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# Topological phenomena in classical optical networks

Contributed by H. J. Kimble, September 13, 2017 (sent for review June 5, 2017; reviewed by Mohammad Hafezi and Mikael C. Rechtsman)

## Significance

We introduce a unique scheme to investigate topological behavior, using optical-passive elements and Kerr nonlinearities. Compared with previous proposals, the topological band gaps are dramatically broadened, leading to very robust edge modes. Our setup displays intriguing phenomena in the nonlinear regime, including instabilities and the production of squeezed light in the edge modes. This proposal promises unique avenues for engineering phases of light with topological character.

## Abstract

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 *Nodes* and *Light Propagation in Fibers*, respectively. In *Full Networks*, 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 *A* and *B*, 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 L. The fibers 1 and 4 are connected to the beamsplitter A, and the fibers 2 and 3 are connected to the beamsplitter B.

Since the polarizations of light are always orthogonal to the propagation direction, the directions of vertical polarization V in the horizontal and vertical fibers are different. As shown in Fig. 1*B*, the directions of horizontal polarization H are chosen to be pointing out of the 2D plane, while the directions of vertical polarization V 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 *B*) is established by the scattering matrix (S matrix) of the node. Here, the two-component amplitudes *B*, in the inner cavity, the input and output amplitudes of the beamsplitter A are

The relation**1**, 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 **2**. To cancel this Fresnel effect, two birefringent elements E and F in close proximity to the beamsplitters (**1**.

Eliminating the inner-cavity fields

### 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*C* shows that in each horizontal fiber, three birefringent elements described by the Jones matrices *D*), which induces a nontrivial

The interaction of light in the fiber, induced by a Kerr nonlinearity (Fig. 1*C*), 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

In the Kerr medium of the fiber connecting nodes *Light Propagation in Single-Segment Nonlinear Fiber* and Figs. S1 and S2, we give the solution of the motion Eqs. **4** and **5** in detail.

The wave Eq. **4** has the solution*C*,

The solution **6** results in the relation

The same analysis is applied to the light propagation in the vertical fiber connecting the nodes **5** results in the relation*C*), where the wave vectors are**3** of the node and the relations **8** and **9** 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 *A* and *B*). The boundary conditions are

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 *C*, where the driving light of frequency *C*), and

For the open planar network, we drive the network through the partially transmissive mirror next to the node *D*. The boundary condition is

Using the S-matrix **3** at the node, the relations **8** and **9**, and the boundary conditions **14** and **16**, we can establish the scattering equations for the entire network in the different geometries. The details are shown in *Scattering Equations on Different Geometries*.

## Linear Regime

In this section, we use the scattering equation to study the topological phenomena in the linear network without the Kerr medium. In *Topological Band Structures in Closed Networks*, we study the photonic spectra E 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 *Probe Edge and Bulk Modes in Open Networks*, 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 *Robustness of the Edge Modes in Open Networks*.

The photonic spectra E 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 equation**8** and **9**, where the free S-matrix

### Topological Band Structures in Closed Networks.

Incorporating the boundary conditions **11** and **12** to the scattering Eq. **17**, we determine the eigenstates and the corresponding spectrum. Due to the translational symmetry, the eigenstate

Fig. 3 *A* and *B* shows the spectra of the networks in the torus and cylinder in the FSR

As shown in Fig. 3*A*, in the torus the photonic bands spread over the whole FSR

As a consequence of time-reversal symmetry [as in the case of *B*, 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

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 *Robustness of the Edge Modes in Open Networks*, 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 *C* and *D* shows that for random phase fluctuations

### 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 *C* and *D*.

For the cylindrical network (Fig. 2*A*), the input light**19**. The boundary condition **14** and the scattering Eq. **17** result in

The solution of the scattering Eq. **22** determines the output amplitude**21** gives the amplitude*A*, for *A*, which isolates a single

For the planar network, circularly polarized driving light with amplitude *C*). The boundary condition **16** and the scattering equation

The solution of the scattering Eq. **27** determines the reflection and transmission amplitudes*B*, the transmission spectrum *C* (Fig. 4*D*), the intensities

### 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 *Robustness of Broadband Setups* 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.

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

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 (

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 δ and

## 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 *Topological Band Structures in Closed Networks* that bulk and edge steady states are both generated. In *Probe Edge and Bulk Modes in Open Networks*, 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 **8** and **9** give rise to the scattering equation*Scattering Equations on Different Geometries*.

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 **29** has the form **19**.

By numerically solving Eq. **29** with the boundary condition **21**, we show the total light intensity *A* and *B* for two driving frequencies *A*) and *B*), respectively, where *Bogoliubov Excitations in Nonlinear Optics*, large domains of the steady-state solutions in Fig. 7 *A* and *B* 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) *Light Propagation in Single-Segment Nonlinear Fiber* and Fig. S2).

For driving frequencies *C* and *D* shows that distinct light distributions *C*, 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

The additional weak probe light

To establish the scattering equation for the fluctuation amplitudes **4** and **5**, we obtain*Light Propagation in Single-Segment Nonlinear Fiber*.

The solution of linearized Eqs. **31** and **32** has the form**31** and **32** result in the relations*Propagation Matrices of Bogoliubov Excitations*.

Additionally, the boundary values of the fields in adjacent fibers are related by the input–output formula**36** and **37** and the input–output relation **38** 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 condition**21**, and the output field is**36**–**38** and **40**, we establish the linearized scattering equation

By solving the fluctuation Eq. **43**, we mark the stable regimes by black circles in the *A* and *B*. We find that the steady states shown in Fig. 7 *C* and *D* are in the stable regime.

Eq. **42** and the solution **43** lead to the input–output relation

We note that squeezing of light in topological insulators has been investigated in the context of optical parametric down-conversion systems (24), where the

Around the stable edge steady state (Fig. 7*C*), the probe field with momentum *A*. 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 *B*, 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 *D*), edge fluctuations can also be generated by a probe light. However, due to the small light distribution along the boundary, the counterpart

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

For the nonlinear network, the relevant parameter is the unitless phase

For the electronic Kerr effect with small

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. *Light Propagation in Single-Segment Nonlinear Fiber* is devoted to analyzing the light propagation in the fiber with a Kerr medium. In *Steady-state solutions and fluctuations in fibers*, 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 *Nonlinear Fabry–Perot cavity*, we analyze the stability of steady states in a single nonlinear Fabry–Perot cavity as a paradigmatic example.

Using the S matrices at each node (*Steady-State Solutions* in the main text) and in fibers (*Light Propagation in Single-Segment Nonlinear Fiber*), in *Scattering Equations on Different Geometries* we derive a nonlinear scattering equation in the network with different geometries, which determines the steady-state properties.

In *Robustness of Broadband Setups*, 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 *Propagation Matrices of Bogoliubov Excitations*, 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 *Steady-state solutions and fluctuations in fibers*, the light propagation in a fiber with the nonlinear Kerr medium is analyzed. In *Nonlinear Fabry–Perot cavity*, 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. **4** and **5** in *Light Propagation in Fibers* of the main text are**6** in the main text.

The fluctuation field **S5** leads to the relation

The same analysis is applied to light propagation in the vertical fiber connecting nodes **9** in *Light Propagation in Fibers* in the main text. By linearizing the motion equation in the vertical fiber around the steady-state solution

#### 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

The relations**8** in the main text, where L is the cavity length, and

By eliminating the output amplitude **S12** and **S14**, we obtain the nonlinear equation**S15** determines the intensity-dependent frequency

For different driving frequency *A* and *B*, where

Fig. S2 *A* and *B* shows that for a given **S7** that the fluctuation fields satisfy

The fluctuation Eq. **S19** leads to the relation

For the good cavity limit *C* and *D*, we show the two curves given by Eq. **S27** for different driving intensities *D*, the positive coordinates *A* and *B*, the stable regimes are marked by the black circles, where these stable solutions are in the positive slope regimes of

### Scattering Equations on Different Geometries.

In this section, we use Eqs. **8** and **9** in the main text to derive the scattering equation for the steady-state amplitudes

Combining Eqs. **8** and **9** and the node S-matrix **3** in the main text, we obtain the scattering equation**S28** becomes Eq. **17** 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. **11**–**13** in *Full Networks* in the main text. For the networks in the torus and cylinder, due to the translational symmetry, the solution has the form **19** in the main text, and the scattering Eq. **S28** becomes**11** and **12** in the main text, the scattering equation for the entire closed network in the torus and cylinder can be written as

Similarly, by the boundary condition **13** in the main text, the scattering equation for the closed networks in the plane reads

In the main text, we numerically solve Eqs. **32** and **33** for the linear closed network, i.e.,

#### Open network.

For the open network in the cylinder shown in Fig. 2 *A* and *C* of the main text, the nonlinear scattering equation for the amplitude **S34** determines the outgoing amplitude **24** in the main text.

Similar to the case for the linear network, when the driving frequency **25** and **26** 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 **S34** for the network with size *C* and *D* of the main text.

For the open network in the plane shown in Fig. 2 *B* and *D* of the main text, the scattering equation for the amplitude **S35** determines the reflection and transmission amplitudes by Eq. **28** 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. **S35** with *B* 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*C* 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

By the same procedure introduced in *Nodes* in the main text, we obtain the S-matrix

The scattering equation at the bulk node in the linear network is**11**–**13** 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 *A*–*C*, where the chiral edge modes appear in the bandgaps. When

The steady-state configuration of edge modes in the open linear network can be obtained by Eq. **S35**, where the pump field drives the network through the node *D*–*F*, for the network with the same intrinsic loss, the steady edge state completely circulates around the boundary in the broadband setup with

### Propagation Matrices of Bogoliubov Excitations.

In this section, we define the propagation matrices of Bogoliubov excitations in *Bogoliubov Excitations in Nonlinear Optics* in the main text. The propagation matrices

## Acknowledgments

This work was funded by the European Union Integrated project SIQS. H.J.K. acknowledges support as a Max-Planck Institute for Quantum Optics Distinguished Scholar; as well as funding from the Air Force Office of Scientific Research Multidisciplinary University Research Initiative (MURI) Quantum Many-Body Physics with Photons; the Office of Naval Research (ONR) Award N00014-16-1-2399; the ONR Quantum Opto-Mechanics with Atoms and Nanostructured Diamond(QOMAND) MURI; National Science Foundation (NSF) Grant PHY-1205729; and the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: tshi{at}mpq.mpg.de or hjkimble{at}caltech.edu.

Author contributions: T.S., H.J.K., and J.I.C. designed research; T.S., H.J.K., and J.I.C. performed research; T.S. contributed new reagents/analytic tools; T.S. analyzed data; and T.S., H.J.K., and J.I.C. wrote the paper.

Reviewers: M.H., University of Maryland; and M.C.R., Pennsylvania State University.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1708944114/-/DCSupplemental.

- Copyright © 2017 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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