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Coherent control of topological states in an integrated waveguide lattice

Topological photonics holds the promise for enhanced robustness of light localization and propagation enabled by the global symmetries of the system. While traditional designs of topological structures rely on lattice symmetries, there is an alternative strategy based on accidentally degenerate modes of the individual meta-atoms. Using this concept, we experimentally realize topological edge state in an integrated optical nanostructure based on the array of silicon nano-waveguides, each hosting a pair of degenerate modes at telecom wavelengths. Exploiting the hybrid nature of the topological mode formed by the superposition of waveguide modes with different symmetry, we implement coherent control of the topological edge state by adjusting the phase between the degenerate modes and demonstrating selective excitation of bulk or edge states. The resulting field distribution is imaged via third harmonic generation allowing us to quantify the localization of topological modes as a function of the relative phase of the excitations. Our results highlight the impact of engineered accidental degeneracies on the formation of topological phases, extending the opportunities stemming from topological nanophotonic systems.

In crystalline topological systems, the nontrivial topology of the bands and the associated edge or interface states originate due to the special choice of lattice symmetries that ensure topological degeneracies (e.g. Dirac points), as well as the opening of topological bandgaps for suitable lattice parameters. Prominent examples are one-dimensional (1D) zigzag arrays [20][21][22][23][24][25] , 2D breathing honeycomb 11,13 and breathing kagome [26][27][28] lattices. However, since lattice geometries are challenging to modify dynamically, the topological properties of such systems, e.g., the existence of topological states and their localization length, are difficult to control in real time.
To overcome this limitation, it has recently been suggested to utilize the accidental degeneracy of modes in the individual meta-atoms of a simple lattice 29,30 . In such case, the interference of the near fields of the degenerate modes can regulate the coupling between neighboring meta-atoms, while the detuning between these modes controls the topological transitions [29][30][31] . Experimental demonstrations of this strategy to create topological structures at the nanoscale have remained elusive so far.
In this Article, we implement this strategy experimentally realizing an integrated optical structure based on an array of multimode nanostructured waveguides that supports topological edge state in the telecom range. By optimizing the design of the waveguides, we ensure that they host a pair of modes with the same propagation constants but different symmetry of the near field profiles [ Figure 1a]. Such accidental degeneracy is crucial for the formation of a Floquet topological phase 7 and the emergence of topological edge states. In contrast to the traditional implementations of topological physics, these modes have hybrid origin being a superposition of two waveguide modes with different symmetry of the near field. As we demonstrate, this feature enables a coherent control of the topological edge state.
Generally, the goal of coherent control phenomena is to tailor the relative phases of multiple excitation signals to control the response of the system in real-time [32][33][34] . In photonics, this is often achieved through the interference of excitations from multiple ports 35,36 . In our case, coherent control is ensured by manipulating the relative phase between two waveguide modes, which allows us to switch between the excitation of edge and bulk modes of the lattice. To image the modes of the fabricated waveguide nanostructure, we excite it with short laser pulses and collect the third harmonic signal, which provides a sensitive tool for detection and localization.  Figure 1: (a) Scanning electron microscopy image (SEM) of the fabricated structure with schematic profiles of the near field distributions for two quasi-degenerate modes of the structured waveguide (yellow curves). Depending on the relative phase difference with which these modes are initiated through the directional couplers, either a topological edge mode (red curve) or bulk modes (green curves) of the waveguide array are excited. (b) Schematic of the |E y | 2 field distribution for the two modes of the neighboring waveguides. The coupling between them gives rise to the hybrid modes with asymmetric field profiles.

Design of the Waveguide Lattice
The key ingredient necessary to obtain the topological features in our setup is the degeneracy of two modes with distinct symmetries of their near field distributions as sketched in Fig. 1b.
If two degenerate modes E y1 and E y2 exhibit different parity of the field relative to reflection in Oyz plane, their linear combinations become strongly asymmetric relative to the waveguide axis. This immediately affects the magnitude of the evanescent coupling between the adjacent waveguides [ Fig. 1b].
In an array of waveguides, hybridization of their modes gives rise to bulk and edge states.
If two degenerate modes of an edge waveguide are excited in phase, their linear combination E y1 + E y2 will feature the hot-spot at the edge of the structure decoupled from the rest of the lattice. On the contrary, if the same modes are excited out of phase, their linear combination E y1 − E y2 will feature the hot-spot on the other side of the waveguide ensuring good coupling to the bulk of the structure [ Fig. 1b]. This reasoning suggests the emergence of an edge-localized state with a hybrid nature.
To achieve the desired functionality and support our qualitative reasoning, we design subwavelength structured waveguides etched from crystalline silicon on sapphire substrate as shown in Figure 2a. Such nanostructuring of the waveguides provides the periodic permittivity modulation essential for dispersion engineering of the two modes 37 . In our case, by tuning the width of the gaps cut in an initially homogeneous silicon ridge waveguide, we tailor the dispersion of the two modes of interest -a fundamental hybrid HE 11   (c) Intensity distributions of the y component of the electric field |E y | 2 in the unit cell of the individual periodic waveguide. Left two panels: |E y | 2 for hybrid HE 11 and quasi-TE 01 modes, respectively, at their intersection frequency, "+" and "-" indicate the phase of the field. Right two panels: |E y | 2 for the superposition of two modes |E y1 ± E y2 | 2 . (d-f) Calculated dependence k z (k x ) for the periodic array at three representative wavelengths λ marked in panel (b) by the horizontal dashed lines.   Figure 2c] which effectively decouples the edge state from the rest of the array. In contrast, the field of the bulk states characterized by a π phase difference between the waveguide modes, E bulk ≈ E 1 − E 2 , is strongly suppressed at the edge of the array being coupled to the rest of the lattice.
The hybrid nature of the topological mode provides a useful tool for its coherent control.
Since the fields of bulk and edge modes are strongly asymmetric relative to the center of the edge waveguide, they can be selectively excited by launching the excitation through the auxiliary homogeneous single-mode bus waveguide (SMWG) placed either from the left or from the right edge from the edge waveguide (Fig. 1a). Here, the width of the SMWG is adjusted in such a way that the dispersion of its fundamental mode crosses the intersection point of the two modes of the multimode waveguide.
To quantify the degree of coherent control of our system, we simulate the field distribution at the output facet of the array for the various relative phases φ between the two waveguide modes launched into the edge waveguide. To achieve that, we assume that the left and right single-mode bus waveguides are excited simultaneously with π/2 phase shift between the excitations: E tot = E r cos φ 2 + i E l sin φ 2 . Since excitations of right and left couplers translate into E 1 + E 2 and E 1 − E 2 superpositions of the waveguide modes, the output field

Results
The designed array of nano-waveguides was fabricated on a silicon on sapphire substrate with a 600 nm device layer using standard CMOS-compatible fabrication techniques (see Methods section). The optical image of one of the fabricated devices, as well as zoom-in SEM images of its elements, are provided in Figure 4b. To excite the structure from the far field, we have added linear tapers and fully etched grating couplers to the designed SMWGs.
As a convenient tool to visualize the field of bulk and edge states, we exploit the thirdharmonic generation (THG) process. Since the topological states are highly confined and the third harmonic signal is proportional to the cube of the field at the fundamental frequency, this technique provides excellent spatial resolution to capture the finest details of the field distribution 24,39 allowing to outcouple light from the waveguide structure for far-field detection.
The scheme of the experimental setup is shown in Figure 4a. Linearly polarized laser pulses with magnetic field oriented along the grating bars (TM-polarization) are focused on one of the grating couplers marked by blue in Figure 4b from the substrate side. The pulses are generated by an optical parametric amplifier (see Methods for details), which allows us to tune the excitation wavelength within a broad spectral range. Light coupled to the planar waveguide via the grating passes through the taper region and excites one of the SMWGs.
The energy from the SMWG is further transferred to the extended structured waveguide (marked by green in Figure 4b) in the form of in-phase or out-of-phase superposition of the two waveguide modes depending on the chosen excitation port: right (1) or left (2) coupler.
The area of the sample containing the directional coupler and the waveguide array is then imaged on a CCD camera. The detected third harmonic signal allows us to visualize the propagation of the modes in the waveguide array.
In agreement with our simulations, the coupling of light through coupler 1 leads to the excitation of a topological edge mode, which manifests itself as a strong THG signal localized at the edge of the waveguide array (Figure 4c-e). In contrast, excitation through the coupler 2 reverses the phase difference between the modes of the extended waveguide, which causes the diffraction of waves into the bulk of the array (Figure 4f-h). The topological edge mode emerges in the range of wavelengths from 1530 nm to 1600 nm, consistent with our numerical results; the localization of the edge mode at a wavelength of 1560 nm appears to be especially pronounced (Figure 4d), which indicates the optimal combination of the directional coupler efficiency and the proximity to the degeneracy point of the two waveguide modes.
Notably, the third harmonic signal from the waveguide array decreases with the increase of the detuning between the propagation constants of the waveguide, which is mostly due to the limited bandwidth of the directional coupler. Even larger detunings of the excitation from the degeneracy point lead to the closure of the topological gap. In this regime, bulk modes are excited both when the structure is fed through coupler 1 or coupler 2. This trivial case is illustrated in Supplementary Materials.

Conclusions
In conclusion, our experiments demonstrate a novel avenue to design highly controllable topological insulator structures and meta-devices at the nanoscale. Owing to the multimode nature of the lattice, the observed topological modes have hybrid nature exhibiting different excitation efficiencies for the different input ports of the optical signal. This feature gives an immediate access to their coherent control and switching in real-time.
At the same time, the topological properties of our structure can be manipulated by changing the wavelength of excitation which determines not only the localization length, but also the very existence of the topological edge mode. We believe that the proposed strategy could be also beneficial for other classes of topological meta-devices such as topological waveguides, cavities and lasers.  acknowledges partial support by the Foundation for the Advancement of Theoretical Physics and Mathematics "Basis".