Splitter engineering through optimizing topological adiababtic passage

Abstract

Topologically protected states are important in realizing robust optical behaviors that are quite insensitive to local defects or perturbations, which provide a promising solution for robust photonic integrations. Here, we propose to implement fast topological beam splitters and routers via the adiabatic passage of edge and interface states in the cross-linking configuration of Su–Schrieffer–Heeger (SSH) chains with interface defects. The channel state does not immerse into the band continuum during the adiabatic cycle, making the adiabatic restriction less stringent and the transport process more efficient. Based on the accelerated topological pumping, the beam splitters and routers exhibit improved robustness against losses of the system yet degraded resilience to fluctuation of coupling strengths and on-site energies compared with the conventional topological splitting and routing schemes. In addition, we confirm that the model demonstrates good scalability when the system size is varied. The simulation results of topological beam splitting in coupled waveguide arrays are in good consistency with theoretical analysis. This topological design provides a robust way to control photons, which may suggest further application of topological devices with unique properties and functionalities for integrated photonics.

Graphical abstract

Keywords

topological effects in photonic systems / Su−Schrieffer−Heeger / topological photonics / waveguides

1 Introduction

Topological insulators (TIs), rooted in global geometric properties characterized by topological order rather than relying upon traditional Landau–Ginzburg theory, describe a novel class of solids [1-3]. Topological pumping, as a transport phenomenon employing topological properties of a modulated system, includes Thouless pumping via topological bulk bands [4, 5] and adiabatic pumping via topological edge modes [6, 7]. The former occurs when system parameters are varied cyclically and slowly so that the system is frozen in its ground state, exhibiting an intrinsic geometric character with the charge transport throughout one cycle determined by the Chern number and leading to the quantization of the excitation transport. Besides, owing to topological nonequivalence of its energy band structures in the momentum space as compared with traditional insulator, topological insulator possesses insulating bulk states and conducting boundary states at the same time [8, 9]. One hallmark of TI is the existence of the conductive boundary states that are not affected by localized defect and external perturbation, demonstrating lossless and one-way propagation immune from backscattering and thus finding promising applications in quantum information processing and low-power electronics [10, 11]. In finite systems with open boundaries sustaining topologically protected edge states, topological edge pumping has been exploited to implement robust excitation transfer through adiabatic evolution of the edge states [12, 13]. Non-trivial topological edge states (TESs) have been well investigated in both low-dimensional and high-dimensional photonic topological insulators [14, 15]. As the simplest one-dimensional lattice sustaining nontrivial topological boundary states, the Su−Schrieffer−Heeger (SSH) model [16, 17] depicting a dimerized chain with alternating strong and weak coupling strengths is widely used for topologically protected edge pumping and has been generalized and exploited to construct diverse functional devices, such as topological beam splitters [18-21], topological routers [22-24], and topological lasers [25-29]. Bound by adiabatic requirements, these devices generally require long evolution time for successful excitation transfer which limits their practical applications. Existing methods for fast topological pumping focus mainly on improving the transport efficiency in the point-to-point pumping, which lags far behind the need for information distribution in classical and quantum communication systems [13, 30-34]. Therefore, it is of urgent need and great significance to engineer relevant devices with more diversified functions and accelerate topological edge pumping based on the SSH model.

As an essential element in integrated optics, waveguide is a commonly used device for electromagnetic waves propagation and transmitting signals and energy of light. However, integrated photonic devices based on coupled waveguides usually exhibit high sensitivity and degraded performance to structural imperfections. Therefore, precise requirement is demanded in fabrication, which poses severe challenges to current manufacturing technology. Attempts to address this long-standing problem include compensation process by Mach–Zehnder interferometer [35], programmable solution for on-demand fabrication [36], and adiabatic passage designs [37-40], which usually require external intervention or rigorous parameter condition. More recently, the introduction of topological order has been theoretically proposed to enable robust and unidirectional reflectionless waveguiding, and robust light propagation based on topological edge states has been demonstrated on silicon platform [41-43]. Moreover, when multiple waveguides are arranged to form an array, the evanescent coupling between the waveguides expands the single waveguide mode into a band, which can be linked with the topological phases. Tunable band structure and topological boundary states can be constructed in waveguide arrays by adjusting the waveguide parameters, which allows for flexible manipulation of optical field, generating various phenomena such as adiabatic mode conversion [44], anomalous refraction and diffraction [45], and discrete solitons [46]. More interestingly, based on the similarity between the coupled mode equation of waveguides and the Schrödinger equation, the evolution of electromagnetic waves in waveguide arrays can be utilized to observe quantum and condensed matter phenomena at the macroscopic scale, such as Bloch oscillation [47-49], Anderson localization [50, 51], and Rabi oscillation [52, 53], where the transmission of light on optical platforms is investigated to reveal the quantum mechanical evolution of electron wave functions in solid materials. Among different classical platforms, the waveguide arrays, also known as photonic lattices, have been widely used to demonstrate novel topological phenomena, including guiding light by artificial gauge fields [54-58], topological insulators [59-61], and Floquet solitons [62-64], by adjusting the waveguide geometrical parameters and complex refractive index.

In this work, based on a cross-linking configuration composed of multiple SSH chains with an interface defect, we present a scheme of fast topological beam splitters and routers whose number of output ports and output ratios can be conveniently adjusted by crosslinking different numbers of SSH chains and elaborately tailoring their linking strength to the mutual site. The pumping scheme via adiabatic passage based on the SSH chains with an interface defect is identified to be more robust against nonadiabatic effects compared to conventional beam splitter based on ordinary SSH chains, since the channel state remains in the gap and does not get immersed into the band continuum, which provides sound adherence to the requirement of adiabaticity and promises more efficient state transfer. Based on the condition of global adiabaticity and numerical analysis of the instantaneous spectrum, we propose optimized protocols for modulation of the coupling strengths for the beam splitters and routers via adiabatic passage. For the chain adopting the 3-step linear modulation protocol with size 31 for example, excitation transfer can be faster than its ordinary counterpart without adiabatic passage. Additionally, fast topological pumping in the SSH chain with an interface defect adopting the 3-step linear modulation protocol improves the scalability of the beam-splitter with a high fidelity. Furthermore, we confirm through numerical sampling that although the three pumping schemes, the Gaussian protocol, the 3-step Gaussian protocol and the 3-step linear protocol, via adiabatic passage exhibit improved robustness against environment induced loss than their ordinary counterpart, accelerated beam splitting and routing is attained at the expense of degraded robustness against disorders in both coupling strengths and on-site energies. Finally, we map the proposed topology beam splitter onto a coupled waveguide array platform and conduct simulation calculations using finite element method (FEM), and the simulative results echo theoretical calculations, confirming again the advantage of the beam-splitting SSH chain with an interface defect in device compactness.

2 Physical model and engineering of topological pumping

2.1 Topological edge pumping in the odd-sized SSH chain and in the even-sized Rice−Mele model

Topologically protected states are important in realizing robust excitation transfer between distant sites in photonic lattices. Here, we propose an efficient and robust transfer of photons in a scalable one-dimensional waveguide array emulating the SSH model. Schematic illustration of the odd-sized SSH model is shown in Fig.1(a), which describes a one-dimensional dimerized lattice composed of $N$ unit cells. The Hamiltonian of system reads

Fig.1 (a, b) Diagrammatic sketch of (a) the odd-sized SSH model composed of $N$ unit cells and (b) the Rice−Mele model with alternate on-site energies $V_{a,b}$ for adiabatic topological pumping of edge states. Each dimerized unit cell (the gray dashed rectangle) contains an a-type (blue dot) site and a b-type (purple dot) site with intracell (orange line) coupling strength $J_1$ and intercell (green line) coupling strength $J_2$ between two adjacent sites. The size of the SSH chain is $2N-1$ and the size of the Rice−Mele chain is $2N$. (c, d) Instantaneous energy spectrum (upper panel) of the time-varying (c) SSH lattice with 21 sites and (d) Rice−Mele lattice with 20 sites under trigonometric modulation as well as the distributions of amplitudes of the channel edge state labeled by the red curve (lower panel) at $0.22T$ and $0.78T$, respectively. The total evolution time $T$ is chosen to unity, and other parameter takes $J_0=0.5$.

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(1)$H={\displaystyle \sum _{n=1}^N(J_1{a}_n^{†}{b}_n+J_2{a}_{n+1}^{†}{b}_n+\mathrm{H}.\mathrm{c}.),}$

where $J_1$ and $J_2$ are the staggered coupling strengths, $a_n(a_n^{†})$ and $b_n(b_n^{†})$ are the annihilation (creation) operators of a particle at the $n$th a- and b-type sites, respectively. For periodic boundary conditions (PBC), performing the Fourier transform, we have $a_n=\frac{1}{\sqrt{N}}{\sum }_k{\mathrm{e}}^{\mathrm{i}kn}{a}_k$ and $b_n=\frac{1}{\sqrt{N}}{\sum }_k{\mathrm{e}}^{\mathrm{i}kn}{b}_k$, with the wavenumber $k\in \left\{\frac{2\pi }{N},\frac{4\pi }{N},\cdots ,\frac{2N\pi }{N}\right\}$ being from the first Brillouin zone. Therefore, we can rewrite the bulk Hamiltonian in the bulk momentum space $H_{\text{bulk}}=\left[\left(J_1+J_2{\mathrm{e}}^{-\mathrm{i}k}\right)a_k^{†}{b}_k+\text{H.c.}\right]$. For each wavenumber $k$ from the first Brillouin zone, under the basis of $(a_k,b_k{)}^{\mathrm{T}}$ with the superscript “T” denoting transposition, the matrix form of the Hamiltonian in the bulk momentum space reads

(2)$H_k=\left(\begin{array}{cc}0& J_1+J_2{\mathrm{e}}^{-\mathrm{i}k}\\ J_1+J_2{\mathrm{e}}^{\mathrm{i}k}& 0\end{array}\right).$

By diagonalizing $H_k$, we can obtain the eigenvalues

$E_{\pm }(k)=\pm \sqrt{J_1^2+J_2^2+2J_1{J}_2\cos k},$

corresponding to eigenstates

(3)$|{\psi }_{\pm }(k)⟩={\displaystyle \frac{1}{\sqrt{2}}}{\left[E_{\pm }(k)/\left(J_1+J_2{\mathrm{e}}^{\mathrm{i}k}\right),1\right]}^{\mathrm{T}}.$

The system is in the topologically trivial or nontrivial phases when intracell and intercell coupling strengths satisfy $J_1>J_2$ or $J_1<J_2$, which can be characterized by the winding number of eigenstates defined as

(4)$w_{\pm }={\displaystyle \frac{\mathrm{i}}{\pi }}{\displaystyle {\int }_{-\pi }^{\pi }⟨{\psi }_{\pm }(k)\mid {\mathrm{\partial }}_k\mid {\psi }_{\pm }(k)⟩.}$

Substituting eigenstate $|{\psi }_{\pm }⟩$ from Eq. (3) into Eq. (4), we can obtain $w=0$ and $w=1$ for $J_1>J_2$ and $J_1<J_2$, respectively. In addition to the winding number in bulk momentum space, the topological properties of the SSH model can also be characterized by the eigen-energy spectrum and edge states in real space. For an odd-sized SSH model composed of finite sites under open boundary condition (OBC), as demonstrated in Fig.1(c), there always exists a zero-energy edge state separating the upper and lower bands due to chiral symmetry $\hat{\mathrm{\Gamma }}\hat{H}\hat{\mathrm{\Gamma }}=\hat{H},\hat{\mathrm{\Gamma }}={\sigma }_z\otimes I$, with eigenvector

(5)$|{\psi }_0⟩=|1,0,-J_1/J_2,0,{\left(-J_1/J_2\right)}^2,\cdots ,{\left(-J_1/J_2\right)}^N⟩,$

whose localized position depends on the ratio $J_1/J_2$. As the intracell and intercell coupling strengths are continuously modulated according to the commonly used trigonometric functions $J_1=J_0[1-\cos (\pi t/T)]$ and $J_2=J_0[1+\cos (\pi t/T)]$ [13], where $T$ is interaction time, $J_1/J_2$ gradually increasing from 0 to $+\mathrm{\infty }$, and a topologically protected state transfer from the left edge to the right edge of the chain can be realized. The adiabatically evolving zero-energy eigenstate of the Hamiltonian is expected to remain in the gap and does not touch or leak into the continuum of state, and thus we call such edge pumping channel topologically protected by the band gap $2\mathrm{\Delta }=2\left|J_1+J_2{\mathrm{e}}^{\mathrm{i}\pi (N-1)/N}\right|$ between the gap state and its adjacent bulk eigenstates [14], and this is the origin of immunity to scattering from inherent disorders and local imperfections. It is worth noting that at time $t=T/2$, the gap closes and the zero-energy mode touches the two bands for an extremely large chain size, which can be one of the major hindrances for its scalable application into large-scale systems.

Apart from the half-cycle pumping in the SSH chain, another method for edge state transfer is provided by the edge pumping in the Rice−Mele model, which is comprised of an even number $2N$ of sites with alternate on-site energies $V_a=-V_b$, as shown in Fig.1(b). The real space Hamiltonian reads

(6)$\begin{array}{rl}H=& \sum _{n=1}^N(J_1{a}_n^{†}{b}_n+V_a{a}_n^{†}{a}_n+\mathrm{H}.\mathrm{c}.)\\ & +\sum _{n=1}^{N-1}(J_2{a}_{n+1}^{†}{b}_n+V_b{b}_n^{†}{b}_n+\mathrm{H}.\mathrm{c}.),\end{array}$

whose eigen-energy spectrum is divided into two bands separated by an energy gap of $2\mathrm{\Delta }$. To realize topological edge pumping, the coupling strengths and on-site energies need to be adiabatically varied along a closed loop that encircles the gap closing point in the parameter space of $(J_2-J_1,V_a)$. Assuming trigonometric modulation is adopted, i.e., $J_1=J_0[1-\cos (2\pi t/T)]$, $J_2=J_0[1+\cos (2\pi t/T)]$, and $V_a=-V_b=V_0\sin (2\pi t/T)$ [18], starting from one of the two zero-energy instantaneous eigenstates at $t=0$ with the corresponding eigenstate localized at the left or right edge, after one cycle the initial state is transferred to the other eigenstate, thus achieving topological edge pumping. As illustrated in Fig.1(d), left- to right-edge pumping is realized in the chain comprised of 20 sites through the red channel. It is worth noting that the instantaneous energy of channel state gets immersed into the upper and lower bands, thus getting fully delocalized during the evolution cycle, which brings along more pronounced nonadiabatic effects and degradation of transfer efficiency, and as a result, calls for longer evolution time for adiabatic pumping.

2.2 Topological pumping through coherent tunneling by adiabatic passage with approximate three-level description

Recent works reveal that the defect states localized at domain walls can act as signal amplifiers and exponentially reduce the transfer time for edge pumping between distant nodes [13, 65]. Inspired by this discovery, to achieve efficient topological transfer robust to nonadiabatic effects, we consider a hetero-structured SSH chain with the number of sites L = 2N − 1 attained by interfacing two dimerized SSH chains with distinct topological order through a mutual a-type site, as schematically displayed in Fig.2(a) and (b) corresponding to even and odd $N$, respectively. The system can be described by the following interaction-picture Hamiltonian

Fig.2 (a, b) Schematic (upper panel) of the one-dimensional SSH model with an interface defect obtained by connecting two SSH chains with distinct topological order. The size of the chain is $L=2N-1$, with (a) corresponding to the case of even $N=4$ and (b) corresponding to the case of odd $N=5$. The intracell coupling strengths $J_1$ and $J_1^{\mathrm{\prime }}$ (orange and pink line) are smaller than the intracell coupling strength $J_2$ (green line), so that the chain holds three zero-energy topologically protected bound states (lower panel) localized at sites $n=1$ for $|{\psi }_L⟩$, $n=N$ for $|{\psi }_C⟩$, and $n=2N-1$ for $|{\psi }_R⟩$ (colored in orange, gray, pink, respectively) in the flat-band limit. (c, d) Instantaneous energy spectrum (upper panel) of the time-varying chain with (c) $2N-1=31$ and (d) $2N-1=$$33$ sites under Gaussian modulation $J_1=f(t-\delta /2),J_1^{\mathrm{\prime }}=f(t+$$\delta /2),f(t)=J_0{\mathrm{e}}^{-t^2/{\omega }^2}$ as well as the distributions of amplitudes of the channel edge state labeled by the red curve (lower panel) at $0.22T$ and $0.78T$, respectively. Other parameters take $T=800/J_2,J_2=1,J_0=0.9,\omega =0.1875T,\delta =\omega /3$.

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(7)$\begin{array}{rl}H=& \sum _{n=1}^{(N-1)/2}(J_1{a}_n^{†}{b}_n+J_2{a}_{n+1}^{†}{b}_n+\mathrm{H}.\mathrm{c}.)\\ & +\sum _{n=(N+1)/2}^{N-1}(J_2{a}_n^{†}{b}_n+J_1^{\mathrm{\prime }}{a}_{n+1}^{†}{b}_n+\mathrm{H}.\mathrm{c}.)\end{array}$

for odd $N$ and

(8)$\begin{array}{rl}H=& \sum _{n=1}^{N/2-1}(J_1{a}_n^{†}{b}_n+J_2{a}_{n+1}^{†}{b}_n+\mathrm{H}.\mathrm{c}.)\\ & +\sum _{n=N/2}^{N-2}(J_1^{\mathrm{\prime }}{b}_n^{†}{a}_{n+1}+J_2{a}_{n+1}^{†}{b}_{n+1}+\mathrm{H}.\mathrm{c}.)\\ & +(J_1{a}_{N/2}^{†}{b}_{N/2}+J_2{b}_{N-1}^{†}{a}_N+\mathrm{H}.\mathrm{c}.)\end{array}$

for even $N$, respectively. We assume the intracell coupling strengths $J_1$ and $J_1^{\mathrm{\prime }}$ are much smaller compared with the intracell coupling strength $J_2$, so that the chain sustains a topologically protected defect state $|{\psi }_C⟩$ at the interface site. Along with two localized edge states $|{\psi }_L⟩$ and $|{\psi }_R⟩$ at the left and right sites of the chain, the system supports a total of three localized states that are nearly degenerate zero-energy modes in the flat-band limit $J_1/J_2\to 0,J_1^{\mathrm{\prime }}/J_2\to 0(\mathrm{i}.\mathrm{e}.,X,Y\to 0)$, with shapes of their wave functions colored in orange, gray, pink, respectively in Fig.2(a) and (b). The three bound states can be analytically expressed as

(9a)$|{\psi }_L⟩=N_L{\displaystyle \sum _{n=1}^N{X}^{n-1}|{\psi }_{a_n}⟩,}$

(9b)$|{\psi }_R⟩=N_R{\displaystyle \sum _{n=1}^N{Y}^{N-n}|{\psi }_{a_n}⟩,}$

(9c)$\begin{array}{rl}|{\psi }_C{⟩}_{\text{odd}}=& N_C\left[\sum _{n=1}^{\frac{N-1}{2}}{X}^{\frac{N+1}{2}-n}|{\psi }_{a_n}⟩+\sum _{n=\frac{N+1}{2}}^N{Y}^{n-\frac{N+1}{2}}|{\psi }_{a_n}⟩\right],\\ |{\psi }_C{⟩}_{\text{even}}=& N_C\left[\sum _{n=1}^{\frac{N}{2}-1}{X}^{\frac{N}{2}-n}|{\psi }_{b_n}⟩+\sum _{n=\frac{N}{2}}^{N-1}{Y}^{n-\frac{N}{2}}|{\psi }_{b_n}⟩\right],\end{array}$

where $X=-J_1/J_2,Y=-J_1^{\mathrm{\prime }}/J_2$ are the ratios of coupling strengths and normalization factors given by

(10)$\begin{array}{rl}{N}_L=& \sqrt{\frac{X^2-1}{X^{N-1}-1}},\\ N_R=& \sqrt{\frac{Y^2-1}{Y^{N-1}-1}},\\ N_C=& {\left(\frac{X^{N-1}-1}{X^2-1}+\frac{Y^{N-1}-1}{Y^2-1}-1\right)}^{-1/2}\end{array}$

for odd $N$ and

(11)$\begin{array}{rl}{N}_L=& \sqrt{\frac{X^2-1}{X^N-1}},\\ N_R=& \sqrt{\frac{Y^2-1}{Y^N-1}},\\ N_C=& {\left(\frac{X^N-1}{X^2-1}+\frac{Y^N-1}{Y^2-1}-1\right)}^{-1/2}\end{array}$

for even $N$, respectively. The three states are eigenvectors of $H$ with zero energy in thermodynamic limit of $N\to \mathrm{\infty }$, while in the chain with finite sites, these three eigenstates hybridize with each other. In order to study the dynamical evolution of the system, under the adiabatic elimination of the other eigenstates in the subspace of the three topological states, the instantaneous state vector can be approximately described with the following ansatz

(12)$|\psi (t)⟩\simeq {\displaystyle \sum _n{a}_n(t)|{\psi }_n⟩,}$

where $n=L,C,R$, and $a_n$ denotes the occupation probability of the instantaneous states on the localized eigenstates $|{\psi }_L⟩,|{\psi }_C⟩,|{\psi }_R⟩$. Substituting the ansatz into time-dependent Schrödinger equation $\mathrm{i}\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}|\psi (t)⟩=H(t)|\psi (t)⟩$ yields

(13)$\begin{array}{rl}& \mathrm{i}\sum _l⟨{\psi }_n|{\psi }_l⟩{\dot{a}}_n(t)\\ & =\sum _l⟨{\psi }_n|H(t)|{\psi }_l⟩a_n(t)-\mathrm{i}\sum _l⟨{\psi }_n|\dot{{\psi }_l}⟩a_l(t),\end{array}$

where the dot signifies the derivative with respect to time. According to the analytical form of $|{\psi }_L⟩,|{\psi }_C⟩,|{\psi }_R⟩$ given in Eqs. (9a)−(9c), we can easily obtain

$\begin{array}{rl}& ⟨{\psi }_L|{\psi }_L⟩=⟨{\psi }_C|{\psi }_C⟩=⟨{\psi }_R|{\psi }_R⟩=1,\\ & ⟨{\psi }_L|\dot{{\psi }_L}⟩=⟨{\psi }_C|\dot{{\psi }_C}⟩=⟨{\psi }_R|\dot{{\psi }_R}⟩=0,\\ & ⟨{\psi }_L|{\psi }_C⟩=⟨{\psi }_R|{\psi }_C⟩=0,\\ & ⟨{\psi }_L|\dot{{\psi }_C}⟩=⟨\dot{{\psi }_L}|{\psi }_C⟩=⟨{\psi }_R|\dot{{\psi }_C}⟩=⟨\dot{{\psi }_R}|{\psi }_C⟩=0,\\ & ⟨{\psi }_L|H(t)|{\psi }_L⟩=⟨{\psi }_C|H(t)|{\psi }_C⟩=⟨{\psi }_R|H(t)|{\psi }_R⟩=0,\\ & ⟨{\psi }_L|H(t)|{\psi }_R⟩=0.\end{array}$

Since $|{\psi }_L⟩$ and $|{\psi }_R⟩$ are localized at both end sites of the chain, for a sufficiently large $N$ it can be assumed that $⟨{\psi }_L|{\psi }_R⟩=0$. Furthermore, for sufficiently slow evolution of the Hamiltonian, $⟨{\psi }_L|\dot{{\psi }_R}⟩$ and $⟨{\psi }_R|\dot{{\psi }_L}⟩$ are much smaller when compared with $⟨{\psi }_L|H(t)|{\psi }_C⟩$ and $⟨{\psi }_R|H(t)|{\psi }_C⟩$, and thus can be omitted from Eq. (13). As a result, Eq. (13) is reduced into

(14)$\mathrm{i}\hslash {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}}\left(\begin{array}{c}{a}_L\\ a_C\\ a_R\end{array}\right)=\left(\begin{array}{ccc}0& {\mathrm{\Omega }}_L& 0\\ {\mathrm{\Omega }}_L& 0& {\mathrm{\Omega }}_R\\ 0& {\mathrm{\Omega }}_R& 0\end{array}\right)\left(\begin{array}{c}{a}_L\\ a_C\\ a_R\end{array}\right)=H_{\text{eff}}\left(\begin{array}{c}{a}_L\\ a_C\\ a_R\end{array}\right),$

where ${\mathrm{\Omega }}_L$ and ${\mathrm{\Omega }}_R$ are the effective coupling coefficients between the interface state and edge states which can be readily calculated as

$\begin{array}{rl}{\mathrm{\Omega }}_L=& ⟨{\psi }_L|H(t)|{\psi }_C⟩=N_L{N}_C{J}_1{X}^{(N-1)/2-1},\\ {\mathrm{\Omega }}_R=& ⟨{\psi }_R|H(t)|{\psi }_C⟩=N_R{N}_C{J}_1^{\mathrm{\prime }}{Y}^{(N-1)/2-1}\end{array}$

for odd $N$ and

$\begin{array}{rl}{\mathrm{\Omega }}_L=& ⟨{\psi }_L|H(t)|{\psi }_C⟩=N_L{N}_C{J}_1{X}^{N/2-1},\\ {\mathrm{\Omega }}_R=& ⟨{\psi }_R|H(t)|{\psi }_C⟩=N_R{N}_C{J}_1^{\mathrm{\prime }}{Y}^{N/2-1}\end{array}$

for even $N$ from Eqs. (9a)−(9c). By diagonalizing the effective Hamiltonian, we can obtain the instantaneous eigenvalues

(15)$E_0=0,E_{\pm }=\pm \sqrt{{\mathrm{\Omega }}_L^2+{\mathrm{\Omega }}_R^2},$

and corresponding eigenvectors

(16a)$|{\psi }_0⟩={\displaystyle \frac{({\mathrm{\Omega }}_R,0,-{\mathrm{\Omega }}_L{)}^{\mathrm{T}}}{\sqrt{{\mathrm{\Omega }}_L^2+{\mathrm{\Omega }}_R^2}}}=\cos \theta |{\psi }_L⟩-\sin \theta |{\psi }_R⟩,$

(16b)$\begin{array}{rl}|{\psi }_{\pm }⟩=& \frac{({\mathrm{\Omega }}_L,E_{\pm },{\mathrm{\Omega }}_R{)}^{\mathrm{T}}}{\sqrt{{\mathrm{\Omega }}_L^2+{\mathrm{\Omega }}_R^2+E_{\pm }^2}}\\ =& \frac{\sin \theta }{\sqrt{2}}|{\psi }_L⟩\pm \frac{1}{\sqrt{2}}|{\psi }_C⟩+\frac{\cos \theta }{\sqrt{2}}|{\psi }_R⟩,\end{array}$

where $\theta =\arctan ({\mathrm{\Omega }}_L/{\mathrm{\Omega }}_R)$. Consequently, the SSH chain with an interface defect can be reduced to an approximate three-level system, as depicted in Fig.3(a), and the zero-energy eigenstate $|{\psi }_0⟩$ can be regarded as the dark state, since the topological interface state is not excited. When the system is well designed to satisfy ${\mathrm{\Omega }}_L/{\mathrm{\Omega }}_R\to 0$ at initial time $t=0$ and ${\mathrm{\Omega }}_L/{\mathrm{\Omega }}_R\to \mathrm{\infty }$ at final time, or in other words the parameter $\theta$ increases continuously from 0 to $\frac{\pi }{2}$, then the dark state adiabatically evolves from $|{\psi }_L⟩$ to $|{\psi }_R⟩$, and a topologically protected state transfer from the left edge to the right end of the chain can be realized via the dark state.

Fig.3 (a) The approximate three-level system for dynamical evolutional description of the SSH chain with an interface defect in the subspace of three topological bound states. (b) Time evolution of the occupation probabilities of the leftmost and rightmost sites in a chain of size $L=31$ under Gaussian modulation by solving the time-dependent Schrödinger equation (solid line) and by the approximate three-state description (dashed line). Parameter values and modulation of coupling strengths are the same as those in Fig.2.

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For example, if the intercell coupling strength $J_2$ is fixed to be 1 and the intracell coupling strengths $J_1$ and $J_1^{\mathrm{\prime }}$ vary in time based on the following Gaussian modulation scheme

(17)$J_1=f(t-\delta /2),J_1^{\mathrm{\prime }}=f(t+\delta /2),f(t)=J_0{\mathrm{e}}^{-t^2/{\omega }^2},$

with $T=800/J_2$, $J_2=1$, $J_0=0.9$, $\omega =0.1875T$, and $\delta =\omega /3$, we can plot evolution of the instantaneous energy spectrum of a chain, with size $2N-1=31$ and $2N-1=33$ for example, in the upper panels of Fig.2(c) and (d) with the channel state marked in red, and localization site of the channel state is shifted from the left edge site to the right, as shown in the lower panels. In the band gap exists three hybridized topological states, so the adiabatic pumping resembles the stimulated Raman adiabatic passage. Different from other topological pumping schemes such as topological edge pumping in the odd-sized SSH chain and in the even-sized Rice−Mele model, the energies of localized states in the SSH chain with a defect state remain in the gap and do not leak into the band continuum so that edge and interface states remain localized throughout the adiabatic cycle, ensuring strict compliance with adiabatic constraints and efficient edge pumping. Time evolution of the occupation probabilities of the leftmost and rightmost sites under Gaussian modulation are depicted in Fig.3(b), and the numerical results calculated by solving the time-dependent Schrödinger equation and by the approximate three-state description verifies the effectiveness of the zero-energy state as an edge pumping channel.

2.3 Beam splitter and router with tunable probabilities via adiabatic passage in the cross-linking configuration

Inspired by topological edge pumping in the SSH model with an interface defect, a topological beam splitter with equal output probabilities can be obtained by cross-linking two identical SSH chains with an interface defect, which comprises $L=2M+1$ lattice sites in total, where $M=2N_n$, as schematically depicted in Fig.4(a) and (b) corresponding to the case of odd $N_n=3$ and even $N_n=4$, respectively. In order to highlight the accelerated topological splitting process in the SSH chain with interface defect states, we have also considered here a simple beam splitter obtained by cross-linking two ordinary odd-sized SSH chains as shown in Fig.4(c). In Fig.4(d), (h), and (l), we show how the instantaneous spectrum evolves over time with the coupling strengths under Gauss modulation of Eq. (17) for the beam splitter with interface defects with odd $N_n=11$ and even $N_n=10$ and the ordinary splitter attained by linking two SSH chains with chain size $L=41$ under the commonly used trigonometric modulation $J_1=J_0[1-\cos (\pi t/T)]/2$ and $J_2=J_0[1+\cos (\pi t/T)]/2$, from which we can identify the existence of a zero-energy mode in the real space energy spectrum since the system Hamiltonian respects chiral symmetry. Probability amplitude of the zero-energy mode marked in red for the 3 cases at the start and end of the beam splitting process are demonstrated in Fig.4(e), (i), and (m) corresponding to $|{\mathrm{\Psi }}_i⟩=|0,0,0,\cdots ,0,1,0,\cdots ,0,0,0⟩$, and Fig.4(f), (j), and (n) corresponding to $|{\mathrm{\Psi }}_T⟩=\frac{1}{\sqrt{2}}|1,0,0,\cdots ,0,0,0,\cdots ,0,0,1⟩$. Distributions of the zero-energy mode marked in red on different sites of the lattice chain versus the time for the 3 cases are also displayed in Fig.4(g), (k), and (o), which provide an intuitive demonstration of state population evolution in topological splitters based on SSH lattice chain.

Fig.4 (a, b) Schematic illustration of the topological beam splitter with equal output probabilities obtained by cross-linking two identical SSH chains with an interface defect. The black dot denotes the linking site shared by both chains with on-site energy $V_a$. The splitter consists of $L=2M+1$ lattice sites in total, where $M=2N_n$ with (a) corresponding to the case of odd $N_n=3$ and (b) corresponding to the case of even $N_n=4$, respectively. (c) Another splitter chain obtained by linking two odd-sized SSH chains with ordinary trigonometric coupling strengths $J_1=J_0[1-\cos (\pi t/T)]/2,J_2=J_0[1+\cos (\pi t/T)]/2$. (d, h, l) Instantaneous energy spectrum as a function of the normalized time for the splitter chain with an interface defect and odd $N_n=11$ in (h), the splitter chain with an interface defect and even $N_n=10$ in (h), and the ordinary beam splitter with $L=41$ in (l). (e, f, i, j, m, n) Distribution of the red gap state at the start and end of the evolution process for the aforementioned 3 cases. (g, k, o) Evolution of probability distribution of the red gap state for the aforementioned 3 cases. Parameter values and modulation of coupling strengths are the same as those in Fig.2.

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We have shown to implement topologically protected state transfer with equal probabilities in a SSH chain with 2 output ports. The essential reason of the equal output probabilities is that, the same linking strengths $T_{\sigma }$ enable the central linking site to distribute the same proportion of excitations to each constituent SSH chain, so that the initial state prepared at the central site can be pumped toward the two end sites with equal probabilities. Following the essence of this framework, through elaborate adjustment of the linking strengths $T_{\sigma }$ of the mutual site to the linking site in different constituent chains, topological beam splitters of tunable output ratios can be constructed. In addition, we can also implement upgradation in terms of the number of constituent chains in the cross-linking structure and construct $K$-port routers by crosslinking multiple SSH chains, as schematically illustrated in Fig.5(a). The interaction of the cross-linking structure formed by $L=MK+1$ sites can be described by the following interaction-picture Hamiltonian

Fig.5 (a) Schematic illustration of the topological router with diverse numbers of output ports and different output ratios obtained by cross-linking multiple SSH chains with an interface defect with different linking strengths $T_{\sigma }(\sigma =1:K)$ through one mutual site. The router consists of $L=KM+1$ lattice points in total, where $M=2N_n$. (b−g) Evolution of the gap state when the topological router possesses the unequal linking strengths $T_{\sigma }$. (b) $T_1:T_2:T_3=1:1:1$. Probability distribution of the evolved gap state satisfies $P_1:P_{M+1}:P_{2M+1}=1:1:1$. (c) $T_1:T_2:T_3=2/\sqrt{6}:1/\sqrt{6}:1/\sqrt{6}$. Probability distribution of the evolved gap state satisfies $P_1:P_{M+1}:P_{2M+1}=4:1:1$. (d) $T_1:T_2:T_3=\sqrt{15}/5:2\sqrt{2}/5:\sqrt{2}/5$. Probability distribution of the evolved gap state satisfies $P_1:P_{M+1}:P_{2M+1}=15:8:2$. (e) $T_1:T_2:T_3:T_4=1:1:1:1$. Probability distribution of the evolved gap state satisfies $P_1:P_{M+1}:P_{2M+1}:P_{3M+1}=1:1:1:1$. (f) $T_1:T_2:T_3:T_4=\sqrt{11}/5:2\sqrt{2}/5:2/5:\sqrt{2}/5$. Probability distribution of the evolved gap state satisfies $P_1:P_{M+1}:P_{2M+1}:P_{3M+1}=11:8:4:2$. (g) $T_1:T_2:T_3:T_4=3/4:\sqrt{5}/4:\sqrt{2}/4:0$. Probability distribution of the evolved gap state satisfies $P_1:P_{M+1}:P_{2M+1}:P_{3M+1}=9:5:2:0$. $N_n$ takes 11 in (b−g). Parameter values and modulation of coupling strengths are the same as those in Fig.2.

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(18)$\begin{array}{rl}H=& \sum _{\sigma =1}^K\sum _{n=1}^{\frac{N_n-1}{2}}[J_1{a}_n^{\sigma †}{b}_n^{\sigma }+J_2{a}_{n+1}^{\sigma †}{b}_n^{\sigma }\\ & +J_1^{\mathrm{\prime }}{b}_{n+\frac{N_n-1}{2}}^{\sigma †}{a}_{n+\frac{N_n+1}{2}}^{\sigma }+J_2{b}_{n+\frac{N_n+1}{2}}^{\sigma †}{a}_{n+\frac{N_n+1}{2}}^{\sigma }+\text{H.c.}]\\ & +\sum _{\sigma =1}^K\left[J_1{a}_{\frac{N_n+1}{2}}^{†}{b}_{\frac{N_n+1}{2}}+T_{\sigma }{b}_{N_n}^{†}{a}_{K{N}_n+1}+\mathrm{H}.\mathrm{c}.\right]\end{array}$

for odd $N_n$ and

(19)$\begin{array}{rl}H=& \sum _{\sigma =1}^K\sum _{n=1}^{\frac{N_n}{2}-1}[J_1{a}_n^{\sigma †}{b}_n^{\sigma }+J_2{a}_{n+1}^{\sigma †}{b}_n^{\sigma }\\ & +J_2{a}_{n+\frac{N_n}{2}}^{\sigma †}{b}_{n+\frac{N_n}{2}}^{\sigma }+J_1^{\mathrm{\prime }}{a}_{n+\frac{N_n}{2}+1}^{\sigma †}{b}_{n+\frac{N_n}{2}}^{\sigma }+\text{H.c.}]\\ & +\sum _{\sigma =1}^K[J_1{a}_{\frac{N_n}{2}}^{†}{b}_{\frac{N_n}{2}}+J_2{a}_{\frac{N_n}{2}+1}^{†}{b}_{\frac{N_n}{2}}\\ & +J_2{a}_{N_n}^{†}{b}_{N_n}+T_{\sigma }{b}_{N_n}^{†}{a}_{K{N}_n+1}+\mathrm{H}.\mathrm{c}.]\end{array}$

for even $N_n$, respectively, where $M=2N_n$, $a_n^{\sigma }(a_n^{\sigma †})$ and $b_n^{\sigma }(b_n^{\sigma †})$ are the annihilation (creation) operators of a particle at the $n$th a- and b-type sites in a constituent SSH chain indexed by $\sigma$, and $T_{\sigma }$ is the linking strength between the mutual site to the linking site in the constituent chain indexed by $\sigma$. If we regard the mutual site as the input port and the end sites as output ports, the cross-linking structure is functionally equivalent to a topological router, through which an excitation injected into the mutual site can be transferred to end sites with different probabilities adjustable with $T_{\sigma }$ parameters. In order to sustain our claim, we conduct a series of numerical simulations. As demonstrated in Fig.5(b)−(d), in which the linking strengths $T_{\sigma }$ satisfies $T_1:T_2:T_3=1:1:1$, $T_1:T_2:T_3=2/\sqrt{6}:1/\sqrt{6}:1/\sqrt{6}$, and $T_1:T_2:T_3=\sqrt{15}/5:2\sqrt{2}/5:\sqrt{2}/5$ in a cross linking chain composed of 3 SSH chains with an interface defect and single chain size $N_n=10$, and the probability distribution of the evolved gap state at the end of the routing process satisfies $P_1:P_{M+1}:P_{2M+1}=1:1:1$, $P_1:P_{M+1}:P_{2M+1}=4:1:1$, $P_1:P_{M+1}:P_{2M+1}=15:8:2$, respectively. In a 4-output port topological router with linking strengths $T_1:T_2:T_3:T_4=1:1:1:1$, $T_1:T_2:T_3:T_4=\sqrt{11}/5:2\sqrt{2}/5:2/5:\sqrt{2}/5$, $T_1:T_2:T_3:T_4=3/4:\sqrt{5}/4:\sqrt{2}/4:0$, as demonstrated in Fig.5(e)−(g), the routing ratios are $P_1:P_{M+1}:P_{2M+1}:P_{3M+1}=1:1:1:1$, $P_1:P_{M+1}:P_{2M+1}:P_{3M+1}=11:8:4:2$, $P_1:P_{M+1}:P_{2M+1}:P_{3M+1}=9:5:2:0$, respectively, which agrees well with theoretical prediction. As a consequence, by elaborate designation of the linking strengths $T_{\sigma }$ of the mutual site to the linking site in the cross-linking SSH chain, the special zero-energy channel can be engineered to implement a topological router with flexible output ratio and immune to the mild local perturbation, which extends the state transfer in one-dimensional SSH model from point-to-point distribution to multi-port distribution and will endow the applications of the topological order in large-scale classical and quantum communication.

3 Fast state transfer with high robustness and scalability

3.1 Fast state transfer in the topological beam splitter and router

We have discussed in the above section the possibility of constructing topological beam splitters and routers on with tunable probabilities via adiabatic passage in the cross-linking configuration. The realization of the topological beam splitting and routing is essentially restricted by adiabatic constraints, and the system needs to be driven slowly enough so that the initial state always evolves along the channel state. As previously noted, the pumping scheme via adiabatic passage based on the SSH chains with an interface defect is more robust against nonadiabatic effects since the channel state remains in the gap and does not get immersed into the band continuum, which provides sound adherence to the requirement of adiabaticity and promises more efficient state transfer. In this section, we prove its superiority in terms of beam splitting and routing efficiency through numerical calculation and optimize the modulation function of coupling strengths to realize state transfer as fast as possible.

Intercell and intracell coupling strengths versus time under Gaussian modulation adopted earlier is displayed in Fig.6(a) with $\omega =0.1875T$ and $\delta =\omega /3$. Taking the beam splitting SSH chain with an interface defect with $T_1:T_2=1:1$ and $L=41$ as an example, we plot transport fidelity defined as $F={\left|⟨{\mathrm{\Psi }}_T\mid \psi (T)⟩\right|}^2$, i.e., the square of the modulus of the inner product of the ideal final state $|{\mathrm{\Psi }}_T⟩$ and the evolved final state obtained by solving the time-dependent Schrödinger equation, versus total evolution time $T$ under the Gaussian modulation protocol in Fig.7(a). For very small values of $T$ the system is in the non-adiabatic regime, the population of the zero-energy state quickly transits to other eigenstates, resulting in wrecked beam splitting channel and fidelity close to 0. As $T$ increases the system enters into the quasi-adiabatic regime, the fidelity curve shows mild fluctuations versus time and the first peak of fidelity over 0.99 appears at $J_{2}T=116$. As total evolution time approaches infinity, the fidelity approaches unity and an excitation injected to the interface site can be transferred to two-end sites with equal probabilities, suggesting that the transfer process satisfies the adiabatic requirements perfectly so that the system state always evolves along the channel state without leaking to others. Since the total evolution time $T$ is always finite in practice, the non-adiabatic effect introduces mild oscillation between the zero-energy channel state and neighboring eigenstates. Nevertheless, the population oscillation between different eigenstates in the quasi-adiabatic regime can lead to sufficiently high splitting fidelity with a small evolution time $T$ before reaching the adiabatic regime, as long as the condition of global adiabaticity, expressed by the area condition [66]

Fig.6 (a, c, e) Dynamic control of the intercell and intracell coupling strengths versus time under Gaussian modulation in which $\omega =0.1875T,\delta =\omega /3$, 3-step Gaussian modulation in which $t_1=0.1,t_2=0.497,{\delta }_1=t_1\times T,{\delta }_2=t_2\times T,{\omega }_1=0.353\times$${\delta }_1,{\omega }_2=0.4\times {\delta }_2$, and 3-step linear modulation in which $t_1=0.1,t_2=0.445$. (b, d, f) Detailed behavior of the eigen-energies of the three localized modes in the SSH model with an interface defect versus time (solid curves). The dashed green curves display the eigen-energies of the localized modes as predicted by the three-mode approximation. The zero-energy topological state evolves adiabatically provided that the pulse area $A$ marked by the shaded area is far greater than $\pi /2$. The SSH chain size is $L=2N-1=31$ and other parameters take $T=1,J_2=1,J_0=0.9$.

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Fig.7 (a) Transport fidelity between the ideal final state and the evolved final state versus total evolution time for the beam-splitting SSH chain with an interface defect adopting the Gaussian modulation protocol, the 3-step Gaussian protocol, the 3-step linear protocol, and the ordinary beam-splitting SSH chain adopting the trigonometric protocol. (b−d) Population of the instantaneous state on the eigenstates during the evolution process for the beam-splitting SSH chain adopting the 3-step linear protocol when $J_{2}T=15$ (b), $47$ (c), $1000$ (d), where the probability of ${\psi }_0$ is highlighted with thick lines. Chain size is chosen as $N_n=10,L=2M+$$1=41$ and modulation of coupling strengths is the same as those in Fig.6.

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(20)$A={\displaystyle {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\mathrm{d}t\sqrt{{\mathrm{\Omega }}_L^2(t)+{\mathrm{\Omega }}_R^2(t)}}$

is satisfied, where the area $A$ corresponds to the shaded pulse area in the eigen-energy spectrum of the SSH model with an interface defect as illustrated in Fig.6(b) for model size $L=2N-1=31$. For the SSH model with an interface defect under Gaussian modulation, pulse area equals $0.0221T{J}_2$. Obviously, owing to the exponential dependence of ${\mathrm{\Omega }}_{L,R}$ on $N$, longer transfer times are required to ensure adiabatic evolution along the channel state when the number of sites in the chain is increased. Besides, the minimum $T_{0.99}$ for 0.99 fidelity as a symbol of successful state transfer is inversely proportional to the average absolute value of the neighboring eigen-energy to the zero-energy channel mode, so the protocol with large average absolute value of the neighboring eigen-energy, that is to say, larger average intracell and intercell coupling strength under the condition of equal $T$, is in better compliance with the constraint of global adiabaticity.

Enlightened by this, by introducing a stage of constant intracell and intercell coupling strengths kept at maximum value in the modulation functions, we investigate a 3-step Gaussian protocol

(21a)$J_1=\{\begin{array}{ll}{f}_1(t-{\delta }_1),& 0\le t\le t_{1}T\\ J_0,& t_{1}T<t\le (1-t_2)T\\ f_2(t-(T-{\delta }_2)),& (1-t_2)T<t\le T\end{array},$

(21b)$J_1^{\mathrm{\prime }}=\{\begin{array}{ll}{f}_2(t-{\delta }_2),& 0\le t\le t_{2}T\\ J_0,& t_{1}T<t\le (1-t_1)T\\ f_1(t-(T-{\delta }_1)),& (1-t_1)T<t\le T\end{array},$

where $f_1(t)=J_0{\mathrm{e}}^{-t^2/{\omega }_1^2}$, and $f_2(t)=J_0{\mathrm{e}}^{-t^2/{\omega }_2^2}$, and $t_1=0.1$, $t_2=0.497$, ${\delta }_1=t_{1}T$, ${\delta }_2=t_{2}T$, ${\omega }_1=0.353{\delta }_1$, and ${\omega }_2=0.4{\delta }_2$, with the pulse area equal to $0.0812T{J}_2$, as displayed in Fig.6(c) and (d). Here, $t_1(t_2)$ is defined as time interval for the coupling strength $J_1(J_1^{\mathrm{\prime }})$ to increase from the initial to maximum value or $J_1^{\mathrm{\prime }}(J_1)$ to decrease from maximum to end value as a portion of the total transfer time, and the values $t_1=0.1$ and $t_2=0.497$ are attained through parameter optimization which is detailed at the end of this subsection. To further increase pulse area $A$, a 3-step linear protocol is also considered,

(22a)$J_1=\{\begin{array}{ll}{f}_1(t-{\delta }_1),& 0\le t\le t_{1}T\\ J_0,& t_{1}T<t\le (1-t_2)T\\ f_2(t-(T-{\delta }_2)),& (1-t_2)T<t\le T\end{array},$

(22b)$J_2=\{\begin{array}{ll}{J}_{0}t/t_2/T,& 0\le t\le t_{2}T\\ J_0,& t_{1}T<t\le (1-t_1)T\\ J_0(T-t)/t_1/T,& (1-t_1)T<t\le T\end{array},$

where optimized parameters are $t_1=0.1$, and $t_2=0.445$ corresponding to pulse area $0.0941T{J}_2$, as displayed in Fig.6(e) and (f). As a result, the 3-step linear protocol supports larger pulse area while still maintaining $J_1,J_1^{\mathrm{\prime }}\ll J_2$ at the beginning and end of the evolution process. We also plot in Fig.7(a) fidelity versus total evolution time for the beam-splitting SSH chain with an interface defect adopting the 3-step Gaussian and the 3-step linear modulation protocols, and total transfer time $T_{0.99}$ needed for fidelity reaching 0.99 are $47/J_2$ for the 3-step linear protocol and $70/J_2$ for the 3-step Gaussian protocol. To make a comparison in terms of the speed of beam splitting, the ordinary beam-splitting SSH chain adopting the trigonometric protocol is also considered here with $T_{0.99}=535/J_2$. Numerical results reveal that the beam splitter via adiabatic passage adopting the 3-step linear protocol is the most efficient and enables topological pumping process which is about 91.2% times faster than its ordinary counterpart. In Fig.7(b)−(d), we plot the populations of eigenstates $|⟨{\psi }_l(t)|\psi (t)⟩{|}^2$ in the beam-splitting SSH chain under the 3-step linear modulation as functions of t for different total evolution time $T$. For a small $T{J}_2=15$, the population of the zero-energy eigenstate $|{\psi }_0⟩$ transits quickly to other eigenstates and fails to return. For the intermediate $T$, the population oscillates between $|{\psi }_0⟩$ and other eigenstates and returns to $|{\psi }_0⟩$ at the end of the evolution process with mildly reduced transport efficiency, and fidelity reaches the first local maximum at $T{J}_2=47$ as shown in Fig.7(a) which corresponds to the returning peak of the zero-energy state. For an extremely large $T{J}_2=1000$, the instantaneous state $|\psi (t)⟩$ is frozen on $|{\psi }_0⟩$ without exciting other eigenstates and the adiabatic beam splitting can be realized perfectly.

At the end of this section, we present the details of parameter optimization process in the 3-step Gaussian and 3-step linear protocols. As illustrated in Fig.8(a), we investigate fidelity of the topological beam splitter with equal output probabilities and chain size $L=41$ under the 3-step Gaussian modulation versus varying total transfer time $T$ and parameter $t_1$. We notice that fidelity is insensitive to parameter $t_1$, so we choose a relatively small $t_1=0.1$ in order to satisfy $t_1<t_2$ (so as to maintain $J_1>J_1^{\mathrm{\prime }}$ and $J_1<J_1^{\mathrm{\prime }}$ at the beginning and end of the evolution process, respectively). Fidelity of beam splitting versus varying total transfer time $T$ and parameter $t_2$ is displayed in Fig.8(b), in which the 0.99 fidelity contour lines manifest strong oscillations for small values of the parameter $t_2$ which can be attributed to the quasi-adiabatic oscillations between the channel state and other eigenstates under strong driving modulation. To achieve successful beam splitting with 0.99 fidelity in the shortest possible time, we can identify the quasi-adiabatic pulses highlighted by the 0.99 fidelity contour lines and choose the $t_2$ parameter corresponding to smallest $T$, namely $t_2=0.497$ for chain size $L=41$. Analogously, fidelities of the splitter under the 3-step linear modulation in parameter space $(T,t_1)$ with $t_2$ fixed at 0.5 and parameter space $(T,t_2)$ with $t_1$ fixed at 0.1 are demonstrated in Fig.8(c) and (d). Through similar parameter searching procedure, we can get the optimized parameters $t_1=0.1$ and $t_2=0.445$ for the 3-step linear protocol.

Fig.8 Fidelity of the splitter under the 3-step Gaussian (a, b) and 3-step linear (c, d) modulation in parameter space $(T,t_1)$ and $(T,t_2)$. Parameter $t_2$ is fixed at 0.5 in (a) and (c) and $t_1$ is fixed at 0.1 in (b) and (d). The green and red solid lines represent 0.9 and 0.99 fidelity contour lines, respectively. Parameter values and modulation of coupling strengths are the same as those in Fig.6.

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3.2 Size effect and scalability

As the number of sites in the chain increases, the optimal parameters deviate from preset and nonadiabatic effects become more pronounced for fixed state transfer time, as noted in the previous subsection. In order to verify more extensively the effect of the modulation protocols in the topological beam splitter and router proposed in this article, we take a closer look on how they behave when the system size is altered. We plot in Fig.9(a) and (c) fidelity of the splitter with increased sites in each constituent chains $N_n=20,K=2$ in parameter space $(T,t_2)$ with $t_1$ is fixed at 0.1 for the 3-step Gaussian and 3-step linear protocols, and the corresponding optimal $t_2$ parameters are shifted to 0.467 and 0.456, respectively. When the number of constituent chains in the topological router with multiple output ports increases, taking the case of $N_n=10,K=10$ as an example, fidelity versus varying total transfer time $T$ and parameter $t_2$ for the 3-step Gaussian and 3-step linear protocols are manifested in Fig.9(b) and (d), with the corresponding optimal $t_2$ parameters shifted to 0.528 and 0.518, respectively. As a result, the optimal $t_2$ parameter increases with number of chains and decreases with number of sites in each constituent chain. Yet this does not signify the failure of topological router. As illustrated in Fig.10(a), we demonstrate the performance of topological splitters and routers adopting the 3-step linear protocol with evolution time fixed at $J_{2}T=800$ in the simulations and different output ports and ratios as a function of the number of sites in each constituent chain. The $t_2$ parameter is chosen to be the optimal value of 0.445 for the system of $N_n=10,K=2,T_1:T_2=1:1$. Apparently, for the number of sites in each chain larger than 30, nonadiabatic effects become strong and the topological routing process highly degraded. In the size range of $32-48$ the fidelity first decreases and then recovers, which can be attributed to the oscillation in the fidelity contour lines due to the negative correlation between optimal $t_2$ and $M$. To compare the performance of different protocols under various system sizes, we also plot in Fig.10(b) and the total evolution time $T_{0.99}$ needed to achieve beam splitting with equal output ratios as a function of the size of the system for the SSH chain with an interface defect adopting the Gaussian modulation protocol, the 3-step Gaussian protocol, the 3-step linear protocol, and the ordinary beam-splitting SSH chain adopting the trigonometric protocol, respectively. For all 4 cases $T_{0.99}$ increases with system size due to the fact that the area $A$ is an almost exponentially decreasing function of L, where $T_{0.99}$ versus L can be fitted by cubic functions $J_2{T}_{0.99}^{\text{Gauss}}=$$0.0230\times M^3-1.3555\times M^2+39.5944\times M-310.486$, $J_2{T}_{0.99}^{\text{3-Gauss}}=$$0.0015\times M^3+0.0388\times M^2+2.4307\times M+5.1880$, $J_2{T}_{0.99}^{\text{3-lin}}=$$0.0017\times M^3+0.0573\times M^2-1.1096\times M+31.5212$, and $J_2{T}_{0.99}^{\text{trig}}=$$-0.0001\times M^3+1.2238\times M^2+2.1367\times M+3.7853$, respectively, and thus the global condition for adiabaticity calls for longer evolution time. Besides, it is evident that within the range of chain size considered here, the four schemes demonstrate good scalability, and beam splitting the SSH chain with an interface defect adopting the 3-step linear protocol manifests the highest efficiency, which is closely followed by the 3-step Gaussian protocol, leaving the Gaussian protocol and the ordinary model without adiabatic passage far behind.

Fig.9 Fidelity of the splitter under the 3-step Gaussian (a, b) and 3-step linear (c, d) modulation in parameter space $(T,t_2)$. Parameter $t_1$ is fixed at 0.1, chain size parameters are chosen as $N_n=20,K=2$ in (a, c) and $N_n=10,K=10$ in (b, d), and modulation of coupling strengths is the same as those in Fig.6.

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Fig.10 (a) Beam splitting and routing fidelities with different number of output ports and ratios versus number of sites $M$ in each configurative chain in the topological router adopting the 3-step linear protocol with fixed evolution time $T=800/J_2$. (b) Total transfer time needed to reach 0.99 fidelity for splitters of different sizes under different modulation protocols. The $t_2$ parameter is chosen to be 0.445 and modulation of coupling strengths is the same as those in Fig.6.

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3.3 Disorder effect

So far, we have demonstrated that via the adiabatic passage of edge and interface states in a SSH chain with an interface defect, a fast and flexible topological beam splitter or router with adjustable number of output ports and output ratios can be fabricated by cross-linking different number of SSH chains and tailoring their linking strength to the mutual site. Owing to the avoidance of delocalization over the entire cycle and quasi-adiabatic oscillation between the channel state and other gap states, pumping efficiency has been significantly improved, and the beam splitting or routing process can be further accelerated to be 91.2% times faster than its ordinary counterpart without adiabatic passage through optimization of coupling modulation. However, a concerning question subsists whether such improvement in transfer speed is followed by degraded performance in other aspects, like robustness against disorder and environment-induced loss on account of the fast evolution process in the system. Due to the existence of manufacturing defects and decoherence effect induced by environment in practical physical systems, coupling strengths and onsite energies that perfectly matches the preset are almost unattainable. Hence in this subsection, using the topological beam splitter with $1:1$ output ratio and chain size of $L=41$, we take a deep dive into the effects of three representative influencing factors on the beam splitting process, namely, disorders in coupling strengths commonly are referred to as off-diagonal disorder, disorders in onsite energies are generally addressed as diagonal disorder, and losses of onsite energies are due to environmental dissipation.

To begin with, we investigate the robustness of the four above-mentioned beam splitting schemes by introducing disorders in coupling strengths and onsite energies, respectively, and the way each disorder implementation is applied to the system parameters can be described by

(23)$\begin{array}{rl}& J_{1(2),n}^i\to J_{1(2),n}^i\left(1+\delta J_{1(2)}^i\right),J_{1,n}^{\mathrm{\prime }i}\to J_{1,n}^{\mathrm{\prime }i}\left(1+\delta J_1^{\mathrm{\prime }i}\right),\\ & V_{1(2),n}^i=0\to \delta V_{1(2)}^i,\end{array}$

where the $i$-th pair of samples $\delta J_{1(2)}^i,\delta J_1^{\mathrm{\prime }i}$ and $\delta V_{1(2)}^i$ acquire random real values uniformly distributed in the interval $[-{\omega }_s,{\omega }_s]$, in which ${\omega }_s$ corresponds to the disorder strength. $\delta J_{1(2)}^i,\delta J_1^{\mathrm{\prime }i}$ and $\delta V_{1(2)}^i$ remain fixed in each disorder implementation since static disorder is considered here. We plot in Fig.11 the statistical distribution of the transfer fidelity obtained from 1000 diagonal and off-diagonal disorder realizations in the beam-splitting SSH chain with an interface defect adopting the Gaussian, the 3-step Gaussian, 3-step linear protocols and the ordinary beam-splitting SSH chain without adiabatic passage adopting the trigonometric protocol, respectively. Disorder strength is set to be a moderate value of ${\omega }_s=0.05$ and the total transfer time for each protocol is fixed to the values of $T_{0.99}$ calculated before under zero disorder in order to check the improvement in transfer speed comes with a deficit for robustness. We can immediately notice that the three schemes via adiabatic passage exhibit reduced robustness against both off-diagonal and diagonal disorder as compared to the ordinary topological splitter under trigonometric modulation. Such a performance degradation can be attributed to the fact that strong oscillation between the topological channel state and the neighboring states in the presence of disorders make evolution process of the system no longer quasi-adiabatic. In other words, we are sacrificing robustness for the sake of transfer speed when employing the adiabatic passage induced by the defect state.

Fig.11 Resilience to imperfect realization of coupling strengths and on-site energies: Statistical distribution of fidelity obtained from 1000 realizations of the off-diagonal (a−d) and diagonal (e−h) disorders in a beam splitter with equal possibilities adopting the Gaussian modulation protocol, the 3-step Gaussian protocol, the 3-step linear protocol, and the trigonometric protocol when total transfer time fixed at $J_{2}T=116,70,48.5,535$, respectively. Disorder strength is fixed at ${\omega }_s=0.05$, chain size is chosen as $N_n=10,L=2M+1=41$ and modulation of coupling strengths are the same as those in Fig.6.

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Furthermore, we also try to reveal the effects of environmental dissipation, which is another influencing factor in practical classical and quantum communication systems. When considering the effect of losses in the lattice chain, non-Hermitian on-site terms should be added to the system Hamilton

(24)$H_{\text{loss}}=H-\mathrm{i}{\displaystyle \sum _n\left({\gamma }_n^a{a}_n^{†}{a}_n+{\gamma }_n^b{b}_n^{†}{b}_n\right),}$

where H is the lossless Hamilton given in Eqs. (18) and (19), and ${\gamma }_n^{a,b}$ denotes the loss rate of each type of sites. For convenience, we assume ${\gamma }_n^a={\gamma }_n^b=\gamma$. The dynamics of the system is governed by the non-Hermitian Liouville equation $\dot{\rho }=-\mathrm{i}\left(H^{\prime }\rho -\rho H^{\prime †}\right)$. We illustrate in Fig.12(a) the final fidelity of the beam splitter adopting the four above-mentioned transfer schemes versus loss rate, with system size $L=41$ and total transfer time for each scheme fixed to the values for 0.99 fidelity calculated before so that beam splitting can be successfully implemented via all schemes when no loss exists. Compared with the ordinary chain under trigonometric modulation of coupling strengths, the fidelities of beam splitters via adiabatic passage have been improved to a different extent, the 3-step linear and the 3-step Gaussian protocols in particular, with the degree of improvement highly dependent on their respective total evolution time. We also investigate the performance of the four schemes in chains of different sizes, as illustrated in Fig.12(b). The numerical calculations indicate that the beam-splitting SSH chain with an interface defect adopting the 3-step linear modulation protocol manifests significant amelioration of robustness against losses of onsite energies induced by environmental dissipation.

Fig.12 (a) Final fidelity as a function of loss rate in a beam splitter with equal possibilities adopting the Gaussian protocol with $J_{2}T=116$, the 3-step Gaussian protocol with $J_{2}T=70$, the 3-step linear protocol with $J_{2}T=48.5$, and the trigonometric protocol with $J_{2}T=535$, respectively. (b) Fidelity as a function of the chain size with fixed loss parameter $\gamma =2.5\times {10}^{-5}{J}_2$ for the four protocols. Modulation parameters are fixed at the optimal values for $N_n=10$ and $K=2$.

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4 Experimental implementation of the splitter model in coupled waveguide arrays

One of the commonly used platforms for presenting peculiar phenomena in condensed matter physics is waveguide arrays, which have already been examplified in photonic [54-64] and acoustic [67-71] systems. Topological states in waveguide arrays can be theoretically described by the coupled mode theory [72, 73]. From classical Maxwell’s equations in a nonmagnetic medium, on the condition that the amplitude of the electromagnetic wave varies slowly in the propagation direction (z direction), the Helmholtz equation can be deduced as

(25)$\begin{array}{rl}& {\mathrm{\nabla }}_{\perp }^2\phi (x,y,z)+2\mathrm{i}{k}_z\frac{\mathrm{\partial }}{\mathrm{\partial }z}\phi (x,y,z)-k_z^2\phi (x,y,z)\\ & +k_0^2{n}^2(x,y)\phi (x,y,z)=0,\end{array}$

where $\phi (x,y,z)$ signifies the electric field envelope, ${\mathrm{\nabla }}_{\perp }$ operator acts on the transverse $(x,y)$ plane, $k_z$ represents the wave vector in the $z$ direction and $k_0=2\pi /\lambda$ with $\lambda$ being the wavelength of the field in vacuum, $n(x,y)$ is the refractive index distribution. This widely-used scalar paraxial optical wave equation can be written in a form mathematically isomorphic to the time-dependent Schrödinger equation $\mathrm{i}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\psi (x,y,z,t)=\left[-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V(x,y,z)\right]\psi (x,y,z,t)$ as following:

(26)$\mathrm{i}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }z}}\phi (x,y,z)=\left[-{\displaystyle \frac{1}{2k_z}}{\mathrm{\nabla }}_{\perp }^2+{\displaystyle \frac{1}{2k_z}}\left(k_z^2-k_0^2{n}^2(x,y)\right)\right]\phi (x,y,z).$

The Helmholtz equation satisfying the optical field in the continuum is extended to discrete model to obtain the coupled mode equation in the coupled waveguide arrays

(27)$\mathrm{i}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }z}}{\phi }_n(z)={\displaystyle \sum _m{c}_{nm}{\phi }_m(z){\mathrm{e}}^{-\mathrm{i}\left({\beta }_n-{\beta }_m\right)z},}$

which describes evolution for the optical field ${\phi }_n$ of the $n$-th waveguide, $c_{nm}$ signifies the coupling coefficients between the $n$-th and the $m$-th waveguides, ${\beta }_{n/m}$ is the propagation constant of the $n$-th/$m$-th waveguide. Under the tight-binding approximation when only the coupling between the nearest neighboring waveguides is considered, the coupled mode equation describing the waveguide array can be obtained by

(28)$-\mathrm{i}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }z}}{\phi }_n(z)={\beta }_n{\phi }_n(z)+c_{n,n-1}{\phi }_{n-1}(z)+c_{n,n+1}{\phi }_{n+1}(z),$

which is also analogous to the Schrödinger equation in form under tight-binding approximation. Note that the coordinate $z$ along the propagation direction plays the role of the temporal coordinate and the refractive index distribution plays the role of optical potential, so the physical phenomena described by the Schrödinger equation can be simulated and experimentally observed by the dynamics of electromagnetic waves in waveguide arrays. For waveguide arrays, the propagation constant of the eigen-modes is functionally equivalent to the eigen-energies of SSH sites. At the same time, coupling between adjacent waveguides due to mode overlap enables photons to transfer between waveguides. By modulating the refractive index and spacing of waveguides, the propagation constant and coupling coefficient can be conveniently arranged, thereby regulating the Hamiltonian of the system. The evolution of optical field in SSH waveguide arrays also follows the coupled mode equation:

(29)$\begin{array}{rl}& -\mathrm{i}\frac{\mathrm{\partial }}{\mathrm{\partial }z}{\phi }_{n,a}(z)={\beta }_a{\phi }_{n,a}(z)+c_2{\phi }_{n-1,b}(z)+c_1{\phi }_{n,b}(z),\\ & -\mathrm{i}\frac{\mathrm{\partial }}{\mathrm{\partial }z}{\phi }_{n,b}(z)={\beta }_b{\phi }_{n,b}(z)+c_1{\phi }_{n,a}(z)+c_2{\phi }_{n+1,a}(z),\end{array}$

where ${\phi }_{n,a/b}$ represents the electric field amplitude, the propagation constant ${\beta }_a$ equals ${\beta }_b$ since in standard SHH model a,b-type waveguides are identical, $c_{1,2}$ are the coupling coefficients between neighboring waveguides. The corresponding bulk Hamilton for Eq. (29) can be written in the form

(30)$H_k=\left(\begin{array}{cc}0& c_1+c_2{\mathrm{e}}^{-\mathrm{i}k}\\ c_1+c_2{\mathrm{e}}^{\mathrm{i}k}& 0\end{array}\right),$

which is consistent with the bulk Hamiltonian in Eq. (2). Therefore, by solving the light propagation in the coupled waveguide system, the same eigenvalues consistent with the dispersion relation and the same eigenvectors consistent with Eq. (3) can be obtained.

Having illustrated the driving protocols for realizing the topological beam splitting and routing in the above sections, in the following we perform simulations in photonic waveguide arrays, which can be experimentally fabricated in borosilicate glass (Eagle XG, with refractive index 1.514) using direct femtosecond-laser-writing technology. Under existing manufacturing technology, it is possible to fabricate waveguide arrays with width of 4 μm and refractive index difference of ${10}^{-3}$ [74]. In the fabricated waveguide array with the 633 nm injected laser, there exists a general relation between the coupling strength and the spacing $J=\eta {\mathrm{e}}^{\gamma d}$, where $\eta$ and $\gamma$ are the parameters to be determined as follows. In a series of two-waveguide systems with the same length $L=$ 9.6 μm and varying $d=[1,15]$ μm with an interval of 2 μm, by simulating the outgoing light intensity in the left waveguide which satisfies $I=I_0{\cos }^2(Jz)$ theoretically, we get the relation of coupling coefficient $J$ with respect to spacing $d$ as displayed in Fig.13(a). A commercial finite-element analysis solver (Comsol Multiphysics 6.0) was employed for full-wave simulations. We can get $\eta =3273$ m⁻¹, $\gamma =0.395$ μm⁻¹ for the fitting curve. The coupling strengths in the beam splitting and routing schemes mentioned above are implemented by the spacings of the adjacent waveguides. For the $z$-dependent $d_1$ and $d_2$ in Fig.13(b), the trigonometric modulations of intracell and intercell coupling strengths for the ordinary beam-splitting SSH chain are achieved as

Fig.13 (a) Simulated inter-waveguide coupling strengths (red dots) as a function of their distance. The blue solid line shows the fitting curve. (b, c, e, f) the spacings $d_1,d_2$ (b) and the coupling strengths $J_1,J_2$ (c) for the ordinary beam-splitting SSH chain adopting the trigonometric protocol with equal output probabilities, and the spacings $d_1,d_1^{\mathrm{\prime }}$ (e) and the coupling strengths $J_1,J_1^{\mathrm{\prime }}$ (f) for the beam-splitting SSH chain with an interface defect adopting the 3-step linear protocol. (d, g) Simulated light propagation in the coupled waveguide arrays composed of 13 sites in total in the trigonometric (d) and 3-step linear (g) protocols through FEM calculation when exciting the central site. (h) Simulated beam splitting efficiencies F as functions of propagation length for the two protocols.

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(31)$\begin{array}{rl}& J_1=J\{0.1+0.4[1+\cos (\pi z/Z)]\},\\ & J_2=J\{0.1+0.4[1-\cos (\pi z/Z)]\},\end{array}$

with the coefficient $J=1400{\mathrm{m}}^{-1}$, as displayed in Fig.13(c), while for the variations of $d_1$ and $d_1^{\mathrm{\prime }}$ with respect to $z$ in Fig.13(e), the 3-step linear modulations of intracell coupling strengths for the beam-splitting SSH chain with an interface defect are achieved as

(32a)$J_1=\{\begin{array}{ll}{J}_0(0.9z/t_1/Z+0.1),& 0\le z\le t_{1}Z\\ J_0,& t_{1}Z<z\le (1-t_2)Z\\ J_0(0.9(1-z/Z)/t_2+0.1),& (1-t_2)Z<z\le Z\end{array},$

(32b)$J_1^{\mathrm{\prime }}=\{\begin{array}{ll}{J}_0(0.9z/t_2/Z+0.1),& 0\le z\le t_{2}Z\\ J_0,& t_{1}Z<z\le (1-t_1)Z\\ J_0(0.9(1-z/Z)/t_1+0.1),& (1-t_1)Z<z\le Z\end{array},$

where $J_2=J,J_0=0.9J,t_1=0.1,t_2=0.445$, as displayed in Fig.13(f). Note that in order to avoid loss caused by the large bending of the waveguides, both $J_1$ and $J_2$ for the trigonometric protocol and $J_1$ and $J_1^{\mathrm{\prime }}$ for the 3-step linear protocol have added a factor of 0.1 compared with the theoretical proposals in the above sections. We further calculate light propagation in the coupled waveguide arrays composed of 13 sites adopting the trigonometric protocol with waveguide length $Z=2.154\mathrm{c}\mathrm{m}$ as shown in Fig.13(d) and the 3-step linear protocol with $Z=0.5348\mathrm{c}\mathrm{m}$ as shown in Fig.13(g). It is easy to notice that in both cases, when the light is injected to the central site, it is pumped adiabatically along the zero-energy channel state and finally arrives at the output port localized at the $1$st (leftmost) and $13$th (rightmost) sites with equal probabilities, which is consistent with the previous theoretical analysis. We further introduce a relative quantity

$F={\left|{\displaystyle \frac{1}{\sqrt{2}}}⟨1,0,\cdots ,0,1\mid \psi (z=Z)⟩\right|}^2/{\left|\psi (z=Z)\right|}^2,$

i.e., the ratio of the intensity of the leftmost and rightmost waveguide at the output facet to that of the total outgoing intensity, which can effectively measure the localization of light at two end waveguides and thus characterize beam splitting efficiency. By simulating light propagation in various waveguide arrays with different total propagation length $Z$, we measure $F$ as functions of propagation length $Z$ for both modulation protocols when the injected light is launched at the central waveguide as shown in Fig.13(h). For a shorter $Z=0.5348\mathrm{c}\mathrm{m}$, the beam splitting efficiency for the 3-step linear protocol reaches as high as 0.9, compared with $Z=2.154\mathrm{c}\mathrm{m}$ for the trigonometric protocol, exhibiting again the superiority of the beam-splitting SSH chain with an interface defect via adiabatic passage in device compactness.

5 Conclusion

In summary, in this work we proposed fast, robust, and flexible topological beam splitting and routing via the adiabatic passage of edge and interface states in the cross-linking configuration of SSH chains with interface defects in configuration. We combined the concept of topological pumping and coherent tunneling by adiabatic passage to realize efficient excitation transfer among edge and interface states, which is more robust against nonadiabatic effects since the channel state remains in the gap and does not get immersed into the band continuum. The number of output ports and output ratios of the splitters and routers can be conveniently adjusted by cross-linking different number of SSH chains and elaborately tailoring their linking strength to the mutual site. Based on the condition of global adiabaticity and numerical analysis of the instantaneous eigen-spectrum, we propose optimized protocols for modulation of the coupling strengths to further accelerate the beam splitting and routing process and the beam splitter in a chain with an interface defect adopting the 3-step linear modulation protocol can be further accelerated to be 91.2% times faster than its ordinary counterpart without adiabatic passage. Additionally, by investigating the scalability of the SSH chain with an interface defect adopting different modulation protocols and the ordinary beam-splitting SSH chain adopting the trigonometric protocol when the system size is varied, we can conclude that within the range of chain size considered here, the four schemes demonstrate good scalability. Furthermore, robustness of different protocols via adiabatic passage and the ordinary chain under trigonometric modulation is extensively discussed by taking into consideration the influence of diagonal and off-diagonal disorders and environment-induced losses. We confirm through numerical sampling that increased efficiency of the beam splitter via adiabatic passage adopting the Gaussian, the 3-step Gaussian, and the 3-step linear protocols is attained at the expense of degraded robustness against disorders in both coupling strengths and on-site energies. Besides, topological beam splitters and routers based on the SSH chains with an interface defect adopting the 3 analyzed protocols exhibits improved robustness against environment induced loss than their ordinary counterpart. Last but not least, we propose coupled waveguide arrays as a feasible platform to implement compact and robust beam splitter discussed in this article and simulate light propagation within the device. The fast and robust topological splitter and router via adiabatic passage manifest sound properties and provides a typical example of topological photonic device, which could induce further research into efficient integrated optics device and the construction of light communication networks.

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Declarations

The authors declare no competing interests and no conflicts.

Acknowledgements

The authors acknowledge the financial support by the National Natural Science Foundation of China (Grant Nos. 62075048 and 12304407) and China Postdoctoral Science Foundation (Grant Nos. 2023TQ0310 and GZC20232446).

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