Multiband topological heterojunctions on the surface nanoscale axial photonics platform

Abstract

Analogue Hamiltonian simulation (AHS) in photonic systems can be an enticing alternative to direct experimental study of complex Hamiltonian systems, leading to the emergence of the field of topological photonics. In this work, we leverage the ultra-low loss operation and sub-angstrom fabrication precision of the Surface Nanoscale Axial Photonics Platform to experimentally demonstrate the first topologically nontrivial SNAP devices by coupling adjacent SNAP microresonators to form Su-Schrieffer-Heeger (SSH) lattices. Each axial mode of the microresonator array manifests distinct topological band structures, enabling us to observe behavior close to and far from the topological-trivial phase transition. We further expand the scope of topological SNAP systems to junctions between higher-order, multiband lattices with dissimilar topological phases created by coupling up to 21 well-matched SNAP microresonators. Analyzing these previously unexplored “heterojunctions" necessitated our development of generalized topological polarization methods. We thus demonstrate the exceptional promise of the SNAP platform for AHS of topological insulators.

The Surface Nanoscale Axial Photonics (SNAP) platform provides low-loss resonators with sub-angstrom precision. Here, authors realize topological analogue Hamiltonian simulation in SNAP microresonator arrays, demonstrating multiband SSH lattices and heterojunctions for scalable studies of topological systems.

Introduction

Analogue Hamiltonian simulation (AHS) has emerged as a powerful paradigm to study the nontrivial physics—in particular topological physics—hosted by a plethora of lattice models^1–3. By engineering the couplings between the lattice sites of an easily controlled physical system (the simulator) to mimic the couplings in the Hamiltonian of another, more difficult system (the simulated system), AHS enables the study of the simulated system with a relaxed set of experimental restrictions⁴. This approach has seen rapid advances across several quantum platforms, such as superconducting circuits⁵, trapped ions^6,7, and photonic circuits^8,9, each of which offers distinct advantages in control, scalability, and implementation. Hamiltonians with topological properties are of particular interest, as those properties can be simultaneously robust to local fluctuations¹⁰ but sensitive to global effects¹¹. For example, topological systems have been used to demonstrate back-scattering immune transport^12,13 and to measure fundamental constants with unmatched precision¹⁴. Several key factors determine a particular platform’s suitability for simulating topologically nontrivial phenomena: most obviously, it must be possible to manipulate the simulator system in such a way that the dynamics of its degrees of freedom align with those of the simulated system. But also, the simulator must accurately mimic the dynamics as opposed to succumbing to loss or decoherence as a result of nonideal factors like fabrication imperfections, noise, or unintentional interactions.

Photonic devices are among the most common platforms used to perform AHS¹⁵ of topologically nontrivial lattice models^16–20 as a result of the photons’ inherent mobility and lack of interparticle interactions relative to other platforms. Both of these features enable photons to traverse large optical microresonator lattices or networks with minimal noise and loss^20–23. The simulators take the form of either arrays of coupled microresonators²⁴ or waveguide arrays^25,26 where the propagation direction takes the role of time. Additionally, naturally independent photonic degrees of freedom, e.g., frequency or orbital angular momentum, can often be effectively manipulated as synthetic dimensions^19,27 to replace or complement the physical dimensions of the simulated system. Synthetic dimensions can improve the scalability of optical systems²⁸, and can also be used to analyze systems that are impossible to physically create since synthetic dimensions often lack restrictions like locality²⁹, unitarity³⁰, or dimensionality^31,32. However, one primary challenge of photonic AHS devices is the requirement that microresonator arrays be both low in loss (and thus narrow in linewidth), but also well-matched in frequency. In photonic integrated circuits (PICs), this often necessitates active optical elements^33,34.

The surface nanoscale axial photonics (SNAP) platform has emerged as a promising option to tackle this challenge³⁵. SNAP devices are made by engineering nanoscale effective radius variations (ERV) into the air-glass interface of the surface of optical fibers (1a)³⁶. They routinely demonstrate superior propagation losses (<0.001  dB/cm)^35,37 compared to typical PIC performance (approximately 0.01 dB/cm to records near 0.001 dB/cm³⁸) and intrinsic quality factors beyond 10⁸³⁹. SNAP shows substantial advantages in equipment costs and fabrication speed compared to PIC systems often used in investigations of topological photonics²⁴. For example, a single optical fiber can host dozens of identical SNAP resonators with sub-angstrom fabrication precision⁴⁰ since SNAP microresonators can easily be post-processed after characterization. This post-processing or trimming is non-volatile and local to each microresonator, that is, it does not disturb other parts of the device⁴¹, and has negligible additional manufacturing costs compared to PICs fabricated by planar lithography, which require active tuning to compensate for nanometer-scale fabrication errors. Hence, it is natural to ask whether topological photonics can be experimentally investigated on the SNAP platform.

Fig. 1. SNAP SSH2 lattices.

a visualization of the CO₂ laser annealing fabrication process and the tapered optical microfiber scanning method (feature sizes not to scale). b A “polymer” depiction of an SSH2 lattice with 10 sites and staggered couplings v₁ and v₂. Unit cells are labeled by an integer n ∈ {0, 1,…,5}, with the depicted structure having 4 full unit cells and 2 half unit cells. Each of the sites within are labeled by an integer s = 1, 2. Spectrograms from a c computational model and d experimental data of the above SNAP device, presenting several axial orders corresponding to the band structure of lattices analogous to that of (b). Blue lines represent the estimated ERV profile along the fiber axis and thus an optical potential. Red arrows point to locations of loss-inducing localized defects.

Here, we answer this question in the affirmative by reporting the first SNAP-based demonstrations of several topologically nontrivial Hamiltonians. Furthermore, we leverage the speed and relative ease with which new SNAP devices can be fabricated to showcase a more complex and previously unstudied structure—a junction between distinct higher-order Su-Schrieffer-Heeger (SSH)⁴² lattices. Such interfaces have not been well-studied theoretically⁴³, which we address by generalizing topological polarization analysis methods to higher-order SSH lattices, revealing how “fake” topological interface modes contingent on the interface’s implementation may emerge. The structure is composed of 21 frequency-matched adjacent SNAP microresonators with evanescent couplings controlled by the intersite spacings and dependent on which axial mode resonances within the SNAP resonator are utilized. We thus demonstrate SNAP’s excellent suitability for the investigation of Hamiltonians with interesting topological properties and conclude with a discussion on prospects for expanding such investigations to two- or higher-dimensional topological models.

SSH lattices^42,44–56 consist of 1D arrays of sites $\mid z_j⟩$ for j = 1, 2, 3,…, L which are coupled in a local and periodic manner (we refer to a lattice with unit cells of length ℓ as an SSHℓ lattice), and are hallmarked by their topologically protected edge modes^26,43,48,57. The general Hamiltonian of the SSH-like lattices we consider can be written as:

$$\widehat{H}={\sum }_{j=1}^{L}u\left(j\right)\mid z_j⟩⟨z_j\mid +v\left(j\right)\left(\mid z_{j+1}⟩⟨z_j\mid +\mid z_j⟩⟨z_{j+1}\mid \right),$$ 1

where v(j) = v(j + ℓ) are the adjacent couplings and u(j) = u(j + ℓ) are self-couplings. While SSH models only consider the ideal case that u(j) = 0 or constant, we include them while analyzing our Hamiltonian models, as they will appear unintentionally and require additional effort to compensate for them. While the conventional 2-band SSH has been extensively explored theoretically and experimentally^31,50,58, less work has focused on its ℓ-band extensions with ℓ > 2^44–46.

Figure 1 depicts the relationship between (a) the fabrication of a SNAP device, and (b) the abstract 1D lattice model instantiated by the completed device. Annealing a section of the fiber by CO₂ laser exposure induces positive ERV (dilations)^35–37,41, localizing a discrete series of optical modes to sites along the fiber’s axis (Fig. 1a). Adjacent resonators couple evanescently, which also couples and hybridizes their whispering gallery modes (WGMs). The unit cells resemble small molecules that are connected into polymer-like structures, motivating the analogous naming convention used in Fig. 1b, which depicts an SSH2 lattice, composed of 4 whole unit cells (referred to as “dimers”) and 2 incomplete unit cells on the edges (called “monomers”).

Results and discussion

In this work, the SNAP microresonators that serve as the building blocks of SSH lattices are initially formed using 30 ms pulses from a CO₂ laser operating at 5 W. The laser annealing of fused silica optical fibers induces momentary localized heating of the fiber surface through optical absorption³⁷. The fiber can thus recrystallize into a lower-stress state for which the effective radius (which depends on both the geometric radius and index of refraction) in the exposed section of the fiber is increased (Fig. 1a). This thermally-induced stress release is inconsistent⁴¹, so it is difficult to predict the resulting ERV after any particular exposure. However, a unique benefit of the SNAP platform is the expedience with which SNAP devices can be characterized and post-processed to correct fabrication errors^41,59. To ensure uniformity, the resonators are first scanned using a tapered microfiber to gauge the size and location of its imperfections, after which, the lowest ERV resonators are adjusted by firing additional laser pulses lasting between 20 and 25 ms. This process was repeated about 10 times until the resonances of the relevant axial mode order(s) of the microresonators were matched to within the desired tolerance.

The measured spectrogram of a 5-site SSH2 (the standard SSH) structure is shown in Fig. 2, accompanied by a best-fit computed spectrogram obtained using Eq. (6) with a potential as described in Supplementary Note 2. It consists of 5 sites separated by alternating spacings of 125 and 100 μm, just like Fig. 1. By fitting this data based on the model in Supplementary Note 1, we were able to design the structure depicted in Fig. 3 and obtain the parameters used for the resonators in Fig. 4. This structure develops a single topological mode concentrated on the monomer whose intensity will decay exponentially as it leaks into the bulk at a rate of $2ln\left(v_1/v_2\right)$ per unit cell⁶⁰.

Fig. 2. 5-site SSH2 lattice.

measured (a, b) and model (c, d) spectrograms of the 5-site SSH model. Purple rectangle corresponds to focused views (b, d) showing the q = 4 and q = 5 axial mode families. The lower-order mode families (higher in resonance wavelength) are too localized to observe coupling between the lattice sites. False color orange overlays highlight topological edge modes for q = 4, 5, and dark (light) green arrows indicate the positive (negative) energy bands.

Fig. 3. SSH2-SSH4 lattice interface.

band structure of a a SSH2 lattice and of a b SSH4 lattice. Upper (lower) bands are red (blue), and inner (outer) bands are dashed (dotted). c Bare tight-binding Green’s function⁸⁰ intensity (arb. units) of tight-binding model of a large SSH2 lattice on the left (v_L(s) = [1, 0.5], N = 400 unit cells) conjoined to an SSH4 lattice on the right (v_R(s) = [1, 0.7, 1, 0.3], N = 400 unit cells). Zero- energy topological bound modes are tinted orange, non-zero energy bound modes are tinted green, and yellow arrows indicate the quasi-bound modes. Colored brackets next to the bands indicate the corresponding bands in (a, b). d Polymeric diagram of a 21-site lattice. The site on the (dashed purple) domain line can be considered a part of either lattice. e Measured spectrogram of a topological heterojunction. An SSH2 lattice (v_j corresponds to spacings of 100 and 125 μm) is conjoined to an SSH4 lattice (v_j corresponds to spacings of 100, 105, 100, and 130 μm). This measured spectrogram closely follows the tight-binding spectrogram in (a) barring the expected local self-coupling shifts, also present in Figs. 1 and 2. Arrows and brackets indicate correspondence between modes and bands as in (a–c).

Fig. 4. SNAP resonator axial mode coupling.

SNAP resonator axial mode coupling: a computed spectrogram of two adjacent sites with Δz = 115 μm showcasing hybridization of axial modes (color coded). The q = 1 fundamental modes are unhybridized, while the high-order modes are almost completely hybridized. b Wavelength splitting induced as a function of intersite spacing for q = 1 to 8 axial modes of the SNAP resonators. At large Δz, the expected exponential decay is observed. At small Δz, a saturation of the wavelength splitting, and hence of the coupling between the sites, is observed.

As in Fig. 4, the dimers of lower-order axial modes (q ≤ 3) are effectively isolated from each other, and even the sites within each dimer are isolated for q ≤ 2. There is some mode coupling apparent between the q = 3 modes, but the gap between the modes arises primarily from the residual resonator mismatch (c.f. modeled coupling from Fig. 4), thus we omit analysis of the q = 3 mode. For axial mode families with q ≥ 6, the axial mode lattices can no longer be distinguished because their potential wells have merged and the couplings have saturated. These two limits correspond to the fully-dimerized and the nearly-critical regions of the SSH model’s topological phase diagram, respectively⁶⁰. We focus on the axial mode lattices with q = 4 and q = 5 (Fig. 2c, d), where the key characteristics of the SSH model can be observed most directly. In those axial mode lattices, the positive (dark green arrow) and negative (light green arrow) bands of modes are distributed throughout the bulk and separated by a bandgap. We observe edge modes that hallmark the successful creation of topological models. While both topological edge modes are localized on the left side of the SNAP lattice, the q = 5 edge mode was significantly less localized, such that its residual transmission after exponential decay from the edge can be seen on the right site of the n = 1 cell around z = 300–400 μm. Based upon the locations of the resonances, we estimate that the couplings of the fifth modal order are approximately v₂ ≈ 12 pm and v₁ ≈ 36 pm (equivalent to 1.5 and 4.5 GHz, respectively), which are comparable to the predictions from Fig. 4 of v₂ = 8 pm and v₁ = 30 pm. The corresponding decay lengths are thus ${\left(2ln\left(v_1/v_2\right)\right)}^{-1}\approx 0.46$ unit cells for the predicted couplings, and  ≈ 0.38 unit cells for the actual lattice. We can also infer that the decay lengths for q < 3 are much less than 1 unit cell (implying the edge states are effectively just position eigenstates), and the decay lengths for q > 6 are more than 5 unit cells (implying the edge states diffuse across the entire bulk), explaining why those modes do not noticeably drop in intensity as they leak into the bulk. The small gap between the experimental couplings and the simulations’ estimates can be attributed to a necessary assumption made in the fitting process that consistently underestimates the ERV profile between resonators (explained in Supplementary Note 2). With this consideration, the SNAP device successfully replicates the physics expected from the SSH lattices.

Given the relative ease of producing the SNAP SSH lattices depicted in Figs. 1 and 2, we next showcase the platform’s versatility by instantiating the first heterojunction of higher-order (multiband) SSH lattices, which has not been accomplished on other platforms to the best of our knowledge. Specifically, we demonstrate an interface between an SSH2 section of a lattice and an SSH4 section. Figure 3 shows the band structure of SSH4 lattices in comparison to SSH2 lattices (a, b), direct tight-binding simulations (c), and experiments (e), for which numerous localized modes emerged at the domain boundary. The overall lattice can be considered either as an 11-site SSH2 lattice attached to a 10-site SSH4 lattice, or a 10-site SSH2 lattice attached to an 11-site SSH4 lattice, since the allocation of the shared site at the center is ambiguous (such ambiguity of the sites can not be avoided without imposing distinct on-site potentials/self-couplings patterns for both lattices). Either way, the last site of the SSH2 region is strongly coupled to the first unit cell of the SSH4 region, resulting in an effective “pentamer” at the junction between the two regions.

Figure 3c shows the bare Green’s function of a similar SSH lattice expressed in the site position (j) basis rather than the real-space (z_j) basis, computed by diagonalizing a large (L = 800 sites) but finite tight-binding Hamiltonian. With no intrasite intensity variations and a wide bulk for the free modes to distribute themselves across, it is easy to spot the interface modes and compare them to the continuum modes. There are three real modes localized to this interface: a standard zero-energy mode in the central bandgap (orange), and two nonzero-energy bound modes just outside the highest and lowest bands (green). While there are 4 bands on the right (SSH4) side, the auxiliary bandgaps are closed on the left (SSH2) side. Because of this, two quasi-modes—lossy resonances distributed across a range of energies (wavelengths) as opposed to a single energy—appear within the auxiliary bandgaps. These quasi-modes are artifacts of the SSH4 lattice’s localized auxiliary bandgap modes after having been coupled to the SSH2 lattice. While the topology of the right lattice forbids the auxiliary bandgap modes from entering it, the topology of the left side permits it, allowing leakage into its bulk. The small distance between the centers of the quasi-modes (localized on sites 3 and 4 of the pentamer) and the SSH2 side prevents them from being totally delocalized.

In the measured spectrogram from Fig. 3e, the fundamental modes of the depicted resonators are equalized in ERV (wavelength) to within a standard deviation of about 0.5 nm (40 pm), which can be considered as an upper bound on the detuning noise in this lattice. The post-processing of the device was focused on optimizing the depicted q = 3 axial mode lattice. There is a substantial amount of uncertainty, considering the adjacent couplings are of approximately the same order of magnitude or less. Despite this, the main topological feature of the lattice can easily be observed: the zero-energy topological mode (orange) that is present at the center and distributed mostly on the sites adjacent to the domain wall (dashed line). Additionally, there is a distinct opening of auxiliary band gaps when crossing from the left to the right sides. While the negative-energy interface bound mode (green) lies firmly below the bands on either side, the positive energy one is nearly at the same wavelength as the top of the band of the right, partly attributable to the Fano resonance lineshape and the lower uniformity of couplings for longer wavelength modes. Lastly, there exist resonances around the right lattice’s auxiliary bandgaps (yellow) where the quasi-modes are expected to be, but they are close in energy with the bulk modes on the left and spatially merge into them, making these modes difficult to isolate.

The topological nature of the interface between SSH models with dissimilar unit cells is only beginning to be explored^43,57, necessitating that we fill this gap in order to determine whether the modes observed in Fig. 3 are topological. The primary obstacle to generalizing standard topological theory to these systems is that the symmetries upon which topological invariants are defined are imperfect or only locally true, as opposed to being globally true. In other systems, imperfect symmetries have been dealt with by basing the definition of the topology on the scattering phases⁶¹ of incident particles reflected from the system in question’s boundaries. This definition can be employed in a significantly more general class of systems, such as those with interparticle interactions or inexact symmetries, as it only requires determining the boundary Green’s function^62–64, which can be obtained through, e.g., transfer-matrix^48,53 methods. However, current forms of this method are inherently focused on midgap zero-energy topological properties of external boundaries, which is only a part of the topology of the systems we are interested in.

By basing the topology on relative phases of the Bloch modes, we can analyze the topological properties of systems that lack complete chiral and inversion symmetry (i.e., these symmetries are only present in the bulk) due to a non-integer number of unit cells or domain boundaries^65–67. In the bulk of a lattice (or for periodic boundary conditions), Bloch’s theorem can be applied to group the eigenmodes of $\widehat{H}$ into momentum modes $\mid k⟩$. This folds the Hamiltonian into an ℓ × ℓ matrix $\widehat{H}\left(k\right)=⟨k\mid \widehat{H}\mid k⟩$ acting on the sublattice for each k^45,47. Each of the ℓ eigenvectors of $\widehat{H}\left(k\right)$ will define a band μ of eigenmodes $\mid k_{\mu }⟩=\left.∣\right)u_{\mu }\left(k\right)\left.⟩\right)\otimes \mid k⟩$, where we define these discrete Bloch modes and momentum eigenmodes as:

$$\left.∣\right)u_{\mu }\left(k\right)\left.⟩\right)={\sum }_{s=1}^{\ell }\frac{a_{\mu }\left(k,s\right)}{\sqrt{\ell }}\exp \left[-i{\theta }_{\mu }\left(k,s\right)\right]\mid s⟩,\mid k⟩=\sqrt{\frac{2\pi \ell }{L}}{\sum }_{n=n_1}^{n_L}\exp \left(ikn\right)\mid n⟩.$$ 2

The basis vectors are factored as $\mid z_{n\ell +s-s_0}⟩=\mid s⟩\otimes \mid n⟩$ with sublattice index s ∈ [1, ℓ] and unit cell index n ∈ [1, N] such that s_j (n_j) is the sublattice (unit cell) index assigned to site j. Wherever nℓ + s corresponds to a nonexistent site, the wavefunction is taken to be zero instead. The sublattice phases θ_μ(k, s) and amplitudes a_μ(k, s) control the wavefunction’s dependence on s.

Depending on the bulk’s symmetries, isolated bound modes can appear within bandgaps and at the boundaries of SSH-like lattices. While SSH2 only allows modes within the central bandgap, non-zero energy modes can also emerge within the auxiliary bandgaps (those not centered at zero) when ℓ > 2, increasing the number of possible topological phases^45,52. When trying to form the eigenmodes of an open-boundary lattice by linearly combining Bloch modes⁴⁵, the requirement that wavefunctions vanish just outside the lattice induces a condition that the phase of propagation between virtual sites j = 0 and j = L + 1 must be an integer multiple of π:

$$m_{\mu }\left(k\right)=\frac{\left[k{n}_{L+1}-{\theta }_{\mu }\left(k,s_{L+1}\right)\right]-\left[k{n}_0-{\theta }_{\mu }\left(k,s_0\right)\right]}{\pi }\in {\mathbb{Z}}_N,$$ 3

Depending on the values of s_L+1 and s₀, the total number of unique solutions to Eq. (3) between all the bands may fall short of L, as the bound modes lie in the complex momentum plane at the time reversal invariant momenta $\mathrm{Re}k=\pm \pi$. We can therefore determine the number of bound modes from the band Q by the difference between the total modes L and the number of free modes (the modes within a reduced Brillouin zone⁴⁵, equivalent to $1/2{\int }_{-\pi }^{\pi }{m}_{\mu }^{\prime }\left(k\right)dk-1$ for each band), giving the formula:

$$Q=\left(s_{L+1}-1\right)-\left(s_0-\ell \right)+{\sum }_{\mu }\oint \frac{dk}{2\pi }\frac{\partial }{\partial k}\left[{\theta }_{\mu }\left(k,s_{L+1}\right)-{\theta }_{\mu }\left(k,s_0\right)\right].$$ 4

This reduces to the winding number and Zak phase^49,60,63,68 when ℓ = 2, and will be discretized in the general case because the discrete translational symmetry forces θ_μ(+π, s) = θ_μ(−π, s) mod 2π regardless of other symmetries. Eq. (4) is gauge- and basis-independent, and can be considered a discrete version of Levinson’s theorem⁶⁹ composed of the previously defined “normalized sublattice phases”⁴⁵ with additional terms to remove the intracellular Zak phase⁵¹. Physically, the integral counts the winding of the wavefunction phase between the s_L + 1 sites and s₀ sites as one circumnavigates a Brillouin zone, implying it could be measured through interferometric techniques and coherent detection. Having the freedom of unit cell basis can reduce counting topological modes to counting the dangling sites on the edges of the lattice, as it is possible for all the normalized sublattice phases to be zero, making Eq. (4) depend only on the starting/ending sites. All unit cell bases in this work are chosen this way, except in Supplementary Note 3.

Converting Eq. (4) into the position basis, Gauss’s law can be used to convert the expression from a sum over the boundaries to a sum over the bulk, making the total bound modes:

$$Q=\left(\ell -1\right)+{\sum }_{j=0}^L\left[\right)D\left(j+1\right)-D\left(j\right)\left]\right)\mathrm{with}D\left(j\right)=s_j-{\sum }_{\mu }\oint \frac{dk}{2\pi i}\frac{⟨s_j\mid {\partial }_k\mid u_{\mu }\left(k\right)⟩}{⟨s_j\mid u_{\mu }\left(k\right)⟩}.$$ 5

The divergence of D(j) averages to 0 across each unit cell, making changes in D(j) across an integer number of unit cells signify the presence of bound modes. We will refer to D(j) as the sublattice displacements, as it can be considered a generalization of chiral displacement/polarization^50,68 for non-chiral systems. It is a local^70,71 but basis- independent property, making it serve as a convenient definition of the topological invariant for systems with local but not global symmetries, such as the SSH2-SSH4 junction of Fig. 3. Furthermore, the sublattice displacements at either ends of the lattice can be directly related to the amount of bound states at that end of the lattice (Supplementary Note 4).

When the number of bands changes between the left and right sides of a junction, the two sides do not naturally belong to the same topological classification system. Furthermore, we will see that non-topological bound modes—modes that may disappear after mild deformations of the Hamiltonian—appear, which motivates us to identify topological changes with the minimum number of bound modes and/or quasi-modes under topology-preserving variations (referred to as the net topological modes). We propose two methods of predicting the net topological modes in such scenarios.

The first method is to guess and check topology-preserving lattice deformations (those that do not cause the merger, splitting, or crossing of bands⁶⁰ and can thus be performed adiabatically) to manually search for the minimal number of interface modes. Notably, unit cells can be formed or removed in groups of ℓ sites around the junction, potentially changing the number of interface modes by ℓ (e.g., Supplementary Fig. 5a relative to Fig. 3a), which is why the topological invariant can only be defined mod ℓ. In a simple system like this, it is not difficult to guess a configuration that minimizes the number of modes- one can break up the pentamer at the junction into two dimers and a monomer. At this perturbed interface, the only mode is the zero-energy mode (Supplementary Fig. 4c), implying that the 2 extra modes and quasi-modes apparent in Fig. 3b, c are non-topological as the bulks have been left untouched.

We can more systemically approach the problem by bringing each part of the lattice under the same classification system and reconsidering both lattices as sharing a common period equal to their least common multiple (e.g., ℓ = 2 and ℓ = 4 lattices are both ℓ = 4 lattices). However, the bands of the system can not be uniquely reconsidered as multiple conjoined bands. Additionally, this puts the system directly between two topological phases (a topological critical point), making Eq. (5) ill-defined. To deal with this, we consider symmetry-preserving minute deformations of the Hamiltonian⁷² to characterize the topology by the properties that persist on either side of the critical point (e.g., the sublattice displacements and bound modes).

Specifically, we take the first 4 cells of the lattice in Fig. 3a as a unit cell of an SSH4 lattice with couplings v_L(j) = [1, 0.5, 1, 0.5], and then consider perturbations that split the auxiliary bands (Supplementary Fig. 4) of the form $v^{\prime }\left(j\right)=\left[1,0.5+\delta ,1,0.5-\delta \right]$ (note that both the chiral and inversion symmetries of the bulk of the SSH4 model^47,57 are preserved). The original junction (when δ = 0) will have different interface modes in comparison to when δ > 0 or when δ < 0, as well as different sublattice displacements. Table 1 compares the values of the sublattice displacement on a representative unit cell of the left lattice for δ < 0, δ = 0, and δ > 0 with the corresponding values on the right lattice and the minimum number topological modes as determined by the manual search method in Supplementary Note 3. For the δ = 0 system, the changes in the sublattice displacements (computed in SSH2 basis) for each possible sublattice index do not give a consistent result: depending on how the lattice sites are assigned into groups, there can either be one or three interface modes, indicating a failure of the sublattice displacements to predict the number of topological modes. However, there are consistent predictions for the perturbed systems, albeit to two distinct common answers: the two possible predictions for the δ = 0 system. This seems to suggest that the inconsistency in the predictions for the δ = 0 system arises because it lies exactly at the critical point of SSH4 systems. Alternatively, it can be recognized that D_L should only be considered to refer to unique phases mod 2, making the same hold for D_L − D_R, in which case the predictions of 1 and 3 modes become equivalent, making 1 the true number of net topological modes.

Table 1.

Analysis of the number of topological modes using sublattice displacements of perturbed lattices

jL (jR)	DL			DR	DL − DR mod 4
	δ < 0	δ = 0	δ > 0		δ < 0	δ = 0	δ > 0
1 (17)	3	1	1	4	3	1	1
2 (18)	4	2	2	1	3	1	1
3 (19)	1	1	3	2	3	3 (1)	1
4 (20)	2	2	4	3	3	3 (1)	1
			Topological modes		3	1	1

Sublattice displacements D(j) on a representative unit cell (j_L = 1, 2, 3, 4) from the left lattice for δ = 0, δ < 0, and δ > 0 perturbations, as well as the corresponding (j_R = 17, 18, 19, 20) values on the right lattice. The last three columns indicate their differences mod 4 (mod 2), which gives the number of topological modes except at δ = 0. When δ = 0, only distinct values of D_L mod 2 correspond to distinct topological modes, making D_L − D_R only correspond to unique phases mod 2, which is why the number of interface modes is written as 3 (1).

Both methods—manually testing local deformations and computing sublattice displacements—have thus concluded that there must be a bound mode at the interface within the central bandgap, corresponding to a change in the topology across said interface. While the former takes a heuristic path contingent on guessing the result of changing certain couplings, the latter can be employed as a brute-force solution to determining the possible configurations of the interface and the net number of topological modes. Alternatively, the sublattice displacements can educate and affirm guesses of local deformations minimizing the number of interface modes, making the combination of the two methods a versatile and reliable approach to analyzing the topological properties of lattice heterojunctions that posses multiple bands. While this method was only applied to a 1D system, it can likely be extended to higher-dimensional systems, akin to how chiral methods⁶³ extend from 1D to higher dimensions.

In summary, we have identified how the ultra-low loss and high fabrication precision of the SNAP platform make it substantially promising for 1D AHS. We demonstrate its unique capability to achieve non-volatile matching between more than 20 non-uniformly coupled resonator sites. The platform naturally supports multiple axial mode orders at each site, meaning our devices host an intrinsic parameter sweep over 5 distinctly-behaving coupling configurations, both far and close to the central topological critical point. We not only demonstrate the first topological lattices on the SNAP platform, but do so with angstrom-level precision. Following this, we ventured beyond what has been accomplished on other platforms by experimentally investigating multiband topological interfaces. To analyze this more general class of topological systems, we developed a methodology based on extending notions of topological charge and polarization to lattice models with arbitrary numbers of bands and without global symmetries. Notably, this analysis revealed that the structure possessed both a topological interface mode as well as two non-topological interface modes per axial mode lattice.

Our work lays a foundation for exploration of topological light control and its robustness to system disorder⁷³, with exciting future extensions of this work involving cavity-enhanced nonlinear-optical effects to couple light between resonant frequency modes^74–76. While this work is focused on 1D Hamiltonians, it serves a stepping stone towards implementing two- and higher-dimensional topological lattices. SNAP fibers’ resonances’ sensitivity to radii makes frequency- and dispersion-matching distinct fibers difficult, affecting the scalability in coupling numerous SNAP devices by either evanescently coupling adjacent fibers^77,78 or by using additional tapered fibers as interconnects. However, the small mode volume and low losses of SNAP resonators enable nonlinear effects to manifest at low intensities^39,74, opening the possibility for both the axial quantum numbers and the non-axial quantum numbers to be utilized as a synthetic dimension⁸. Specifically, different frequency modes (due to either a difference in axial, azimuthal, radial, or polarization quantum numbers) can be coupled together through Kerr-induced four-wave-mixing⁷⁹ by exciting the resonator with auxiliary lasers that satisfy the requisite energy and momentum conservation conditions. Finally, the theoretical topological framework developed in this work could potentially be extended to analyze such higher-dimensional systems as well as to analyzing topological insulators lacking symmetries in general.

Methods

The optical spectrum of the resulting SNAP device is then probed by placing a tapered optical microfiber oriented perpendicular to the sample on the surface such that it is evanescently coupled to WGMs in the sample, and measuring the transmission spectrum through the tapered microfiber. The WGMs localized in the SNAP devices are modeled using the Green’s function of the SNAP system in the energy-position basis, with wavelength playing the role of energy (Eq. 6)⁸⁰. The bare Green’s function models the WGM structure in the absence of any coupled tapers. The renormalized Green’s function includes the effects induced by the presence of a taper upon transmission. Destructive interference causes transmission dips to occur when the input light is resonant with WGMs in the SNAP device. By resolving the transmission in both wavelength λ and position z, the mode spectrum is obtained and visualized in spectrograms, with representative examples shown for the model and the experimental measurements in Fig. 1c, d, respectively.

SNAP resonators can support multiple axial modes simultaneously, each with a different evanescent tail extent beyond the microresonator, which allows the elegant realization of several SSH lattices with distinct coupling coefficients within a single 1D SNAP array. These axial modes, labeled by q ∈ {1, 2,...}, exhibit Fano lineshapes⁸¹, corresponding to the multiple vertically offset transmission dips seen within each microresonator in the 1D array of Fig. 1c, d. The presence of the tapered microfiber renormalizes modal wavelengths and linewidths in proportion to the local modal intensity. Hence, the measured resonances most closely approximate the bare resonances (i.e., WGM resonances in the absence of any coupling to a tapered microfiber) near the zeros/nodes of the measured mode profiles. The ERV curve is estimated by fitting data from measured spectrograms to that of the computational model of the SNAP device⁸² as described in Supplementary Note 2.

The fitted ERV curve exhibits a highly uniform array, evidencing the precision with which we can fabricate a large number of SNAP devices through iterative characterization and localized post-processing (Section “Results and Discussion”). On top of this uniform background, we observe localized distortions in the measured spectrogram near the positions z = 550 μm and z = 1150 μm, indicated by red arrows in Fig. 1d. Resonances that intersect those positions (particularly the q = 1, 2, and 4 modes of the n = 4, s = 2 resonator) exhibit linewidth broadening. These are indicative of localized defects (e.g., an impurity, such as dust on the surface of the SNAP device) inducing scattering losses; however, the defects did not cause notable changes to the edge modes, which attests to the topological protection they inherit from the underlying SSH lattice.

To simulate a Hamiltonian with a system of coupled resonators, we analyze the behavior of its supermodes in relation to the modes of the individual resonators. By identifying the uncoupled modes of each resonator with the basis vectors of the lattice $\mid z_j⟩$, perturbations to the optical system correspond to terms within a Hamiltonian, and the light’s dynamics mimic time evolution under that Hamiltonian.

Within a lone SNAP microresonator, for each polarization, azimuthal, and radial mode of the unperturbed fiber, there will be similar but independent axial mode series. The axial mode series can be determined by solving a 1D Schrödinger equation in the fiber’s axial dimension z, where the ERV profile Δr(z) plays the role of the potential, and the shift Δλ_q of the qth axial mode from the cutoff wavelength $\overline{\lambda }$ plays the role of energy⁸⁰:

$$\left(-\frac{1}{2{\beta }_0^2}\frac{{\partial }^2}{\partial z^2}-\frac{\mathrm{\Delta }r\left(z\right)}{\overline{r}}\right){\psi }^q\left(z\right)=-\frac{\mathrm{\Delta }{\lambda }_q}{\overline{\lambda }}{\psi }^q\left(z\right).$$ 6

Here, β₀ is the longitudinal component of the material wavenumber, the base fiber radius is $\overline{r}$ at zero ERV, and ψ^q(z) is the axial mode profile. In the tight-binding approximation^51,83, the Hamiltonian of the SNAP microresonator array can be rewritten as:

$$\widehat{H}\left(z^{\prime },z\right)\approx {\sum }_{j^{\prime },q^{\prime }}{\sum }_{j,q}\left.∣\right)z_{j^{\prime }}^{q^{\prime }}\left.⟩\right)H_{q,j}^{q^{\prime },j^{\prime }}\left.⟨\right)z_j^q\left.∣\right),$$ 7

where the axial mode profiles and thus basis vectors $\left.∣\right)z_j^q\left.⟩\right)\equiv {\psi }^q\left(z-z_j\right)$ correspond to a fixed ERV profile Δr(z − z_j), shifted to the location of the jth resonator, and $H_{q,j}^{q\prime ,j\prime }$ are the matrix elements of the overall Hamiltonian in this basis. This methodology—utilizing the bound modes of a complicated physical system to approximate idealized sites on a lattice—can be considered a reversal of the widely used tight-binding approximation, as a discrete lattice is approximated by the bound states of a physical system, as opposed to modeling the bound states of a physical system with a discrete lattice. In most tight-binding models, the Hamiltonian matrix elements can be determined directly by overlap integrals of the independent site potentials and their wavefunctions since the overall potential will simply be the sum of the individual potentials⁸³ corresponding to each “atom” of the system. Since the laser annealing process features a nonlinear response to the laser dose⁴¹, the cumulative ERV profile after multiple nearby exposures will not follow the same rule (e.g., a single exposure typically creates a super-Gaussian ERV profile, but a subsequent nearby exposure with the same power and duration does not result in a system that is well modeled by the sum of two super-Gaussians). More details are in Supplementary Note 2 and in Supplementary Figs. 1 and 2.

As a stepping stone on the path to building a full SSH lattice, we first investigate the effects of evanescent coupling between nearby resonators by directly solving Eq. (6) (see also Supplementary Note 2). This 2-site computation allows us to estimate the tight-binding matrix elements between adjacent lattice sites as a function of the intersite spacing Δz. A characteristic spectrogram obtained by these computations is displayed in Fig. 4. While the lower-order axial modes (q ≤ 2, higher wavelength) remain effectively degenerate at the unperturbed wavelength ${\lambda }_q^{\left(0\right)}$, the higher order axial modes hybridize with their twin into symmetric and antisymmetric combinations with wavelengths of ${\lambda }_q^{-}$ and ${\lambda }_q^{+}$ respectively. Past the 5th axial order, the site potentials are mostly merged, and the only remaining feature signifying a separate “left” and “right” mode is the minute increases in the intensity near their midpoint. To first order, the coupling between adjacent microresonators is given by the splitting of the resonances, $v^q\left(\mathrm{\Delta }z\right)=\left({\lambda }_q^{+}-{\lambda }_q^{-}\right)/2$ and the self-couplings will be given by the shifting of their mean $u^q\left(\mathrm{\Delta }z\right)=\left({\lambda }_q^{+}+{\lambda }_q^{-}\right)/2-{\lambda }_q^{\left(0\right)}$. This approximation is valid whenever the axial modes remain well separated in wavelength, i.e., when Δz is much greater than the modal widths Δw_q. Figure 4b shows v_q as a function of Δz for each q, revealing how coupling strengths follow a decaying exponential when Δz ≫ Δw, but saturate when Δz ~ Δw_q (the curves are truncated after saturation as certain modeling assumptions explained in Supplementary Note 2 fail for Δz < Δw_q). Δw_q increases with axial order, which explains why the higher-order modes couple more strongly. Coupling strength increases for either high q or small Δz, but very importantly, the variations in coupling strength (i.e., their derivatives in respect to Δz) decrease in the same limit. We can thus expect the couplings between higher-order modes to be both stronger and more uniform than the lower-order modes. For example, v₅(100 μm)/v₅(120 μm) ~3 but v₆(100 μm)/v₆(120 μm) ~1.5.

The couplings are nearly zero for spacings beyond 200 μm, which allows us to ignore non-adjacent couplings throughout this work, as the smallest adjacent spacings used are 100 μm. The only relevant terms within Eq. (7) are thus the self and adjacent couplings between modes of the same axial order ($q=q^{\prime },\mid j-j^{\prime }\mid \le 1$), i.e., those present in Eq. (1). By controlling the spacing between sites in a lattice, the v terms can be customized to any desired value between 0 and the saturation point, but not without inducing self-couplings. Therefore, a necessary consideration of making non-uniformly coupled resonator arrays is that the resonances will be detuned if we do not compensate for the associated nonuniform self-couplings. To compensate for the nonuniformity and also undo the effect of fabrication errors, we adjust the height of individual microresonators. This can be implemented with an increased initial dose from the CO₂ laser or by firing additional pulses during the post-processing step, as explained in Section “Results and Discussion.” Note that if non-zero self-couplings were desired (e.g., to make lattices with on-site detunings, such as the Rice–Mele model^60,84), one can use these steps to intentionally induce self-couplings.

To summarize, when building a lattice from the q^* axial mode of a SNAP microresonator array, the $u_{q^{*}}$ coefficient for each resonator can be chosen almost arbitrarily and the coupling coefficient $v_{q^{*}}$ between each pair of adjacent resonators can be chosen to be any positive value by using the coupling curves of Fig. 4b. For the axial mode lattices corresponding to q greater than q^*, there will be less variations in both the v_q and the u_q coefficients. Therefore, the SNAP platform enables simultaneous investigation of multiple lattice configurations so long as the assumption that axial mode lattices are independent remains valid.

Author contributions

N.F. designed, fabricated, measured, and modeled the SNAP devices depicted in this work, developed the topological heterojunction theory and created initial drafts of this manuscript. D.V. and A.D. conceived the idea, supervised the project, guided the investigation, and contributed to the writing of this manuscript.

Peer review

Peer review information

Nature Communications thanks Wange Song and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

A.D. acknowledges support from the National Science Foundation (NSF) CAREER award (# 2340835). N.F. acknowledges support from NSF (QuSeC-TAQS # 2326792) and Army Research Laboratory through the NSF-DEVCOM INTERN Program and from the DoD SMART Scholarship Program.

Data availability

The authors declare that the main data supporting the findings of this study are available within the article and its Supplementary Information files. Source data are provided with this paper.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains Supplementary material available at https://doi.org/10.1038/s41467-026-76041-1.