Topological phenomena in classical optical networks.

We propose a scheme to realize a topological insulator with optical-passive elements and analyze the effects of Kerr nonlinearities in its topological behavior. In the linear regime, our design gives rise to an optical spectrum with topological features and where the bandwidths and bandgaps are dramatically broadened. The resulting edge modes cover a very wide frequency range. We relate this behavior to the fact that the effective Hamiltonian describing the system's amplitudes is long range. We also develop a method to analyze the scheme in the presence of a Kerr medium. We assess robustness and stability of the topological features and predict the presence of chiral squeezed fluctuations at the edges in some parameter regimes.

The discovery of topological insulators (TIs), as well as quantum spin Hall (QSH) insulators (1–9), has opened up a wide range of scientific and technological questions. Their spectra feature a set of bands, connected by chiral edge modes that reflect the topological nature of the material. These modes are robust against perturbations whose energy does not exceed the corresponding bandgap and that do not break the time-reversal (TR) symmetry (10, 11). Electronic interactions give rise to a wide range of phenomena. Although the edge modes persist, their properties are qualitatively modified (12). In addition, they can give rise to other exotic phenomena, like the fractionalization of charges, or the appearance of excitations with fractional statistics (13, 14).

Recent proposals to generate TIs and QSH insulators with light have also attracted a lot of attention (2, 15–27, 28). In fact, the first experimental observations (15, 16) of topological features in optical systems have been recently reported, and several schemes exhibiting intriguing features have been proposed (17–25). There exist different setups where one can realize the optical analog of QSH insulators and observe similar features. In the context of coupled resonator arrays, one can use either differential optical paths in waveguides (26) or an optical active element (27). Despite their success, in the first case it would be desirable to enlarge the bandgaps in the spectrum, which is limited by the small coupling of the local modes in the (high-finesse) resonators (26), to gain robustness. To enlarge the bandwidth and bandgaps, recently, several proposals in the Floquet systems (29), microwave networks (30), and strongly coupled spoof-plasmon systems (31) have been studied. In the second one, photon absorption in the active media also limits the operationality of the scheme. In other schemes, like the one based on bianisotropic metacrystals (28), the realization of long-lived edge modes in a broader frequency range is challenged by the weak bianisotropy in metamaterials (32, 33). To enhance the bianisotropy, an alternative realization has been proposed for metallodielectric photonic crystals in the microwave regime (34). The effects of interactions, including the stability of edge modes, edge solitons, and the quantum dynamics, in those optical models have been also investigated recently (24, 35–42).

In this work we propose and analyze a scheme to realize the optical version of the QSH insulator and investigate the effects produced by Kerr nonlinearities. Our scheme uses beam splitters and birefringent materials that are optically passive and thus circumvent the problem of photon absorption. Our scheme features several distinct phenomena compared with some of the previous proposals.

In the linear regime, the Hamiltonian description of our setup features long-range hopping, which leads to a dramatic increase of the spectral bands and bandgaps. This results in a more robust behavior of the edge modes against perturbations. We analyze quantitatively the robustness of our scheme against losses and compare it to other models not displaying long-range Hamiltonian descriptions, as well as to recent experiments (15).

In the nonlinear regime, we obtain the following results: (i) In a closed network, an arbitrarily small Kerr interaction induces instability, a phenomenon we explain in terms of a simple model. (ii) Opening the network and driving it in the appropriate regime stabilizes the system. By tuning the frequency of the driven light, stable bulk and edge modes are both generated. (iii) The small excitations around the edge modes are themselves chiral and thus protected. (iv) The edge modes, apart from being chiral, are squeezed.

In this paper we also introduce theoretical frameworks based on the S-matrix approach to describe our model both for the linear and for the nonlinear regimes. The reason why standard approaches do not apply in the linear regime is that the energy spectrum spreads over the whole free spectral range (FSR), so that the energy bands in two adjacent ranges connect to each other. Thus, one cannot use an effective Hamiltonian description in each FSR. Furthermore, since the energy spectrum is not lower bounded, the nonlinear behavior is very different from that of lower-bounded Hamiltonians. The analysis of such behavior cannot be carried out with standard Bogoliugov techniques, but requires a sophisticated method based on a nonlinear S-matrix formalism.

Model Setup

In this section, we construct (non)linear classical optical networks that exhibit nontrivial phenomena. The light propagation in the nodes and (non)linear fibers is investigated in and , respectively. In , we analyze the boundary conditions for the closed and open networks in the torus, cylinder, and open plane.

We consider a toy model, i.e., a network of size with optically passive elements. As shown in Fig. 1 , at each node of a square lattice, two beamsplitters and two perfect mirrors form a “bad cavity” to change the propagation direction of incoming light in the optical fibers with length . The fibers and are connected to the beamsplitter , and the fibers and are connected to the beamsplitter .

Fig. 1.

Scheme of our optical network. (A) The planar network. (B) A node connects horizontal and vertical links. Here, and denote the input and output amplitudes, and denotes the amplitudes in the cavity. The horizontal and vertical polarizations (,) in fibers are shown by the red circles and arrows. (C) The horizontal and vertical links with the Kerr medium and birefringent elements, where the Kerr medium is put on the right-hand side of three birefringent elements in the horizontal link, where the birefringent elements are assumed to have no Kerr nonlinearity (). From left to right, the three birefringent elements are described by the Jones matrices , , and in the linear polarization basis . (D) The polarized light acquires the phase () by propagating (anti)clockwisely in each plaquette, while the polarized light acquires the phase () by propagating (anti)clockwisely in each plaquette.

Since the polarizations of light are always orthogonal to the propagation direction, the directions of vertical polarization in the horizontal and vertical fibers are different. As shown in Fig. 1, the directions of horizontal polarization are chosen to be pointing out of the 2D plane, while the directions of vertical polarization are pointing up and right in the horizontal and vertical fibers, respectively.

Nodes.

In this subsection we study the light propagation in the node, where the relation of input and output amplitudes and (Fig. 1) is established by the scattering matrix ( matrix) of the node. Here, the two-component amplitudes and are defined in the linear polarization basis . As shown in Fig. 1, in the inner cavity, the input and output amplitudes of the beamsplitter are and , and the elements and are perfectly reflecting mirrors.

The relationbetween amplitudes , , and is determined by the -matrix ,of beamsplitters and , where the real reflection and transmission coefficients are and . In Eq. , we have assumed that the size of the node is much smaller than the wavelength of light such that the free propagation phase in the node can be neglected. In principle, for a node cavity of small dimensions compared with the optical wavelength, the diffraction effect should be considered and a full finite-difference time-domain method might be required. In practice, one should consider designing coupling directly the fiber to the cavities and a design such that losses are negligible. Under such conditions, the node will be characterized by a set of parameters, i.e., the reflection and transmission coefficients, that could be adjusted to match the toy model considered here. We expect that our toy model can capture the main physics in the system. A similar treatment was used in ref. 27.

Due to the Fresnel reflection rule, for the incoming vertical polarized light from the fiber 1 (3), the reflecting light in the fiber 4 (2) changes the sign. The sign change for the vertical polarization is described by the Pauli matrix in Eq. . To cancel this Fresnel effect, two birefringent elements and in close proximity to the beamsplitters () in the fibers and have been introduced and are described by the Jones matrix in Eq. .

Eliminating the inner-cavity fields , we derive the scattering equation at the node, where the -matrixrelates the input and the output amplitudes and . At the node, the effective reflection and transmission coefficients are and , respectively. In our scheme, we choose , such that maximizes the topological bandwidth and bandgap defined later on.

Light Propagation in Fibers.

In this subsection, we use a wave equation to study the light propagation in the nonlinear fiber connecting two adjacent nodes, where the birefringent elements are introduced to induce the artificial gauge field for the light. By solving the wave equation, we obtain the input–output relation of the amplitudes at adjacent nodes.

Fig. 1 shows that in each horizontal fiber, three birefringent elements described by the Jones matrices , , and are placed close to the node on the left side of the horizontal fiber, where are Pauli matrices in the polarization basis . The element with the row-dependent Jones matrix generates the opposite phase shifts for the - and -polarized light, and the element with the Jones matrix induces the phase shifts for the linear polarized light . The birefringent elements cause the light to acquire a phase matrix by propagating around each plaquette. Eventually, circular polarizations () experience oppositely directed “magnetic” fields with fluxes (Fig. 1), which induces a nontrivial topology in this TR-invariant system.

The interaction of light in the fiber, induced by a Kerr nonlinearity (Fig. 1), leads to the additional phase proportional to the light intensity. We set to zero the cross-phase modulation between orthogonal circular polarizations in the Kerr nonlinearity, as discussed in ref. 43, chap. 6 and ref. 44, chap. 4. As a result, the two polarizations are decoupled, and we are able to treat the polarizations independently, as long as these polarizations are separately excited by the external input (i.e., only or polarization circulating in the fiber links).

In the Kerr medium of the fiber connecting nodes and , the right- and left-moving fields (denoting the left or right polarized light) obey the motion equations (45–47) (we use the matrix convention to label the sites, where is the row index and is the column index; we note that the matrix convention is different from the coordinate convention that labels the row and column by and , respectively)andwhere is the distance along the fiber and describes the self-focusing Kerr interaction (43, 48). In and Figs. S1 and S2, we give the solution of the motion Eqs. and in detail.

Fig. S1.

A single Fabry–Perot cavity with Kerr nonlinearity and an anisotropic phase plate placed next to the left end mirror to mimic the horizontal link, where the driving field with frequency is applied.

Fig. S2.

Steady-state solutions and stability analysis, where is taken as a unit. (A and B) The relation of the light intensity in the cavity and the driving field intensity in steady state for the driving frequencies (A) and (B). Here, the stable regimes are marked by the black circles. The solid (blue), dashed (red), and dashed-dotted (green) curves denote the light intensities for , , and , respectively. (C and D) For , the first and second equations in Eq. are shown by the solid (blue) and dashed (red) curves, where , (C) and , (D).

The wave Eq. has the solutionwhere, as shown in Fig. 1, is the amplitude of the right-moving input field to the node , and is the amplitude of the left-moving input field to the birefringent elements. The wave vectorsof the right- and left-moving fields are determined by the intensities of the fields and the frequency of light in the fiber.

The solution results in the relationwhere is the amplitude of the right-moving output field out of the node , is the amplitude of the right-moving field on the right-hand side of the birefringent elements, is the amplitude of the left-moving output field out of the node , is the amplitude of the left-moving input field to the node , and for the two orthogonal circular polarizations.

The same analysis is applied to the light propagation in the vertical fiber connecting the nodes and . The steady-state solution of Eq. results in the relationfor the amplitudes of input fields and output fields (as shown in Fig. 1), where the wave vectors areThe -matrix of the node and the relations and determine the light distribution in the bulk of the network.

Full Networks.

In this subsection, different boundary conditions are studied for the closed networks in the torus, cylinder, and open plane. To generate the nontrivial topological states in the network, we drive the open network by external light through the boundary.

To realize the cylindrical and planar geometries, perfect mirrors are placed along the boundaries to form the closed network, where the distance between the boundary mirror and the boundary node is (Fig. 2 ). The boundary conditions arefor the network in the torus,for the closed cylindrical network with the periodic boundary condition along the direction, andfor the planar network with boundary perfect mirrors.

Fig. 2.

Scheme for the boundary mirrors in the cylindrical and planar networks. (A and B) Perfect and partially transmissive mirrors are put along the boundaries of the cylindrical (A) and planar (B) networks. Here, the partially transmissive mirrors are placed along the top boundary in the cylindrical network and at the top left and bottom right corners in the planar network. The driving light (red arrows) is applied to generate the excitations and the transmitted light (blue arrow). (C) Driven cylindrical network through each of the partially transmissive mirrors on the top boundary. (D) Light reflection and transmission through the boundary mirrors next to the nodes and in the planar network (i.e., nodes in the top left and bottom right corners of the planar network).

Through the partially transmissive mirrors at the boundary of the open networks, nontrivial topological states can be generated by the external optical driving field. For the open cylindrical network, we drive the network through the top boundary mirrors with the reflection (transmission) coefficient (), as shown in Fig. 2, where the driving light of frequency has the amplitude . The corresponding boundary conditionis determined by the -matrixof the transmissive mirrors, where is the amplitude of the output field above the boundary mirror (Fig. 2), and and are the amplitudes of the down-moving input field and the up-moving output field at the top of the cylinder [i.e., the boundary node ].

For the open planar network, we drive the network through the partially transmissive mirror next to the node with light of frequency and detect the transmission to the node , as shown in Fig. 2. The boundary condition iswhere denotes the input amplitude to the network, and () is the reflection (transmission) amplitude. The amplitudes of right-moving input and left-moving output fields at the node are and , while the amplitudes of left-moving input and right-moving output fields at the node are and .

Using the -matrix at the node, the relations and , and the boundary conditions and , we can establish the scattering equations for the entire network in the different geometries. The details are shown in .

Linear Regime

In this section, we use the scattering equation to study the topological phenomena in the linear network without the Kerr medium. In , we study the photonic spectra by solving the scattering equation for the closed networks in the torus, cylinder, and open plane. We find that the edge states appear in the bandgaps covering a very wide frequency range. In , we show that the edge and bulk modes can be generated by the external driving light and detected by the spectroscopic analysis of the transmitted light. The robustness of the edge modes against losses and imperfections is analyzed in .

The photonic spectra of the closed linear networks in different geometries exhibit nontrivial topological phenomena, which are described by the scattering equation. For the bulk degrees of freedoms, the scattering equationfollows from Eqs. and , where the free -matrixconnects the right-, up-, left-, and down-moving input fields at the node with the input amplitudes , , , and at the four nearest-neighbor nodes. We note that the eigenstates of have well-defined polarization or .

Topological Band Structures in Closed Networks.

Incorporating the boundary conditions and to the scattering Eq. , we determine the eigenstates and the corresponding spectrum. Due to the translational symmetry, the eigenstatehas well-defined quasi-momenta , where .

Fig. 3 shows the spectra of the networks in the torus and cylinder in the FSR around a large central frequency , where is a positive integer, and .

Fig. 3.

The energy spectra in the torus and cylinder, where , the network size is , and is taken as a unit. Here, we use and to denote the polarizations on the top and bottom boundaries, respectively. (A and B) The energy spectrum in the torus (A) and cylinder (B) without phase randomness and losses of linear elements, where and denote the bottom and top boundaries, respectively. (C and D) The real part of the energy spectrum in the torus (C) and cylinder (D) with phase randomness and nonzero losses of linear elements.

As shown in Fig. 3, in the torus the photonic bands spread over the whole FSR and display large bandgaps. For instance, for (15), the FSR is . In contrast to the standard narrow-band schemes (26, 27), the wide-band spectrum results from the large hopping strength (comparable with ) between nodes beyond nearest neighbors in the bad cavity regime, . In each FSR, this long-range hopping behavior is characterized by the spatially nonlocal Hamiltonian ln rather than the Hofstadter (tight binding) model (49). Here, we emphasize that even though the matrix contains only the nearest-neighbor couplings, the effective Hamiltonian could show long-range hopping behavior between cavity modes since it is determined by the logarithm of the matrix. One can introduce the creation (annihilation) operators of the eigenmodes in the band to express the effective Hamiltonian as , where denotes the dispersion relation in the band .

As a consequence of time-reversal symmetry [as in the case of -protected topological insulators (10)], helical edge modes arise at the broad topological bandgaps. In Fig. 3, for the cylindrical geometry, the spectrum displays four edge modes between the bandgaps, where the chiralities of two edge modes on each boundary are locked to the polarizations. We focus on the right polarized mode. The Chern number associated to the right polarized mode in each subband can be properly defined. The lowest and highest subbands in each FSR have Chern number 1. From the second to the eighth subbands, the Chern number changes alternatively between two values, . As a result, the right polarized edge mode localized at the top boundary of the cylinder changes its chirality alternatively in different midgaps. This band structure is similar to that found in Floquet systems (50).

These helical edge modes are robust to local perturbations that do not break the time-reversal symmetry, as long as the bandgap remains open. As we shall see in , the effects of randomness and losses are strongly suppressed due to the broadness of the spectrum as a consequence of the low finesse of the cavities. Fig. 3 shows that for random phase fluctuations around and a 10% optical loss in each element, the bandgaps in the energy spectrum Re are still open in the torus, and the helical edge modes survive in the cylinder with lifetime /Im. The losses are described by the nonunitary matrix of the optical element.

Probe Edge and Bulk Modes in Open Networks.

To generate and detect the edge and bulk modes, we consider the open networks in the torus and cylinder driven by an external light, as shown in Fig. 2 .

For the cylindrical network (Fig. 2), the input lightwith amplitude and frequency is applied through the transmissive top-boundary mirror. Due to the translational symmetry in the driven network, the steady-state solution has the form . The boundary condition and the scattering Eq. result inandfor the field , where is the amplitude of the output field , is obtained by replacing the diagonal matrix element of the -dimensional identity matrix with ,and is composed of the -dimensional null vector and .

The solution of the scattering Eq. determines the output amplitudeWhen the driving frequency is resonant with an eigenfrequency of the closed system, the boundary condition gives the amplitudeand the input–output relation , where the phase shiftRemarkably, the relative phase jumps from to when sweeps across a resonant frequency of the closed network. Thus, the measurement of this phase shift reveals the spectrum. As shown in Fig. 4, for polarized driving light, the peaks of show the band structure and the chiral edge mode on the top boundary. The spatial separation of the bottom edge mode and the driving light makes the first invisible in Fig. 4, which isolates a single polarized chiral edge mode on the top boundary.

Fig. 4.

Detection of topological properties, where , , and is taken as a unit. (A) For the cylindrical geometry, the contourplot of shows the eigenspectrum for the network of size . (B) For the open plane of size , the eigenspectrum for the closed network and the transmission spectrum. (C) The light intensity of the bulk mode under the -polarized driving light. (D) The light intensities of the edge mode in the network under -polarized driving light. From the right- (left-) and up- (down-)moving fields shown in the Top (Bottom) row, the chirality of the edge mode can be identified.

For the planar network, circularly polarized driving light with amplitude and frequency is injected through the transmissive mirror at the upper left corner (Fig. 1). The boundary condition and the scattering equationdetermine the light distribution in the open network, where is obtained by replacing the diagonal matrix elements and of the-dimensional identity matrix with , and .

The solution of the scattering Eq. determines the reflection and transmission amplitudesAs shown in Fig. 4, the transmission spectrum of the output light through the mirror at the bottom right corner can be identified with the energy spectrum. For a driving frequency () resonant with the bulk (edge) mode, as shown in Fig. 4 (Fig. 4), the intensities display that the light propagates in the bulk (along the boundary).

Robustness of the Edge Modes in Open Networks.

To analyze the robustness of edge modes, we take into account possible imperfections in the network, including losses and phase fluctuations of the linear elements. The edge modes generated by the external driving field are robust, which is the result of the broad topological bandwidth and bandgap. In and Figs. S3 and S4, we construct another topological network with the tunable spectral width and show that a spectrum spreading over the whole FSR has a dramatic effect on the robustness.

Fig. S3.

The new setup with tunable topological bandgaps. Here, the fiber is constructed similar to that in Fig. 1 of the main text, but now each node is built by four transmissive mirrors A with reflection amplitude and one beam-splitter B with reflection amplitude . Two birefringent elements in close proximity to the mirrors are described by the Jones matrix .

Fig. S4.

(A–C) The energy spectra of the cylindrical network with nodes for . (D–F) The light intensities of the steady edge modes in the open planar network with nodes for , where the node and each birefringent element have intrinsic loss.

To show the differences of our network and that in ref. 15, we use the same input–output configuration, namely, pumping the network through the node and detecting the transmission light at the node . The loss of each element is chosen to be in the linear optical system. In Fig. 5, the light intensity in the steady edge state shows that the edge mode completely circulates around the network.

Fig. 5.

The intensity of the steady edge mode in the open planar network, where the intrinsic loss is , and is the intensity of the pump field.

The short lifetime of the edge modes in the narrow-band setup can be overcome in the broadband setup. In the narrow-band setup, each resonator has a high finesse (), such that the light reflects many times in the resonator, which induces a large decay to undesired modes. In the broadband setup, the low finesse () of the resonator results in a short time of the light in the resonator and the small loss to the undesired modes.

Even though the system is not completely immune to the losses and imperfections that break the TR symmetry and induce backscattering that changes the polarization, the imperfections are strongly suppressed due to the broad width of the spectrum. We note that the birefringent element is a linear element, which cannot break the TR symmetry. The TR symmetry breaking mentioned here amounts to the coupling between two polarizations with opposite chiralities.

For instance, small phase fluctuations and of birefringent elements, i.e., and , can induce the coupling between and polarized light. We consider driving the planar network of size through the node with -polarized light and detecting the transmitted light at the node . In Fig. 6, the intensities of -polarized edge modes are shown for , and ,. For random phase fluctuations ,, -polarized clockwise propagating light in the network is scattered to -polarized anticlockwise propagating light, and -polarized light is detected in the transmitted light. For the larger phase fluctuations ,, -polarized clockwise propagating light in the network is scattered to -polarized anticlockwise propagating light, and -polarized light is detected in the transmitted light. In real experiments, phase fluctuations of linear optical elements can be much smaller than . As a result, in the steady state, -polarized chiral mode may survive at the boundary of the network.

Fig. 6.

A and C show the intensities of the -polarized light in the network for ; B and D show the intensities of the -polarized light in the network for .

Nonlinear Regime

In this section, we study how the topological properties predicted in the previous section get modified in the nonlinear regime. We restrict ourselves to the cylindrical network. By driving the open network from the top boundary, we show in that bulk and edge steady states are both generated. In , we analyze the stability of the steady states by means of a generalized Bogoliubov theory, where it turns out that the Bogoliubov edge mode can be detected by the squeezing spectrum of the reflected light.

The nonlinear Kerr medium generates a self-focusing interaction for . Here, we consider separately the polarizations and thereby avoid the complexity associated to a Kerr nonlinearity for polarizations propagating simultaneously in the fiber links (43, 44). The relations and give rise to the scattering equationfor the bulk degrees of freedom, where the intensity-dependent matrix is defined in .

The effective Hamiltonian provides an insight into the physics in the interacting case. By projecting the system on a certain band, the effective Hamiltonian can be interpreted as describing weak interacting bosons in a topological band. At the mean-field level, we could expect that the steady state is a Bose–Einstein condensate of light, and the fluctuations are described by Bogoliubov modes that give rise to squeezing. In the following, we focus on the steady state and fluctuations in the cylindrical network.

Steady-State Solutions.

To generate the interacting bulk and edge steady states, we consider driving the cylindrical network by an external field. Circularly polarized pump-field with amplitude is applied through the top boundary. Due to the translational symmetry along the direction, the steady-state solution of Eq. has the form .

By numerically solving Eq. with the boundary condition , we show the total light intensity vs. the driving strength in Fig. 7 for two driving frequencies (Fig. 7) and (Fig. 7), respectively, where and the size of the network is , . The vs. curves display that for the given parameters (,), the driving light with amplitude generates multiple light intensities in the steady state of the network. As discussed in , large domains of the steady-state solutions in Fig. 7 are unstable to small perturbations. The qualitative origin of these complex stabilities can be traced to the behavior of a single fiber segment with mirrors (47) and Fig. S2).

Fig. 7.

Light distributions of the nonlinear system in the cylinder, where the size is , , , and is taken as a unit. (A and B) The relation of the total intensity of (A) edge and (B) bulk modes in the network with different reflection indexes and the input intensity of driving light with frequencies (A) and (B) . (C and D) The stable internal intensities of (C) edge and (D) bulk modes for and (red circles in A and B).

For driving frequencies and , Fig. 7 shows that distinct light distributions are generated for the interacting edge and bulk modes, respectively, where the total intensity , and . We emphasize that the topologically protected chiral edge mode survives even in the nonlinear regime, as illustrated in Fig. 7, where the chirality can be gathered from the fact that the right-moving intensity dominates.

Bogoliubov Excitations in Nonlinear Optics.

The stability of steady-state solutions is analyzed in this subsection. Small fluctuations around the driving field induce excitations around the steady state. If excitation is exponentially amplified in the real-time evolution, the steady state is not stable. We develop a generalized Bogoliubov theory to analyze the properties of the fluctuations and the stability of the steady states. We show that around the stable steady state, chiral Bogoliubov edge excitations are squeezed and can be detected by the squeezing spectrum of the reflected light.

The additional weak probe lightwith frequency through the top boundary induces the fluctuation field around the steady-state and the reflected fluctuation field , where and are the quasi-momenta along the direction.

To establish the scattering equation for the fluctuation amplitudes in the entire network, we first study the propagation of fluctuation fields in the fiber. By linearizing the motion in Eqs. and , we obtainandwhich describe the dynamics of Bogoliubov fluctuationsin the horizontal and vertical fibers, respectively. Here, the matrices and are defined in .

The solution of linearized Eqs. and has the formThe input and output fluctuation fields at the nodes modulate along the direction with the quasi-momenta and , where the fluctuation amplitudes are related to the boundary value of the wavefunctions and asThe scattering Eqs. and result in the relationsandconnecting the boundary values of the fields in the horizontal and vertical fibers, respectively, where the propagation matrices and are defined in .

Additionally, the boundary values of the fields in adjacent fibers are related by the input–output formulaat each node, whereThe propagation Eqs. and and the input–output relation result in the scattering equation for the fluctuation amplitudes in the bulk. To analyze the properties of those fluctuations and the stability of the steady states, the boundary conditions for the fluctuation fields are required.

The boundary conditionfor the fluctuation amplitudes follows from Eq. , and the output field iswhere the componentsEliminating the output fluctuation fields in Eqs. – and , we establish the linearized scattering equationfor the fluctuations, whereand . The steady state is stable if the fluctuations are not amplified during the time evolution. This stable condition demands that all of the roots of det have negative imaginary part, i.e., Im.

By solving the fluctuation Eq. , we mark the stable regimes by black circles in the vs. curves for the steady states with in Fig. 7 . We find that the steady states shown in Fig. 7 are in the stable regime.

Eq. and the solution of Eq. lead to the input–output relationwhere the matrixis determined byInduced by the light “condensation” , the Bogoliubov fluctuation couples to the conjugate amplitude . As a result, the probe light with positive frequency, i.e., , induces a chiral Bogoliubov fluctuation, which eventually generates the squeezed reflected light . (We emphasize that our results from the fluctuation analysis for the classical light are also valid in the quantum regime, where the squeezing behaviors of quantum edge fluctuations arise from the interplay of the Kerr nonlinearities and topological effects.) The squeezing behavior is characterized by the squeezing spectra and , where the expression reflects the bosonic nature of light.

We note that squeezing of light in topological insulators has been investigated in the context of optical parametric down-conversion systems (24), where the nonlinearity is treated at the mean-field level and quadratic terms with double creation (annihilation) operators are directly introduced in the Hamiltonian to describe the generation of squeezed light.

Around the stable edge steady state (Fig. 7), the probe field with momentum induces the generation of squeezed light with the spectra displayed in Fig. 8. Here, we chose this quasi-momentum, because the edge mode is more isolated from the bulk modes and has the smallest localization length. The peak around the frequency in the squeezing spectra is the signature of chiral Bogoliubov fluctuations and . As shown in Fig. 8, the large light distribution at the top boundary generates a strong coupling of Bogoliubov fluctuations localized at the edge, which results in comparable magnitudes of and . In the bulk steady state (Fig. 7), edge fluctuations can also be generated by a probe light. However, due to the small light distribution along the boundary, the counterpart and thus the edge Bogoliubov fluctuations hardly respond to the driving field .

Fig. 8.

Bogoliubov fluctuations in the cylinder, where the system size is , . The amplitude of the probe light and are taken as a unit. (A) The squeezing spectra of the probe field with the positive frequency, where the Bogoliubov fluctuations above the stable edge steady states are generated. (B) The distributions of Bogoliubov edge fluctuations above the stable edge steady states, where the Bogoliubov excitation has the frequency , shown by the red arrow in A.

In this nonlinear regime our system displays a set of phenomena that are quite different from the results in previous works (51–55): (i) A closed nonlinear network is unstable, and the system becomes stable by including losses; (ii) around the stable edge steady state, a probe field with a second frequency develops small edge Bogoliubov fluctuations which turn out to be chiral; (iii) the presence of squeezing, a quantum feature, is identified in the edge modes. The reason for the appearance of these phenomena is that the energy bands in different FSRs connect to each other, so that the system cannot be described by a lower-bounded Hamiltonian.

Experimental Parameters

In experimental implementations, one could take a fiber with and cross section , such that the bands spread over the broad FSR with in the linear network.

For the nonlinear network, the relevant parameter is the unitless phase induced by the Kerr interaction. In terms of the experimental parameters, the phase is determined by the wavevector in the optical fiber, the second-order nonlinear refractive index , the intensity (power/area) of the pump field, and the length of the fiber , where the optical fiber with , and the section .

For the electronic Kerr effect with small , a length of the link of the fiber and a power of the pump field should be large enough to realize a phase shift . To realize the large nonlinearity in the small network with , one can use the thermal Kerr effect with . A pump power ranging from to gives rise to nonlinear phase shifts ranging from .

We note that our proposal could also be implemented in the all-in-fiber temporal lattice setup (56).

Conclusions

We have proposed a scheme to display QSH phenomena using classical light and passive optical elements. Compared with previous schemes, ours features broad topological bandgaps. For open networks in the linear regime, chiral edge modes appear at the bandgaps and are very robust against losses and random phase fluctuations. Adding Kerr nonlinearities in the fibers leads to interacting bulk and edge states that also display topological properties. For closed networks, the system becomes unstable. This can be avoided by opening it. We also predict squeezing in the chiral edge modes.

SI Text

This SI Text is divided into four sections. is devoted to analyzing the light propagation in the fiber with a Kerr medium. In , we establish the nonlinear motion equations to describe the light propagation in the horizontal and vertical fibers. By solving the motion equations, we investigate the properties of steady states and Bogoliubov fluctuations. To get insight into the steady-state stability of the whole network, in , we analyze the stability of steady states in a single nonlinear Fabry–Perot cavity as a paradigmatic example.

Using the matrices at each node ( in the main text) and in fibers (), in we derive a nonlinear scattering equation in the network with different geometries, which determines the steady-state properties.

In , we show that broadband models are able to be immune to losses and perturbations. To illustrate the advantages of broadband models compared with the narrow ones, we build a new setup, in which the width of the topological bandgap can be tuned. The edge currents in the networks with the broad and narrow bands are shown to reveal the robustness of edge modes in the broadband network, where the intrinsic losses are the same in the two networks. In , the matrices used in the Bogoliubov fluctuation analysis are defined.

Light Propagation in Single-Segment Nonlinear Fiber.

This section is divided into two subsections. In , the light propagation in a fiber with the nonlinear Kerr medium is analyzed. In , a simple nonlinear system, i.e., the Fabry–Perot cavity, is analyzed, where the stability of steady states is investigated.

Steady-state solutions and fluctuations in fibers.

The formal solutions of Eqs. and in of the main text arewhere and are the fluctuation fields around the steady state. For the closed network, the characteristic frequency is the eigenfrequency, and is the frequency of the driving field applied to the open network. The steady-state solution gives rise to the relation in the main text.

The fluctuation field obeys the linearized motion equationwhere the matrices areandThe Bogoliubov mode with the fluctuation frequency around obeys the equationwhere the time-independent fieldThe formal solution of Eq. leads to the relationof the input and output fluctuation fieldsandaround the steady-state amplitudes () and ().

The same analysis is applied to light propagation in the vertical fiber connecting nodes and , which leads to Eq. in in the main text. By linearizing the motion equation in the vertical fiber around the steady-state solution () and (), we establish the relationof the input and output fluctuation fields () and (), where

Nonlinear Fabry–Perot cavity.

Before studying the steady-state properties and the stability of the light in the whole network, we use a paradigmatic example, i.e., the single Fabry–Perot cavity with nonlinear Kerr medium (47), to show the stability analysis of steady states. Our goal is to understand better the stability analysis for more complex 2D arrays of nonlinear fibers and beam splitters.

As shown in Fig. S1, the cavity with a perfect right end mirror is driven by the light with frequency through a partially transmissive mirror at the left end. In the cavity, the phase plate is placed next to the transmissive mirror. In propagation from left to right, the light acquires the phase factor . Here, () corresponds to the single horizontal (vertical) fiber in the network.

The relationsof input () and output amplitude () follow from Eq. in the main text, where is the cavity length, andAt the end mirrors, the boundary conditions are , andwhere () is the real transmission (reflection) coefficient of the left end mirror, and () is the input (output) amplitude of the cavity.

By eliminating the output amplitude () in Eqs. and , we obtain the nonlinear equationthat determines the amplitude , where , and the output amplitudeof the cavity is determined by the relation . In the good cavity limit , Eq. determines the intensity-dependent frequencyof the closed cavity, where is an integer.

For different driving frequency , the relationof and the input intensity is shown in Fig. S2 , where is taken as a unit and . When the driving frequency is resonant with the intrinsic frequency of the closed cavity, the output field .

Fig. S2 shows that for a given , the driving field with a fixed intensity can generate multiple intracavity intensities. To analyze the stability of these multiple steady states, we investigate the energy spectrum of Bogoliubov fluctuations. It follows from Eq. that the fluctuation fields satisfywhere the matrixfor the single fiber, and the unitary matrix is determined by the 2D identity matrix .

The fluctuation Eq. leads to the relation of and , where the matricesare determined by the propagating matrix , and the diagonal matricesOn the other hand, the boundary conditions at the end mirrors are andBy eliminating the fluctuation field , we obtain the scattering equationwith the driving term , where the matrices andThe zeros of the determinantdetermine the stability of the steady-state solution, where the steady state is stable if all Im.

For the good cavity limit , the momentum , and the eigenfrequency of Bogoliubov fluctuations is , where is an integer. For the open cavity, the condition leads to the two transcendental equationsfor Re and Im. In Fig. S2 , we show the two curves given by Eq. for different driving intensities , where the intersection of two curves determines the solution and . As shown in Fig. S2, the positive coordinates at points of intersection imply an unstable steady state. In Fig. S2 , the stable regimes are marked by the black circles, where these stable solutions are in the positive slope regimes of vs. curves.

Scattering Equations on Different Geometries.

In this section, we use Eqs. and in the main text to derive the scattering equation for the steady-state amplitudes in the nonlinear network. Here, in terms of different boundary conditions, we analyze the scattering equations describing the closed and open networks on three kinds of geometries.

Combining Eqs. and and the node -matrix in the main text, we obtain the scattering equationfor the input amplitudes at the bulk nodes, where the phase shift induced by the Kerr nonlinearity is depicted by the intensity matrixWithout the Kerr nonlinearity, i.e., , the scattering Eq. becomes Eq. in the main text for the linear network.

Closed network.

The boundary conditions for networks in the torus, cylinder, and open plane are given by Eqs. – in in the main text. For the networks in the torus and cylinder, due to the translational symmetry, the solution has the form in the main text, and the scattering Eq. becomeswhere the intensity matrix along the row of the network isBy taking into account the boundary conditions and in the main text, the scattering equation for the entire closed network in the torus and cylinder can be written asin the basis .

Similarly, by the boundary condition in the main text, the scattering equation for the closed networks in the plane readsin the basis .

In the main text, we numerically solve Eqs. and for the linear closed network, i.e., , and show the spectra of the network with different geometries. For the closed nonlinear network, i.e., , the solutions are unstable in general. To generate and stabilize the state of light with Kerr nonlinearities, we drive the network through the top boundary mirrors of the cylindrical open network.

Open network.

For the open network in the cylinder shown in Fig. 2 of the main text, the nonlinear scattering equation for the amplitude readswhere is obtained by replacing the diagonal matrix element of the -dimensional identity matrix by , and is composed of the -dimensional null vector and . The solution of the scattering Eq. determines the outgoing amplitude by Eq. in the main text.

Similar to the case for the linear network, when the driving frequency is resonant with the eigenfrequency of the closed system, and the phase shift are determined by Eqs. and in the main text. In the main text, we consider linear and nonlinear open networks in the cylindrical geometry. In the linear case, we study the detection of the energy spectrum through the phase shift . In the nonlinear case, we numerically solve Eq. for the network with size and show the light distributions for different and in Fig. 7 of the main text.

For the open network in the plane shown in Fig. 2 of the main text, the scattering equation for the amplitude readswhere is obtained by replacing the diagonal matrix elements and of the -dimensional identity matrix by , and is composed of the -dimensional null vector and . The solution of the scattering Eq. determines the reflection and transmission amplitudes by Eq. in the main text.

In the main text, we study the light transmission to the linear network in the open plane. The solution of Eq. with determines the light distribution in the linear network and the transmission probability , which are shown in Fig. 4 of the main text for different driving frequency .

Robustness of Broadband Setups.

In this section, we investigate the robustness of broadband models. To tune the topological bandwidth, we construct a new network, where the construction of the fiber is the same as that in Fig. 1 of the main text. As shown in Fig. S3, the node is built by four mirrors and one beam splitter in the center, where two birefringent elements described by the Jones matrix in close proximity to the mirrors are connected to the horizontal fibers.

By the same procedure introduced in in the main text, we obtain the -matrix for each node, whereare determined by the () matrix () and the reflection and transmission coefficients () and () of the mirrors (beam splitter).

The scattering equation at the bulk node in the linear network iswhereWe focus on the -polarized light with . With the different boundary conditions – in the main text, we can study the energy spectrum in the closed networks in the torus, cylinder, and open plane. We show the energy spectra in the cylindrical networks with different reflectivities in Fig. S4 , where the chiral edge modes appear in the bandgaps. When is increasing, the topological bandwidth becomes narrow.

The steady-state configuration of edge modes in the open linear network can be obtained by Eq. , where the pump field drives the network through the node , and the node and each birefringent element have intrinsic loss. As shown in Fig. S4 , for the network with the same intrinsic loss, the steady edge state completely circulates around the boundary in the broadband setup with ; however, the steady edge states can travel only a half or a quarter of the boundary in the narrow-band setup with or , where is the light intensity at the node , and is the intensity of the pump field.

Propagation Matrices of Bogoliubov Excitations.

In this section, we define the propagation matrices of Bogoliubov excitations in in the main text. The propagation matrices and for the Bogoliubov excitations in the horizontal and vertical fibers are determined by the matricesandwhere .