two actual roundtrips of the resonator. The swapping of the two hybrid modes at each roundtrip is introduced through the discrete application of the operator \({\hat{U}}_{{{{{{{{\rm{swap}}}}}}}}}={\hat{\sigma }}_{x}={e}^{i(\pi /2)({\hat{\sigma }}_{x}1)}\) (with \({\hat{\sigma }}_{x}\) a Pauli matrix), leading to
The first three factors on the righthand side of this expression define \({\hat{H}}_{1}\) which can be obtained exactly as^{41}
Noting that the swapping operation is a singleparticle process, we can substitute \({\hat{J}}_{x}/\hslash\) for \({\hat{\sigma }}_{x}/2\) in the exponentials above and then use the Baker–Campbell–Hausdorff formula. Specifically, we have \({e}^{i\pi {\hat{J}}_{x}/\hslash }\,{\hat{J}}_{z}\,{e}^{i\pi {\hat{J}}_{x}/\hslash }={\hat{J}}_{z}\) while all other terms of \({\hat{H}}_{{{{{{{{\rm{asym}}}}}}}}}\) are left unchanged, leading to
Comparing the expressions for \({\hat{H}}_{{{{{{{{\rm{asym}}}}}}}}}\), Eq. (8), and \({\hat{H}}_{1}\), above, we observe that the sign of the asymmetric term is reversed. It is also clear that \({\hat{H}}_{{{{{{{{\rm{asym}}}}}}}}}\) commutes with \({\hat{H}}_{1}\), which from Eq. (9) leads to \({\hat{H}}_{{{{{{{{\rm{eff}}}}}}}}}=({\hat{H}}_{1}+{\hat{H}}_{{{{{{{{\rm{asym}}}}}}}}})/2\). It follows that the asymmetric term cancels out exactly over two resonator roundtrips, and we have \({\hat{H}}_{{{{{{{{\rm{eff}}}}}}}}}={\hat{H}}_{{{{{{{{\rm{NL}}}}}}}}}\). Our analysis therefore shows that the effective Hamiltonian, which describes propagation every two resonator roundtrips, including two swaps of the hybrid modes, only contains the original—symmetric—nonlinear interactions; it is immune to fluctuations and asymmetries in detunings as foreseen in our qualitative description of the previous Section. This confirms the existence of an exact exchange symmetry for the hybrid modes, ψ_{+} ⇌ ψ_{−}, and the topological robustness of the P2 SSB regime.
We are now in a position to derive meanfield equations of motion for the classical fields ψ_{+}, ψ_{−} based on the effective Hamiltonian \({\hat{H}}_{{{{{{{{\rm{eff}}}}}}}}}\). We have already established that the dissipation guarantees an exact overall roundtrip phase jump of π for the y mode, so we do not need to consider the influence of δ_{π}, the deviation from an exact π phase defect, on the swapping operator \({\hat{U}}_{{{{{{{{\rm{swap}}}}}}}}}\). Still, δ_{π} contributes to the linear detunings through the relation δ_{0} − δ_{y} = π + δ_{π}. This effect can be taken into account by the following roundtrip phase detuning operator,
Note that, for the y mode, we have discarded the π phaseshift that is already incorporated in \({\hat{H}}_{{{{{{{{\rm{eff}}}}}}}}}\). The contribution of the detunings can be distributed along the resonator roundtrip and combined with the effective Floquet Hamiltonian, leading to, in the classical limit,
To these equations, we have added phenomenologically losses (dissipation) associated with the linewidth Δω of the resonances and driving, with strength S and external coupling fraction θ. In the conservative limit applied above, these terms are not included directly. An alternative classical derivation of the meanfield equations (13)–(14), taking into account these dissipative effects, is presented in the Methods for completeness. That derivation confirms, in particular, that the Möbius topology leads, as for the detunings, to a robust symmetrization of the driving. Note that the symmetry of the driving can also be interpreted as stemming from the absence of driving of the y mode; under these conditions, Eqs. (1) indeed entails \({S}_{+}={S}_{}=S={S}_{x}/\sqrt{2}\), where S_{x} is the driving strength of the x mode only. Additionally, we can observe that a deviation δ_{π} from an exact π phase defect simply causes a common shift in detuning as well as some linear coupling between the hybrid modes, none of which alters the exchange symmetry of the equations. Conversely, we can interpret the equality of the detunings as stemming from the absence of linear mode coupling between the x and y modes due to the associated fields having distinct spectral components.
Apart from the absence of kinetic terms, the role of which will be discussed below, Eqs. (13)–(14) have the form of coupled, driven, damped, Gross–Pitaevskii/nonlinear Schrödinger equations. These equations are well known to exhibit SSB, provided the detunings and the driving strengths of the two modes are identical^{25,26,30,33,43,44}. Contrary to previous implementations, in our system, and thanks to the presence of the π phase defect, this condition is automatically and robustly satisfied. Finally, while the equations above describe the stroboscopic meanfield evolution of ψ_{+} and ψ_{−} two resonator roundtrips at a time, it is important to keep in mind that the amplitudes of the two fields swap at each resonator roundtrip due to the underlying Möbius topology.
Experimental observation of P2 SSB
To demonstrate experimentally the topologicallyprotected P2 SSB regime, we have used, as modes x and y, the two polarization modes of a ring resonator constructed from singletransverse mode silica optical fiber. This platform enables straightforward implementation of both the π phaseshift defect and the projection of the x and y modes onto the + and − hybrid modes through manipulations of the polarization state of light with optical fiber polarization controllers (FPCs)^{60}. Note that for x and y modes that are orthogonally linearly polarized, the + and − hybrid modes as defined by Eqs. (1) correspond to circular polarization states of opposite handedness.
A schematic of the experimental setup is depicted in Fig. 2. As we used slightly different resonators with different specifications for different measurements (in part to demonstrate the universality of the methodology), we list here the parameters corresponding to the results presented in this Section. Variations between setups will be highlighted as appropriate. To facilitate comparisons, experimental results will be presented in terms of normalized parameters. Specifically, we will be referring to the normalized roundtrip phase detuning Δ = δ_{0}/α and the normalized driving power \(X=g{t}_{{{{{{{{\rm{R}}}}}}}}}\theta {S}_{x}^{2}/{\alpha }^{3}\), where α = t_{R}Δω/2 is the dissipation rate of Eqs. (13)–(14).
Our first resonator is built around a fiber coupler (beamsplitter) that recirculates 90% of the intracavity light. This coupler is also used for the injection of the coherent driving field (θ = 0.1). Another 1% tap coupler extracts a small fraction of the intracavity field for analysis. Both couplers are made up of standard SMF28 (Corning) optical fiber. The rest of the resonator is made up of a highly nonlinear, lowbirefringence spun fiber (iXblue Photonics)^{61}. At the 1550 nm driving wavelength, that fiber exhibits normal groupvelocity dispersion (corresponding to repulsive interactions or a positive effective mass), which has been selected to avoid scalar parametric instabilities^{57}. Overall, the resonator is 12 m long, corresponding to a roundtrip time t_{R} of 57 ns (FSR = 17.54 MHz), and has a resonance linewidth Δω/(2π) = 650 kHz. The resonator is driven by a 1550 nmwavelength continuouswave laser (NKT Photonics) with an ultranarrow linewidth < 1 kHz, ensuring coherent driving. The laser frequency can be varied with a piezoelectric transducer, which provides the ability to scan the detuning. Alternatively, we can use the transducer to lock the detuning at a set value via a feedback loop using the technique discussed in ref. ^{62}. The driving laser is intensitymodulated with a Mach–Zehnder amplitude modulator driven by a pulse pattern generator to generate flattop 1.1 nslong pulses with a period matching the resonator roundtrip time t_{R}^{63}. Before injection into the cavity, the driving pulses are amplified with an erbiumdoped fiber amplifier followed by an optical bandpass filter, which reduces amplified spontaneous emission noise.
The setup includes three FPCs. The first one is placed before the input coupler (FPC_{in}) to align the polarization state of the driving field with one of the two principal polarization modes of the resonator, henceforth defining the x mode. Note that because of residual birefringence, these principal modes correspond in practice to polarization states, which evolve along the fiber but map onto themselves at each roundtrip. The second FPC, placed inside the resonator (FPC_{π}), defines the π phaseshift defect. We adjust it for δ_{π} to be as close as possible to zero, which is controlled by monitoring the linear cavity resonances while scanning the driving laser frequency. Finally, the third FPC, placed on the output port of the resonator (FPC_{out}) and followed by a polarization beam splitter (PBS), can be set so as to split the output field either in terms of the x and y modes or in terms of the + and − hybrid modes. The intensity of the two components is then recorded with individual 12.5 GHz photodiodes. The total intensity is recorded separately.
To proceed, we record the output intensity levels across our driving pulses over subsequent resonator roundtrips using a realtime oscilloscope. A typical measurement is shown as color plots in Fig. 3a–c in the form of a vertical concatenation (bottomtotop) of a 20round trip sequence of realtime oscilloscope traces plotted against time. Panels a and b correspond to the intensities of the + and − hybrid modes, while (c) is the total intensity. The measurements have been obtained with 11 W of peak driving power (X = 50) and a roundtrip phase detuning locked to δ_{0} = 1.16 rad (Δ = 10). Figure 3a–b reveals that under these conditions, the hybrid mode intensity levels exhibit a broken symmetry (∣ψ_{+}∣^{2} ≠ ∣ψ_{−}∣^{2}) as well as antiphase alternating dynamics: the hybrid modes appear to exchange their intensities from one roundtrip to the next. This alternation is a key signature of the modal Möbius topology. At the same time, the total intensity [Fig. 3c] appears constant and presents no sign of the periodic alternation of the hybrid modes. This points to a high level of symmetry between the two hybrid modes and confirms the realization of the ψ_{+} ⇌ ψ_{−} exchange symmetry. We must note that this remarkable level of symmetry is achieved without any finetuning of parameters and can persist for hours. Numerical simulation results presented as Supplementary Section I and Supplementary Fig. S1 are found to be in excellent agreement with all these observations.
Further insights can be gained by measuring the high and lowintensity levels of the hybrid modes for a range of resonator detunings. The results of such measurements are plotted in Fig. 3d for Δ ranging from −4 to 10 and for the same driving power level as Fig. 3a–c. Blue and orange markers correspond to the + and − hybrid modes, respectively. For low values of the detuning, the two hybrid modes are found to have the same intensity, and the intracavity field appears symmetric in terms of these modes. In contrast, beyond Δ ≃ 2, we observe instead two mirrorlike, asymmetric states, whose relative contrast increases with the detuning. The emergence of these asymmetric states is found to correspond to the start of the roundtriptoroundtrip alternating dynamics described above. These measurements agree with numerically calculated steadystate solutions of Eqs. (13) and (14), which are shown as black curves in Fig. 3d (dashed lines correspond to unstable solutions; discrepancies can be attributed to uncertainties in the experimental parameters). Overall, the data in Fig. 3d exhibit the behavior of a pitchfork bifurcation characteristics of SSB. This result is remarkable in that it is obtained without having to make any effort to realize an exchange symmetry. In fact, we observe this behavior as soon as the ymode phase defect is sufficiently close to π (see Supplementary Section II and Supplementary Fig. S2 for more details on the existence range of P2 SSB).
To complete these observations, we have also made measurements in terms of the x and y modes. The emergence of the P2 SSB regime is characterized by the ymode intensity rising above zero through parametric generation (see also Supplementary Section III and Supplementary Fig. S3), but the intensities of the x and y modes do not otherwise reveal the alternating dynamics characteristics of the underlying Möbius topology. As described above, the associated dynamics for the ymode should be entirely contained in a roundtriptoroundtrip phase step of π, which calls for a phasesensitive measurement. To that end, we have used homodyne detection by mixing the x and y output fields into a singlepolarization 90° optical hybrid (Kylia). A pair of 40 GHzbandwidth balanced detectors then provide inphase, \(I\propto {{{{{{{\rm{Re}}}}}}}}({\psi }_{x}{\psi }_{y}^{*})\), and quadrature, \(Q\propto {{{{{{{\rm{Im}}}}}}}}({\psi }_{x}{\psi }_{y}^{*})\), components. Measurements of I and Q over 1000 resonator roundtrips in the P2 SSB regime are reported in Fig. 3e, with cyan (respectively, purple) points corresponding to odd (even) roundtrips. The arrangement of these two groups of points in diametrically opposite positions in the complex plane reveals unequivocally that the relative phase between the x and y fields varies in the step of π at each roundtrip. The average phase step is found to be (0.995 ± 0.020)π. Again, this has been obtained without fine tuning of parameters and confirms that the system is nonlinearly attracted to a roundtrip phase jump of π as predicted by theory. Note that, in comparison with the rest of the observations reported in Fig. 3, the data in Fig. 3e was obtained with a slightly shorter but otherwise identical resonator (resonator length of 10.5 m, roundtrip time t_{R} = 50.6 ns, and resonance linewidth Δω/(2π) = 825 kHz).
Test of the exchange symmetry
The results presented in the previous Section provide compelling evidence that a πphase defect combined with the Kerr nonlinearity does confer a robust Möbius topology and exchange symmetry to a driven resonator, enabling SSB. In particular, the reproducibility of this behavior and our ability to observe it over long periods of time without any finetuning of parameters strongly hint that the exchange symmetry is exact and protected by the nonlinearity, as predicted by theory. In order to confirm this hypothesis further, we now present a more stringent and quantitative test of the quality of the exchange symmetry realized in our system.
Our test is based on the properties of SSB. Specifically, upon crossing the SSB bifurcation threshold in our resonator, an initially symmetric state will lose its stability in favor of one of two states with broken symmetry. Under perfectly symmetrical conditions, this selection should be truly random. Conversely, a deviation from symmetry will lead to a bias in the state selection process^{33,35,64}. The degree of randomness of the state selection process is, therefore, a measure of the quality of the exchange symmetry. To test for such randomness in practice, we sinusoidally modulated the resonator detuning at 5.2 kHz around a mean (locked) value so as to cross the SSB threshold repeatedly and measured the state selection outcome. This measurement is based on the same resonator as that used to obtain the results of Fig. 3e but with a driving beam modulated so as to have 19 pulses simultaneously present in the resonator. Each pulse undergoes SSB independently, providing multiple simultaneous realizations of the experiment, thus enabling the acquisition of a larger amount of data for better statistical significance. The outcomes of the state selection process are determined by analyzing the output pulses captured with our realtime oscilloscope.
For this experiment, we have accumulated a sequence of 2.4 million individual events and expressed them as zero and one binary values. Due to the limitations of our oscilloscope, this data was assembled from four different batches acquired in quick succession over a few seconds without adjusting any parameters. The randomness of the acquired sequence was then rigorously assessed using the NIST Statistical Test Suite for random number generation (NIST STS2.1.2), which consists of a series of statistical tests used to detect nonrandomness within a given data set^{65}. In each test, a pvalue is generated, which gives the probability of this result occurring under the null hypothesis. We consider a pvalue of 0.01 or lower to be evidence of nonrandomness or failure. NIST recommends that the minimum pass rate for each statistical test should be 96% (the whole sequence is partitioned into 100 individual subsequences for testing). The results are presented in Fig. 4, which shows that all tests are passed with a proportion well over 96%, highlighting that we have no evidence of any bias in the SSB state selection process. Additional data gathered as the phase defect is varied around π further support this finding (see Supplementary Section IV and Supplementary Fig. S4). This confirms the robustness of the exchange symmetry provided by the Möbius topology in our experiment.
Topological localized structures
In the experimental results discussed so far, we have only considered cases where the intracavity field is homogeneous across the nanosecond driving pulses. The spatial extension of the driving provides, however, the opportunity to support more complex, inhomogeneous field configurations. The P2 SSB regime naturally extends to such structures because the mechanism giving rise to the Möbius topology of our resonator and to the tworoundtrip alternating dynamics is purely local. We now discuss this aspect in more detail.
Inhomogeneous field structures can be accounted for by introducing a kinetic term into the Hamiltonian. This corresponds to adding terms of the form −i(ϕ_{2}/2)∂^{2}ψ_{±}/∂τ^{2} to, respectively, the meanfield equation for the + [Eq. (13)] and − [Eq. (14)] hybrid modes. Here τ is a fast time variable describing variations of the fields along the resonator. In the optical context relevant to our experiments, ϕ_{2} represents chromatic dispersion integrated along the roundtrip of the resonator. With these kinetic terms, Eqs. (13) and (14) take the form of coupled drivendissipative GrossPitaevskii equations. Such equations have been used, e.g., to describe exciton–polariton condensates^{38,66,67}. In optics, these equations are known as Lugiato–Lefever equations (LLEs). The LLE was initially introduced as a paradigmatic model of pattern formation in dissipative optical systems^{68}. It is also known to support localized structures referred to as cavity solitons^{69,70}. In the temporal domain^{71}, these solitons underlie the new key technology of microresonator broadband optical frequency combs^{72,73}. In their coupled form, Eqs. (13) and (14) exhibit an even richer dynamics^{62,67,74}. Of particular relevance here are the recent experimental observations of twocomponent localized structures emerging through SSB, namely domain walls between homogeneous symmetrybroken states^{31} and bright symmetrybroken vector cavity solitons^{75,76}. Given the formal equivalence of the equations describing these structures and our system, it should be clear that these structures have a P2 counterpart; in fact, the only difference with what was previously reported should be the roundtriptoroundtrip alternation of the two components.
We start by considering P2 domain walls, which exist in the presence of repulsive interactions (ϕ_{2} > 0). Experimental evidence is presented in Fig. 5a–c. Panel a shows intensity profiles of the + (blue) and − (orange) hybrid modes at a selected odd (left) and even (right) roundtrip, while panels b and c present the roundtriptoroundtrip evolution of those intensities over 20 roundtrips as pseudocolor plots similar to those used in Fig. 3. These plots reveal a field structure whose components are perfectly anticorrelated and, at the same time, alternate roundtriptoroundtrip (see also^{48}). The driving pulses are subdivided into domains realizing, at any one time, one or the other of the two homogenous symmetrybroken states. By symmetry, the other component has the opposite domain structure. The domains are segregated by sharp kinklike temporal transitions, which correspond to dissipative (polarization) domain walls or PDWs^{31,77,78}. A closeup of a domain wall obtained with a 70 GHzbandwidth photodiode and sampling oscilloscope is shown in the inset (this measurement is bandwidthlimited with simulations predicting a rise/fall time of 4 ps). These observations were performed with the same resonator and in the same conditions (X = 50, Δ = 10) as that used for the results of Fig. 3a–c (for which we have ϕ_{2} ≃ + 0.5 ps/THz). We have only added a shallow 10 GHz sinusoidal phase modulation imprinted onto the driving so as to align and trap the PDWs onto a controlled temporal reference grid^{79}. The domain walls were excited spontaneously by ramping up the detuning through the SSB bifurcation threshold. As different parts of the nanosecond driving pulses break their symmetry in different directions, this leads to the emergence of a random pattern of domain walls, which are then trapped by the phase modulation.
Next, we consider P2 bright symmetrybroken temporal cavity solitons. Because these require attractive interactions, i.e., a negative effective mass or, equivalently, anomalous chromatic dispersion (ϕ_{2} < 0), we have used here a different resonator than for all the other observations reported so far in this Article. The resonator was 86 m long and entirely made up of standard SMF28 (Corning) optical fiber with an input coupler that recirculates 95% of the light (θ = 0.05). The roundtrip time t_{R} = 420 ns, the resonance linewidth Δω/(2π) = 57 kHz, and ϕ_{2} = − 1.7 ps/THz. The driving and detection schemes were the same as before. Experimental results are presented in Fig. 5d–f using the same layout as for the P2 domain walls. Here, we can observe a sequence of five bright solitons, which are aligned on a 2.87 GHz driving phase modulation grid^{79}. Spectral measurements (see Supplementary Section V and Supplementary Fig. S5) indicate a pulse duration of 1.6 ps (not resolved by the oscilloscope). These solitons are symmetrybroken, with one hybrid mode component dominating over the other. Due to the high driving level (X = 30, corresponding to 2.5 W peak driving power, and with Δ = 14.5), the contrast between the components is almost 100%. Again, the symmetry implies the existence of two mirrorlike states, i.e., solitons exhibiting a different dominant component, and our data confirms that these simultaneously coexist in the resonator. Finally, we observe a regular alternation of the components, with + modedominant solitons becoming − dominant and vice versa, each roundtrip to the next, as expected from the Möbius topology.
Bright symmetrybroken cavity solitons are also known to go through a Hopf bifurcation for lower detuning values, leading to breathing dynamics. This dynamic is characterized by slow inphase oscillations of the intensities of the two components of the soliton^{76}. In Fig. 5g–k, we provide evidence of this dynamic in the P2 SSB regime. As for the other structures discussed above, panels i and j show color plots of the evolution of the + and − hybrid mode temporal intensity profiles. Additionally, panel k displays the corresponding evolution for the total intensity. The latter clearly reveals the breathing dynamics occurring here with a period of about 9 resonator roundtrips, while the roundtriptoroundtrip alternation characteristics of the P2 SSB regime are only detected in the individual components. The superposition of breathing and alternation is better seen in panel h, with curves highlighting the evolution at the center of the soliton in terms of + (blue), − (orange), and total (green) intensities; these correspond to crosssections through the center of panels i–k, respectively. Note that we have selected here a lower driving level (X = 10) than in Fig. 5d–f, with the detuning set to Δ = 5.2. This choice leads to less contrast between the soliton components [compare Fig. 5d and g], and provides a clearer illustration of the overall complexity of the P2 symmetrybroken breather dynamics.
Finally, we must point out that all the structures reported in Fig. 5 could consistently be maintained and observed for long periods of time, from 30 to 60 min (see Supplementary Section VI and Supplementary Figs. S6 and S7). This confirms that the robustness of the Möbius topology extends to complex inhomogeneous structures.
Discussion
We have described theoretically and experimentally a system characterized by a complex interplay between topology, nonlinearity, and SSB. Specifically, we have shown that a π phase defect can force a driven Kerr resonator with two modes, x and y, toward an attractor arising through nonlinear degenerate parametric interactions so that the system displays a modal Möbius topology. That topology is associated with an exchange symmetry, which, together with the nonlinearity, enables SSB. Because of the Möbius topology, the two symmetrybroken states of our resonator alternate roundtriptoroundtrip, which we refer to as P2 SSB. In the experiment, this alternation is directly seen in the intensity of the + and − hybrid modes. Homodyne measurements have also enabled us to detect the corresponding phase steps in the evolution of the x and y modes of the resonator. Because the Möbius topology arises through a purely local mechanism, it extends to inhomogeneous field distributions, giving rise to a range of P2 localized structures, which we have also been able to observe. Thanks to the robustness associated with the topology, SSB is realized without any bias or finetuning of parameters. This fact has been theoretically confirmed by an effective Floquet Hamiltonian model and experimentally tested through rigorous testing of the randomness of the symmetrybroken state selection statistics. The inherent robustness is also seen in the fact that all the different behaviors we have reported can be obtained without any sensitive adjustments of parameters and are consistently observed over long periods of time, easily exceeding 30 min (typically only limited by our detuning locking feedback loop). As the realization of SSB is often hampered by biases and nonidealities, we believe that our findings can have interest for a number of applications, in photonics, but more broadly in other bosonic systems, including twocomponent atomic gases, cold bosonic atoms trapped in a double well, or spinors (see, e.g.,^{41,80}).
As a final remark, we would like to mention that phasedefectinduced topological protection can be generalized to systems with more than two modes, potentially leading to P3, P4, … regimes of SSB. In what follows, we present brief arguments to illustrate the general idea, restricting our attention to three modes for simplicity. Consider a resonator with modes x, y, and z, where the y and z modes present phase defects of about, respectively, 2π/3 and 4π/3 with respect to the x mode. This could be implemented, e.g., using etched waveguidebased adiabatic mode converters or photonic crystals^{50,81}. The resonance frequencies of the three mode families would be shifted in steps of a third of the FSR. With these modes, it is possible to define hybrid modes α, β, γ, such that the hybrid mode amplitudes would be converted at each roundtrip α → β → γ → α, and display a generalized Möbius topology over a cycle of three resonator roundtrips. This will realize a cyclic \({{\mathbb{Z}}}_{3}\) symmetry. The nonlinear fourwave interactions enabled by energy conservation are of the form x + x → y + z (and permutations). Using the same approach as in the theory presented above, it is easy to show that these interactions make the 2π/3 and 4π/3 roundtrip phase steps robust attractors of the system, making the \({{\mathbb{Z}}}_{3}\) symmetry exact, and protecting the topology. We have further confirmed that parameters exist where this system exhibits three symmetrybroken states, leading to P3 SSB, where these states cycle over three roundtrips.
As a particular application, we must note that the symmetrybroken states described in our Article could potentially be used to realize artificial spins. In the P2 SSB regime, these would correspond to spin1/2 and could be exploited to implement an Ising machine. Such a machine aims to find the ground state of interacting spins described by the Ising Hamiltonian with an appropriate physical system. The interest rests on the fact that many complex combinatorial optimization problems relevant to modern society, such as drug discovery^{82} or the analysis of social networks^{83}, can be mapped onto an Ising Hamiltonian^{84,85}. We must note that coherent optical Ising machines based on degenerate parametric oscillation have been particularly successful in recent years^{36,37}. A P2 SSB Ising machine would work along the same principle, but the inherent topological robustness of the Möbius topology could provide additional benefits. In this context, using more than two modes would offer even more tantalizing possibilities. For example, P3 symmetrybroken states would correspond to spin1 particles and open the way to the realization of a Potts machine^{86}.
Methods
Classical derivation
We present here a classical derivation of the meanfield equations (13) and (14). This derivation complements our Hamiltonian description, which was derived in the conservative limit, by including dissipation and driving explicitly. For completeness, we also consider modes with arbitrary polarization states, i.e., arbitrary values of the A, B, and C coefficients [see Eq. (2)]. We start from the classical equations of motion for the two modes, i.e., Eq. (3) and a similar equation for ψ_{x}. These equations are complemented by boundary conditions,
Here, α = t_{R}Δω/2 is the dissipation rate of the resonator, as introduced before, S_{in} is the total driving amplitude, with the ellipticity angle χ describing a split of the driving between the two modes (χ = 0 corresponds to only driving the x mode as previously considered, and \({S}_{x}={S}_{{{{{{{{\rm{in}}}}}}}}}\cos \chi\)), and the matrix M represents the action of the detunings,
We proceed by restricting ourselves to highQ resonators operated close to resonance and by following an approach similar to that of Refs. ^{87,88}. In these conditions, α, θ, δ, and δ_{π} can be assumed ≪1. Similarly, all the nonlinear terms in the equations of motion are small, and these equations can be integrated over one roundtrip time t_{R} by a single Euler step, as we have done with Eq. (4). Combining the result with the boundary conditions, Eq. (15), and expanding all small terms at first order, we can express the fields at roundtrip m + 1 in terms of those at roundtrip m as
Above, we can observe that the leading contribution for the y mode is a change of sign, as expected from the π phase defect.
The next step is to iterate Eqs. (17) and (18) and seek expressions for \({\psi }_{x,y}^{(m+2)}\) in terms of the mth roundtrip fields. At first order, we can assume that, in the nonlinear terms that result, \({\psi }_{x}^{(m+1)}\approx {\psi }_{x}^{(m)}\) and \({\psi }_{y}^{(m+1)}\approx {\psi }_{y}^{(m)}\). Because of the double action of the π phase defect over two roundtrips, we then find that the changes in the fields over two roundtrips, \({\psi }_{x,y}^{(m+2)}{\psi }_{x,y}^{(m)}\), are first order quantities. This enables us to introduce a slowtime derivative over two resonator roundtrips, defined as \(d./dt=\left[{.}^{(m+2)}{.}^{(m)}\right]/(2{t}_{{{{{{{{\rm{R}}}}}}}}})\), which transforms the difference equations we get into the following differential equations (this latter step is analogous to the stroboscopic Floquet approach used in the main Article),
where we have dropped the roundtrip index m. Here we can observe in particular that the y component of the driving cancels out over two roundtrips. This stems from that contribution being close to antiresonance conditions due to the halfFSR shift between the x and y mode resonances. Accordingly, the driving ellipticity χ simply affects the effective driving power, and any misalignment of the driving polarization with that of the x mode can simply be compensated for by driving the resonator stronger.
The last step is to express the equations above in terms of the + and − hybrid mode amplitudes defined by Eqs. (1). This leads to Eqs. (13) and (14), with
but with additional parametric terms, \(i\,g{t}_{{{{{{{{\rm{R}}}}}}}}}{C}^{{\prime} }{\psi }_{+}^{*}{\psi }_{}^{2}\) for Eq. (13) and \(i\,g{t}_{{{{{{{{\rm{R}}}}}}}}}{C}^{{\prime} }{\psi }_{}^{*}{\psi }_{+}^{2}\) for Eq. (14), where
We can observe that these terms do not break the ψ_{+} ⇌ ψ_{−} exchange symmetry of the equations and overall do not change the phenomenology. When the x and y modes correspond to linear polarization states, \({C}^{{\prime} }=0\) as in Eq. (6).
Note that the above analysis assumes a π phase defect that is purely localized. Additional considerations show that distributing the phase defect over a certain fraction of the resonator length amounts to a phase mismatch for the parametric interaction (nonconservation of momentum). This leads to less efficient generation of the y mode, but P2 SSB remains possible. In our experiments, FPC_{π} typically uses about one meter of fiber, so it is distributed over about 10% of the resonator or less (depending on the resonator length). Simulations indicate that P2 SSB can persist even with the phase defect distributed over more than half the resonator length but typically requires an increase in driving power. When parametric generation cannot overcome the losses of the resonator anymore, the P2 SSB regime eventually disappears.
Kinetic terms (chromatic dispersion) can also be included in our derivation in a straightforward manner. We note that neither these terms nor any higherorder terms like thirdorder dispersion or stimulated Raman scattering that lead to asymmetries in the fasttime variable τ, affect the exchange symmetry of the hybrid modes.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon request.
Code availability
The code that supports the plots within this paper and other findings of this study are available from the corresponding author upon request.
References
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045 (2010).
Lu, L., Joannopoulos, J. D. & Soljačić, M. Topological photonics. Nat. Photon. 8, 821 (2014).
Nicolis, G. & Prigogine, I., Exploring Complexity: An Introduction (St. Martin’s Press, New York, 1989).
Akhmediev, N. N. & Ankiewicz, A. Solitons—Nonlinear Pulses and Beams, 1st ed. (Chapman & Hall, London, 1997).
Anderson, P. W. More is different. Science 177, 393 (1972).
Strocchi, F. Symmetry Breaking. (Springer, Berlin New York, 2008).
Nambu, Y. Nobel lecture: spontaneous symmetry breaking in particle physics: a case of cross fertilization. Rev. Mod. Phys. 81, 1015 (2009).
Landau, L. D. & Lifshitz, E. M. Statistical Physics (Elsevier, 1980).
Korotkevich, E. et al. The apical domain is required and sufficient for the first lineage segregation in the mouse embryo. Dev. Cell 40, 235 (2017).
Qi, X.L. & Zhang, S.C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057 (2011).
Cooper, N. R., Dalibard, J. & Spielman, I. B. Topological bands for ultracold atoms. Rev. Mod. Phys. 91, 015005 (2019).
Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).
Segev, M. & Bandres, M. A. Topological photonics: where do we go from here? Nanophot. 10, 425 (2021).
Huber, S. D. Topological mechanics. Nat. Phys. 12, 621 (2016).
Ozawa, T. & Price, H. M. Topological quantum matter in synthetic dimensions. Nat. Rev. Phys. 1, 349 (2019).
Kitaev, A. Y. Faulttolerant quantum computation by anyons. Ann. Phys. 303, 2 (2003).
Raussendorf, R. & Harrington, J. Faulttolerant quantum computation with high threshold in two dimensions. Phys. Rev. Lett. 98, 190504 (2007).
Bandres, M. A. et al. Topological insulator laser: experiments. Science 359, eaar4005 (2018).
Smirnova, D., Leykam, D., Chong, Y. & Kivshar, Y. Nonlinear topological photonics. Appl. Phys. Rev. 7, 021306 (2020).
Maczewsky, L. J. et al. Nonlinearityinduced photonic topological insulator. Science 370, 701 (2020).
Xia, S. et al. Nonlinear tuning of PT symmetry and nonHermitian topological states. Science 372, 72 (2021).
de Léséleuc, S. et al. Observation of a symmetryprotected topological phase of interacting bosons with Rydberg atoms. Science 365, 775 (2019).
Jürgensen, M., Mukherjee, S. & Rechtsman, M. C. Quantized nonlinear Thouless pumping. Nature (London) 596, 63 (2021).
Hamel, P. et al. Spontaneous mirrorsymmetry breaking in coupled photoniccrystal nanolasers. Nat. Photon. 9, 311 (2015).
Cao, Q.T. et al. Experimental demonstration of spontaneous chirality in a nonlinear microresonator. Phys. Rev. Lett. 118, 033901 (2017).
Del Bino, L., Silver, J. M., Stebbings, S. L. & Del’Haye, P. Symmetry breaking of counterpropagating light in a nonlinear resonator. Sci. Rep. 7, 43142 (2017).
Cao, Q.T. et al. Reconfigurable symmetrybroken laser in a symmetric microcavity. Nat. Commun. 11, 1136 (2020).
Moroney, N. et al. A Kerr polarization controller. Nat. Commun. 13, 398 (2022).
Kaplan, A. E. & Meystre, P. Enhancement of the Sagnac effect due to nonlinearly induced nonreciprocity. Opt. Lett. 6, 590 (1981).
Woodley, M. T. M. et al. Universal symmetrybreaking dynamics for the Kerr interaction of counterpropagating light in dielectric ring resonators. Phys. Rev. A 98, 053863 (2018).
Garbin, B. et al. Dissipative polarization domain walls in a passive coherently driven Kerr resonator. Phys. Rev. Lett. 126, 023904/1 (2021).
Benjamin, T. B. Bifurcation phenomena in steady flows of a viscous fluid. II. Experiments. Proc. R. Soc. Lond. A 359, 27 (1978).
Garbin, B. et al. Asymmetric balance in symmetry breaking. Phys. Rev. Res. 2, 023244/1 (2020).
Garbin, B. et al. Spontaneous symmetry breaking in a coherently driven nanophotonic BoseHubbard dimer. Phys. Rev. Lett. 128, 053901 (2022).
Crauel, H. & Flandoli, F. Additive noise destroys a pitchfork bifurcation. J. Dyn. Differ. Equ. 10, 259 (1998).
Inagaki, T. et al. Largescale Ising spin network based on degenerate optical parametric oscillators. Nat. Photon. 10, 415 (2016).
Honjo, T. et al. 100,000spin coherent Ising machine. Sci. Adv. 7, eabh0952 (2021).
Carusotto, I. & Ciuti, C. Quantum fluids of light. Rev. Mod. Phys. 85, 299 (2013).
Goldman, N. & Dalibard, J. Periodically driven quantum systems: Effective Hamiltonians and engineered gauge fields. Phys. Rev. X 4, 031027 (2014).
Eckardt, A. Colloquium: atomic quantum gases in periodically driven optical lattices. Rev. Mod. Phys. 89, 011004 (2017).
Goldman, N. et al. Floquetengineered nonlinearities and controllable pairhopping processes: From optical Kerr cavities to correlated quantum matter. PRX Quantum 4, 040327 (2023).
Sheikh, J. A., Dobaczewski, J., Ring, P., Robledo, L. M. & Yannouleas, C. Symmetry restoration in meanfield approaches. J. Phys. G 48, 123001 (2021).
Kaplan, A. E. & Meystre, P. Directionally asymmetrical bistability in a symmetrically pumped nonlinear ring interferometer. Opt. Commun. 40, 229 (1982).
Haelterman, M., Trillo, S. & Wabnitz, S. Polarization multistability and instability in a nonlinear dispersive ring cavity. J. Opt. Soc. Am. B 11, 446 (1994).
Pond, J. Möbius dualmode resonators and bandpass filters. IEEE Trans. Microw. Theory Tech. 48, 2465 (2000).
Starostin, E. L. & van der Heijden, G. H. M. The shape of a Möbius strip. Nat. Mater. 6, 563 (2007).
Ballon, D. J. & Voss, H. U. Classical Möbiusring resonators exhibit fermionboson rotational symmetry. Phys. Rev. Lett. 101, 247701 (2008).
Javaloyes, J., Ackemann, T. & Hurtado, A. Arrest of domain coarsening via antiperiodic regimes in delay systems. Phys. Rev. Lett. 115, 203901 (2015).
Maitland, C., Conforti, M., Mussot, A. & Biancalana, F. Stationary states and instabilities of a Möbius fiber resonator. Phys. Rev. Research 2, 043195 (2020).
Xu, X.B., Shi, L., Guo, G.C., Dong, C.H. & Zou, C.L. "Möbius” microring resonator. Appl. Phys. Lett. 114, 101106 (2019).
Wong, G. K. L., Murdoch, S. G., Leonhardt, R., Harvey, J. D. & Marie, V. Highconversionefficiency widelytunable allfiber optical parametric oscillator. Opt. Express 15, 2947 (2007).
Carmichael, H. J. Optical bistability and multimode instabilities. Phys. Rev. Lett. 52, 1292 (1984).
Haelterman, M. & Tolley, M. D. Pure polarization perioddoubling instability in a Kerrtype nonlinear ring cavity. Opt. Commun. 108, 165 (1994).
Schuster, H. G. & Just, W. Deterministic Chaos: An Introduction, 4th ed. (WileyVCH, Weinheim, 2005).
Chembo, Y. K. & Yu, N. On the generation of octavespanning optical frequency combs using monolithic whisperinggallerymode microresonators. Opt. Lett. 35, 2696 (2010).
Menyuk, C. Pulse propagation in an elliptically birefringent Kerr medium. IEEE J. Quantum Elec. 25, 2674 (1989).
Agrawal, G. P. Nonlinear Fiber Optics, 5th ed. (Academic Press, 2013).
Woo, J. & Landauer, R. Fluctuations in a parametrically excited subharmonic oscillator. IEEE J. Quantum Elec. 7, 435 (1971).
Weitenberg, C. & Simonet, J. Tailoring quantum gases by Floquet engineering. Nat. Phys. 17, 1342 (2021).
Lefevre, H. Singlemode fibre fractional wave devices and polarisation controllers. Electron. Lett. 16, 778 (1980).
Barlow, A. J., RamskovHansen, J. J. & Payne, D. N. Birefringence and polarization modedispersion in spun singlemode fibers. Appl. Opt. 20, 2962 (1981).
Nielsen, A. U., Garbin, B., Coen, S., Murdoch, S. G. & Erkintalo, M. Coexistence and interactions between nonlinear states with different polarizations in a monochromatically driven passive Kerr resonator. Phys. Rev. Lett. 123, 013902/1 (2019).
Nielsen, A. U., Garbin, B., Coen, S., Murdoch, S. G. & Erkintalo, M. Invited article: emission of intense resonant radiation by dispersionmanaged Kerr cavity solitons. APL Photonics 3, 120804/1 (2018).
Steinle, T., Greiner, J. N., Wrachtrup, J., Giessen, H. & Gerhardt, I. Unbiased alloptical randomnumber generator. Phys. Rev. X 7, 041050 (2017).
Bassham, L. E. et al. A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications, Special Publication (NIST SP) 80022 Rev 1a (National Institute of Standards and Technology, Gaithersburg, MD, 2010).
Amo, A. et al. Collective fluid dynamics of a polariton condensate in a semiconductor microcavity. Nature (London) 457, 291 (2009).
Amo, A. et al. Exciton–polariton spin switches. Nat. Photon. 4, 361 (2010).
Lugiato, L. A. & Lefever, R. Spatial dissipative structures in passive optical systems. Phys. Rev. Lett. 58, 2209 (1987).
Scroggie, A. J. et al. Pattern formation in a passive Kerr cavity. Chaos Solitons Fractals 4, 1323 (1994).
Barland, S. et al. Cavity solitons as pixels in semiconductor microcavities. Nature 419, 699 (2002).
Leo, F. et al. Temporal cavity solitons in onedimensional Kerr media as bits in an alloptical buffer. Nat. Photon. 4, 471 (2010).
Coen, S., Randle, H. G., Sylvestre, T. & Erkintalo, M. Modeling of octavespanning Kerr frequency combs using a generalized meanfield LugiatoLefever model. Opt. Lett. 38, 37 (2013).
Herr, T. et al. Temporal solitons in optical microresonators. Nat. Photon. 8, 145 (2014).
Weng, W. et al. Heteronuclear soliton molecules in optical microresonators. Nat. Commun. 11, 2402 (2020).
Xu, G. et al. Spontaneous symmetry breaking of dissipative optical solitons in a twocomponent Kerr resonator. Nat. Commun. 12, 4023 (2021).
Xu, G. et al. Breathing dynamics of symmetrybroken temporal cavity solitons in Kerr ring resonators. Opt. Lett. 47, 1486 (2022).
Haelterman, M. & Sheppard, A. P. Polarization domain walls in diffractive or dispersive Kerr media. Opt. Lett. 19, 96 (1994).
Gilles, M. et al. Polarization domain walls in optical fibres as topological bits for data transmission. Nat. Photon. 11, 102 (2017).
Jang, J. K., Erkintalo, M., Coen, S. & Murdoch, S. G. Temporal tweezing of light through the trapping and manipulation of temporal cavity solitons. Nat. Commun. 6, 7370/1 (2015).
Dogra, N. et al. Dissipationinduced structural instability and chiral dynamics in a quantum gas. Science 366, 1496 (2019).
Wang, M. et al. Fractional optical angular momentum and multidefectmediated mode renormalization and orientation control in photonic crystal microring resonators. Phys. Rev. Lett. 129, 186101 (2022).
Kitchen, D. B., Decornez, H., Furr, J. R. & Bajorath, J. Docking and scoring in virtual screening for drug discovery: Methods and applications. Nat. Rev. Drug. Discov. 3, 935 (2004).
Carrington, P. J., Scott, J. & Wasserman, S. eds. Models and Methods in Social Network Analysis, Structural Analysis in the Social Sciences No. 28 (Cambridge University Press, 2005).
Barahona, F. On the computational complexity of Ising spin glass models. J. Phys. A: Math. Gen. 15, 3241 (1982).
Lucas, A. Ising formulations of many NP problems. Front. Phys. 2, 5 (2014).
HonariLatifpour, M. & Miri, M.A. Optical Potts machine through networks of threephoton downconversion oscillators. Nanophot. 9, 4199 (2020).
Haelterman, M., Trillo, S. & Wabnitz, S. Dissipative modulation instability in a nonlinear dispersive ring cavity. Opt. Commun. 91, 401 (1992).
Haelterman, M. Simple model for the study of perioddoubling instabilities in the nonlinear ring cavity. Appl. Phys. Lett. 61, 2756 (1992).
Acknowledgements
We acknowledge financial support from The Royal Society Te Apārangi in the form of Marsden Funding (18UOA310) as well as M.E.’s Rutherford Discovery Fellowship (RDF15UOA015). J.F. thanks the Conseil régional de Bourgogne FrancheComté, mobility (2019710614), financial support from the CNRS, the IRP WallIN project, and FEDER. N.G. is supported by the FRSFNRS (Belgium), the EOS program (CHEQS project), and the ERC (TopoCold and LATIS projects).
Author information
Authors and Affiliations
Contributions
B.G. and J.F. set up the experiment and made the initial observations of P2 SSB and P2 domain walls. J.F. carried on subsequent numerical modeling and made the measurements shown in Fig. 3, as well as the P2 domain walls [Fig. 5a–c]. G.X. performed the experiment related to the P2 bright symmetrybroken cavity solitons and associated breather [Fig. 5d–k]. L.Q. was experimentally implemented and carried out on the randomness statistical test of Fig. 4. N.G. derived the Hamiltonian model. G.L.O., M.E., and S.M. provided extra theoretical and/or experimental support. S.C. suggested the topological interpretation of P2 SSB and derived the classical meanfield description. All authors helped in interpreting the results. S.C. wrote the paper with feedback from all the authors.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Coen, S., Garbin, B., Xu, G. et al. Nonlinear topological symmetry protection in a dissipative system. Nat Commun 15, 1398 (2024). https://doi.org/10.1038/s4146702344640x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s4146702344640x
This article is cited by

Spontaneous symmetry breaking of coupled Fabry–Pérot nanocavities
Communications Physics (2024)

Symmetry broken vectorial Kerr frequency combs from FabryPérot resonators
Communications Physics (2024)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.