Topologically protected subradiant cavity polaritons through linewidth narrowing enabled by dissipationless edge states

The system under investigation comprises of a hybrid cavity QED system based on a WGM ring resonator and a waveguide QED system [65], as figure 1(a) depicts. We choose such a setup for three reasons: (1) The coupled QE-WGM cavity is a relatively mature system for realizing the Jaynes–Cummings model [1] in nanophotonic platform, which is easy to fabricate using the electron beam lithography in experiments [66–68]; (2) It is more efficient to collect the spectrum directionally through a side-coupled bus waveguide than through the scattering of a bare cavity; (3) A topological atom mirror in waveguide instead of on the surface of cavity can avoid the problem of matching the unit cells in topological mirror with the mode profile of cavity in order to achieve efficient coupling. Therefore, our setup is experimentally feasible and is beneficial to unravel the role of topology in manipulating quantum states. Although the ring resonator supports a series of WGM resonances, only a pair of degenerate clockwise (CW) and counterclockwise (CCW) modes with the same WGM order is considered in our model. This simplification is reasonable for a realistic WGM resonator operated in the visible and near-infrared regions, where the linewidth of QE can be much smaller than the free spectral range [21, 34, 69, 70]. A QE is embedded inside the resonator, which is coupled to a waveguide with a topological atom mirror at the right end. The nearest-neighbor interactions between topological atoms change alternately to form 1D diatomic chain in mimicking the Su–Schrieffer–Heeger (SSH) model [41, 71], accompanied by the long-range interactions mediated by waveguide. An extended cascaded quantum master equation is derived in the appendix A.1 and employed to describe the quantum dynamics of the composed system

$$\begin{equation} \dot{\rho} = -i\left[H, \rho\right]+\mathcal{D}\left[\rho\right] \end{equation} \tag{ 1 }$$

with Lindblad operator

$$\begin{equation} \begin{aligned} \mathcal{D}\left[\rho\right]& = \frac{\kappa_R}{2} \mathcal{L}\left[c_\textrm{ccw}\right] \rho+\frac{\kappa_L}{2} \mathcal{L}\left[c_\textrm{cw}\right] \rho+\sum_{j = 0}^N \frac{\gamma_0}{2} \mathcal{L}\left[\sigma_{-}^{\left(j\right)}\right] \rho+\sum_{\lambda = R, L} \sum_{j = 1}^N \frac{\gamma_\lambda}{2} \mathcal{L}\left[\sigma_{-}^{\left(j\right)}\right] \rho \\ & \quad + \sum_{\lambda = R, L} \sum_{\substack{j, l \unicode{x2A7E} 1 \\ k_\lambda x_j>k_\lambda x_l}}^N \gamma_\lambda\left(e^{i k_\lambda\left(x_j-x_l\right)}\left[\sigma_{-}^{\left(l\right)} \rho, \sigma_{+}^{\left(j\right)}\right]+e^{-i k_\lambda\left(x_j-x_l\right)}\left[\sigma_{-}^{\left(j\right)}, \rho \sigma_{+}^{\left(l\right)}\right]\right) \\ & \quad + \sum_{j = 1}^N \sqrt{\kappa_R \gamma_R}\left(e^{i k_R x_j}\left[c_\textrm{ccw} \rho, \sigma_{+}^{\left(j\right)}\right]+e^{-i k_R x_j}\left[\sigma_{-}^{\left(j\right)}, \rho c_\textrm{ccw}^{\dagger}\right]\right) \\ & \quad + \sum_{j = 1}^N \sqrt{\kappa_L \gamma_L}\left(e^{-i k_L x_j}\left[\sigma_{-}^{\left(j\right)} \rho, c_\textrm{cw}^{\dagger}\right]+e^{i k_L x_j}\left[c_\textrm{cw}, \rho \sigma_{+}^{\left(j\right)}\right]\right) \end{aligned} \end{equation} \tag{ 2 }$$

where the first line introduces the dissipation for individuals, the second line describes the waveguide-mediated interaction between atoms, and the third (four) line accounts for the chiral coupling between the atoms and the CCW (CW) mode through the right-propagating (left-propagating) guided mode of the waveguide. $\mathcal{L}[O] \rho = 2 O \rho O^{\dagger}-O^{\dagger} O \rho-\rho O^{\dagger} O$ is the Liouvillian superoperator for the dissipation of operator O. $c_\textrm{ccw}$ ($c_\textrm{cw}$) is the bosonic annihilation operator of CCW (CW) mode, while κ_R (κ_L ) is the corresponding decay rate stemming from the evanescent coupling to the waveguide. The intrinsic decay of cavity modes is omitted in consideration of the high-Q feature of WGM resonators. $k_R = -k_L = k_0$ is the wave vector of photons. $\sigma_{-}^{(j)}$ is the lowering operator of the jth atom located at x_j and particularly, j = 0 represents the atom inside the cavity, which is referred to as QE hereafter to distinguish from the atoms in the mirror. N and x₀ denote the number of atoms and the location of waveguide-cavity junction, respectively. γ₀ and γ_λ ($\lambda = R,L$) stand for the free-space decay and the waveguide-induced decay of atoms, respectively. Throughout the paper, we consider symmetric coupling of atoms ($\gamma_R = \gamma_L = \Gamma$) and cavity modes ($\kappa_R = \kappa_L = \kappa$) to two chiral guided modes of waveguide. Meanwhile, the coherent interaction between atoms can be tailored by adjusting the atom spacing [72, 73]. Without loss of generality, we focus on the case of equal atom spacing, i.e. $x_{j+1}-x_j = d$ for $j \unicode{x2A7E} 1$; the impact of unequal atom spacing is investigated in section 2.3. As we show in the subsequent derivation, the change of atomic spacing solely influences the phase factor φ_j of jth atom (see equation (9) for definition), which serves as an independent parameter in the equations governing the dynamics of each atom, thus does not render the validity of equations derived in the main text. With rotating-wave approximation, the total Hamiltonian H of the composed system reads

$$\begin{equation} H = H_0+H_I+H_{\text {topo }} \end{equation} \tag{ 3 }$$

with the free Hamiltonian

$$\begin{equation} H_0 = \omega_c c_\textrm{ccw}^{\dagger} c_\textrm{ccw}+\omega_c c_\textrm{cw}^{\dagger} c_\textrm{cw}+\omega_c \sigma_{+}^{\left(0\right)} \sigma_{-}^{\left(0\right)}+\sum_{j = 1}^N \omega_j \sigma_{+}^{\left(j\right)} \sigma_{-}^{\left(j\right)} \end{equation} \tag{ 4 }$$

and the interaction Hamiltonian for cavity QED system

$$\begin{equation} H_I = g\left(c_\textrm{ccw}^{\dagger} \sigma_{-}^{\left(0\right)}+\sigma_{+}^{\left(0\right)} c_\textrm{ccw}\right)+g\left(c_\textrm{cw}^{\dagger} \sigma_{-}^{\left(0\right)}+\sigma_{+}^{\left(0\right)} c_\textrm{cw}\right) \end{equation} \tag{ 5 }$$

and the Hamiltonian describing the coherent coupling between adjacent atoms [44, 47, 74]

$$\begin{equation} H_{\mathrm{topo}} = \sum_{j = 1}^{N-1} J_j\left(\sigma_{+}^{\left(j\right)} \sigma_{-}^{\left(j+1\right)}+\sigma_{+}^{\left(j+1\right)} \sigma_{-}^{\left(j\right)}\right) \end{equation} \tag{ 6 }$$

where ω_c is the frequency of cavity modes, which resonantly couples to QE with strength g. ω_j is the transition frequency of the jth atom and we assume $\omega_j = \omega_c$ unless specially noted. The staggered hoppings $J_j = J_{-}$ ($J_{+}$) for an odd (even) j result in the dimerized interactions between atoms (see schematic presented in figure 1(a)). Explicitly, the staggered hoppings can be written as $J_\pm = J_0 [1\pm\mathrm{cos}(\phi)]$, with J₀ and φ being the interaction strength and the tunable parameter of dimerization strength that control the bandgap and localization of edge states, respectively. In the absence of dimerized interactions (J₀ = 0), the band structure of the atom mirror is topologically trivial, which is centrosymmetric with respect to $d = \lambda_0/2$ and plotted in the inset of figure 1(a). The band structure is modified by the dimerized interactions and gives rise to localized edge states in the strong topological regime with $J_0 \gg \gamma_0$, which exhibits the periodicity of $\lambda_0 = 2\pi/k_0$ in d. The inset of figure 1(a) also plots the band structure of topological atom mirror with an odd number of atoms and $J_0 = 8\Gamma$. It shows that a single edge state survives and is isolating from the bulk states due to the presence of energy gap. It also shows that the edge state is exactly protected from the waveguide-mediated interaction for two special values of atom separation, $d = \lambda_0/4$ and $3\lambda_0/4$ (indicated by the vertical dashed line in the inset of figure 1(a)), where the coupling between topological atoms is fully dispersive [72, 73] but no energy shift is observed. This protection stems from the chiral symmetry of SSH chain; however, the topological phases of $d = \lambda_0/4$ and $3\lambda_0/4$ are distinct [47]: the former is dissipative while the latter becomes dissipationless when $J_0 \gt \Gamma / 2$ where topological phase transition occurs and the decay of edge state is reduced to zero. More discussion on topological phase transition can be found in the appendix A.3. Hereafter, atom mirrors with and without the dimerized interactions (i.e. J₀ = 0) are called the topological and trivial atom mirrors, respectively, their interaction with the polaritonic states of strong-coupling cavity QED system yields the topological and non-topological cavity polaritons. In the following study, we consider the Bragg reflection condition $d = \lambda_0 / 2$ for trivial atom mirror since it corresponds to the highest reflectivity and hence the greatest lifetime enhancement of cavity polaritons in this setup. While for topological atom mirror, we focus on the case of $d = 3\lambda_0/4$, where the dissipationless edge state with suppressed decay can produce prominent anisotropic scattering of photon [48], as figure 7(f) shows. Furthermore, we consider an odd N since the topological atom mirror with an even N gives rise to two edge states with opposite polarization [57]. It results in the significant reduction of reflectivity compared to that with an odd N, as figure 1(c) shows. The results of figures 1(b) and (c) also indicate that N = 31 is sufficient to reach the maximal reflectivity for $J_0\unicode{x2A7D}10\Gamma$. In this work, we show that the coupling of cavity QED system to the dissipationless edge state can suppress the cavity dissipation and results in significant linewidth narrowing of cavity polaritons in both weak- and strong-coupling regimes.

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We consider the case of a single excitation in the composed system, where the subradiant single-photon states hold a great promise for applications related to quantum memory and quantum information storage [29, 31]. To better understand how the topological edge state affects the quantum dynamics and photon transport, we derive the effective Hamiltonian from equations (1)–(6) under the open boundary condition for atom mirror, which is given by

$$\begin{equation} H_{\mathrm{eff}} = H_0^n+H_I+H_{\text {topo }}+H_{\mathrm{vp}} \end{equation} \tag{ 7 }$$

with the non-Hermitian free Hamiltonian

$$\begin{equation} \begin{aligned} H_0^n = \left(\omega_c-i \frac{\kappa}{2}\right) c_\textrm{ccw}^{\dagger} c_\textrm{ccw}+\left(\omega_c-i \frac{\kappa}{2}\right) c_\textrm{cw}^{\dagger} c_\textrm{cw} \left(\omega_c-i \frac{\gamma_0}{2}\right) \sigma_{+}^{\left(0\right)} \sigma_{-}^{\left(0\right)}+\sum_{j = 1}^N\left[\omega_c-i\left(\Gamma+\frac{\gamma_0}{2}\right)\right] \sigma_{+}^{\left(j\right)} \sigma_{-}^{\left(j\right)} \end{aligned} \end{equation} \tag{ 8 }$$

and the virtual-photon interaction Hamiltonian accounting for the waveguide-mediated long-range hoppings

$$\begin{equation} \begin{aligned} H_{\mathrm{vp}} = -i \sum_{\substack{j, l = 1 \\ j \neq l}}^N \Gamma e^{i\left|\phi_j-\phi_l\right|}\left(\sigma_{+}^{\left(j\right)} \sigma_{-}^{\left(l\right)}+\sigma_{+}^{\left(l\right)} \sigma_{-}^{\left(j\right)}\right) -i \sum_{j = 1}^N \sqrt{\kappa \Gamma} e^{i \phi_j}\left(\sigma_{+}^{\left(j\right)} c_\textrm{ccw}+c_\textrm{cw}^{\dagger} \sigma_{-}^{\left(j\right)}\right) \end{aligned} \end{equation} \tag{ 9 }$$

where the first and second lines characterize the nonlocal interactions between atoms and between the cavity modes and the atoms, respectively. $\phi_j = k_0 x+(j-1)\varphi$ is the effective phase of light propagating from the waveguide-cavity junction to the jth atom, where $\varphi = k_0 d$. Due to the open boundaries of the system, we directly diagonalize $H_{\mathrm{eff}}$ to obtain the single-photon band structure (the eigenvalues E and the corresponding eigenstates V, i.e. $H_{\mathrm{eff}} = V^{-1} E V$). For the case of N = 31 atoms, figure 2(a) displays the probability distributions of all eigenstates, which are indexed as $m = 1,2,\ldots,34$ by increasingly sorting the decay rates (i.e. $-\mathrm{Im}[E]$, the imaginary parts of eigenenergies E) versus system components, including the QE and two cavity modes of cavity QED system and the dimer cells of topological atom mirror. A remarkable feature is that the probability presents a substantial atom content for most of the eigenstates, while concentrates in the cavity QED system for eigenstates labelled $m = 1,2$ and $32-34$, i.e. the first two eigenstates with smallest decay rate and the last three with fastest decay. The eigenenergies shown in figure 2(b) reveal that the first two eigenstates are essentially the same as the cavity polaritons of bare cavity QED system (i.e. without the atom mirror), but their decay rates are significantly reduced by over an order of magnitude and even smaller than the free-space decay rate γ₀ of QE, referred to subradiant cavity polaritons. On the contrary, the subradiance cannot be generated for a non-topological cavity polaritons and its decay rate is nearly triple to the topological one.

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Figure 2(c) compares the decay rates of topological and non-topological cavity polaritons versus the number of atom N. It shows that the non-topological cavity polaritons has the advantage of slow decay with a few atoms; however, the decay rate of topological cavity polaritons rapidly drops with increased N and becomes smaller than that of non-topological cavity polaritons with around 10 atoms. For $J_0 = 8\Gamma$, 21 atoms are sufficient to reduce the decay rate of topological cavity polaritons to the minimum, and this minimum value is stable as the atom number increases. While the decay rate of non-topological cavity polaritons gradually increases after an optimal N due to the accumulated loss in the system. Figure 2(c) also indicates that there exists an optimal interaction strength $J_0\approx8\Gamma$, denoted as $J_0^\mathrm{opt}$, corresponding to the smallest decay rate of topological cavity polaritons, which is about $\gamma_0/2$, a half of the decay rate of a bare atom. It implies that the reduced decay is achieved by suppressing the dissipation of cavity modes, since on resonance the decay rate of cavity polaritons is the average of atom and photon components. Therefore, cavity polaritons can acquire permanent coherence with a QE whose free-space decay vanishes (i.e. γ₀ = 0). This claim is confirmed by the results presented in the inset of figure 2(c), where it shows that in such a case, the decay of topological cavity polaritons can be completely suppressed for different κ. It reveals the formation of bound cavity polaritons in a fully open architecture, which is not found with a trivial atom mirror and cannot be interpreted as a result of classical destructive interference, since the reflectivity of trivial atom mirror at $d = \lambda_0 / 2$ is even higher than the topological atom mirror at $d = 3\lambda_0 / 4$ while the lifetime enhancement of the latter is greater, see the figures 7(e) and (f) of appendix A.3. The results indicate that two requirements of atom mirror should be satisfied to produce significant lifetime enhancement of cavity polaritons: one is to have high reflectivity and the other is that the quantum states resonant with cavity QED system are subradiant. As we show in figure 7(f), the highest reflectivity of non-topological atom mirror corresponds to the Bragg reflection condition, however, in this case all quantum states of non-topological atom mirror are resonant with cavity QED system, see the inset of figure 1(a), where one of them is superradiant with decay rate $N\Gamma$. Therefore, the non-topological atom mirror only satisfies one of the requirements and the non-topological cavity polaritons cannot be bound. While for topological cavity, comparable high reflectivity can be achieved at $d = 3\lambda_0/4$ since it corresponds to the dissipationless topological phase. At the same time, the edge state undergoes topological phase transition and becomes subradiant when $J_0\gt0.5\Gamma$. We can see that the topological atom mirror meets both requirements for significant enhancement of lifetime, where the existence of dissipationless topological phase plays a key role in achieving high reflectivity and forming a subradiant edge state.

Besides the decay rate, the probability distributions of non-topological and topological cavity polaritons are also distinct, as figure 2(d) shows. For non-topological cavity polaritons, the probability is uniformly distributed at each cell of atom mirror due to the translational symmetry of homogeneous chain, while localizes at the left boundary for topological cavity polaritons with $J_0 = 8\Gamma$ and converges to zero after 10 cells. Furthermore, the probability distributions of topological cavity polaritons manifest the behavior of exponential decay from the left boundary, except for the first two cells since the chiral symmetry is broken in our model. These features indicate the formation of edge state in topological atom mirror and its efficient coupling to the cavity QED system, which are the foundation to realize the topology-engineered cavity polaritons. Similar phenomena are also observed for $J_0 = 10\Gamma$, but the probability distribution is less concentrated and extended closer to the right boundary with a slow decay. The strong delocalization of probability distributions for a large J₀ is a consequence of waveguide-mediated long-range hoppings between topological atoms [48, 57, 75]. The delocalization tends to populate all topological atoms and leads to the significant decline of reflectivity when J₀ exceeds a critical value where the probability is extended to the last cell of topological atom mirror, as illustrated in figure 7(b) of appendix A.3. As a result, the delocalization weakens the ability of topological atom mirror in suppressing the leakage of cavity modes. In this situation, more atoms are required to hinder the extension of probability to the right boundary and reduce the decay of topological cavity polaritons. On the other hand, figure 2(c) also shows the increased decay of topological cavity polaritons for a small J₀. It is attributed to the coupling of topological cavity polaritons to bulk states and yields $J_0^\mathrm{opt}$ for the minimum decay, which we will discuss in the later part of this work.

The subradiance of topological cavity polaritons enables to enhance the quantum coherence, demonstrating the features of slow population decay and linewidth narrowing in spectrum. To investigate the quantum dynamics, we derive the equations of motion from the extended cascaded quantum master equation (equations (1)–(6))

$$\begin{equation} \frac{\textrm{d}}{\textrm{d} t} \tilde{\sigma}_{-}^{\left(0\right)} = -\frac{\gamma_0}{2} \tilde{\sigma}_{-}^{\left(0\right)}-i g\left(\tilde{c}_\textrm{cw}+\tilde{c}_\textrm{ccw}\right) \end{equation} \tag{ 10 }$$

$$\begin{equation} \frac{\textrm{d}}{\textrm{d} t} \tilde{c}_\textrm{ccw} = -\frac{\kappa}{2} \tilde{c}_\textrm{ccw}-i g \tilde{\sigma}_{-}^{\left(0\right)} \end{equation} \tag{ 11 }$$

$$\begin{equation} \frac{\textrm{d}}{\textrm{d} t} \tilde{c}_\textrm{cw} = -\frac{\kappa}{2} \tilde{c}_\textrm{cw}-i g \tilde{\sigma}_{-}^{\left(0\right)}-\sqrt{\kappa \Gamma} \sum_{j = 1}^N \tilde{\sigma}_{-}^{\left(j\right)} e^{i \phi_j} \end{equation} \tag{ 12 }$$

$$\begin{equation} \begin{aligned} \frac{\textrm{d}}{\textrm{d} t} \tilde{\sigma}_{-}^{\left(j\right)} = & -\frac{\gamma_0}{2} \tilde{\sigma}_{-}^{\left(j\right)}-\sqrt{\kappa \Gamma} e^{i \phi_j} \tilde{c}_\textrm{ccw}-\Gamma \sum_{l = 1}^N \tilde{\sigma}_{-}^{\left(l\right)} e^{i\left|\phi_j -\phi_l\right|} -J_{j-1} \tilde{\sigma}_{-}^{\left(j-1\right)}-J_{j+1} \tilde{\sigma}_{-}^{\left(j+1\right)} \end{aligned} \end{equation} \tag{ 13 }$$

where the substitution $\langle O\rangle = \operatorname{Tr}[O \rho] = \tilde{O} e^{-i \omega_c t}$ is applied and the single-photon approximation $\left[\sigma_{+}, \sigma_{-}\right] = \sigma_z = -1$ is imposed [63, 76]. Figures 3(a) and (b) plot the population dynamics $|\tilde{\sigma}_{-}^{(0)}|^2$ of an initially excited QE with parameters of the weak- and strong-coupling regimes for bare cavity QED system, respectively. By reducing the losses associated with the leakage of cavity modes through the topological atom mirror, a weak-coupling cavity QED system can enter into the strong-coupling regime, which is evidenced by Rabi oscillation in the population dynamics of both QE and cavity modes and shown in figure 3(a). While for a bare cavity QED system already in the strong-coupling regime, figure 3(b) shows that the period of Rabi oscillation is almost unchanged after introducing the topological atom mirror, while its decay is strongly suppressed and even slower than a bare QE in the free space. It implies that the coupling strengths of QE-cavity interaction are comparable in two configurations but the linewidth of cavity polaritons is significantly narrowed. As a consequence, the lifetime of cavity polaritons can be prolonged by over an order of magnitude, see the lifetime enhancement $\tau_\mathrm{TO}/\tau_0$ shown in figures 3(b) and (c), where $\tau_\mathrm{TO}$ and τ₀ are the lifetimes of topological and non-topological cavity polaritons, respectively. We find that for topological cavity polaritons, $\tau_\mathrm{TO}$ allows more than 11 cycles of energy exchange between the QE and the cavity, while the non-topological cavity polaritons cannot accomplish a complete period of Rabi oscillation within τ₀. We also find that $\tau_\mathrm{TO}/\tau_0$ depends on the choice of ϕ and there is a narrow window of ϕ for significant enhancement of lifetime, offering a degree of tuning tolerance to fabrication errors and experimental uncertainties. The maximum enhancement of lifetime, or equivalently, the greatest linewidth narrowing, is only found at $\varphi = 3\pi/2$. The reason is that the topological atom mirror exhibits the highest reflectivity at $\varphi = 3\pi/2$ ($d = 3\lambda_0/4$), meanwhile the edge state is subradiant for $J_0\gt0.5\Gamma$, as we mentioned before. These features are not simultaneously satisfied with other values of φ.

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To observe the phenomenon of linewidth narrowing associated with the subradiance of topological cavity polaritons, we calculate the spectra of the composed system for two kinds of excitation configurations, the reflection and transmission for left-incident planewave and the fluorescence of QE addressed through the free space, which are both experimentally relevant. For the first configuration, the dynamics of system can be written in a compacted form (see appendix A.2 and also [78] for detailed derivation)

$$\begin{equation} \frac{\textrm{d} }{\textrm{d} t}\langle\mathbf{s}\left(t\right)\rangle = -i V^{-1} E V \langle\mathbf{s}\left(t\right)\rangle-P_\textrm{in} \end{equation} \tag{ 14 }$$

with

$$\begin{equation} \mathbf{s}\left(t\right) = \left[\sigma_{-}^{\left(0\right)}\left(t\right), c_\textrm{ccw}\left(t\right), c_\textrm{cw}\left(t\right), \sigma_{-}^{\left(1\right)}, \cdots, \sigma_{-}^{\left(N\right)}\left(t\right)\right]^T \end{equation} \tag{ 15 }$$

and

$$\begin{equation} P_{i n} = \left[0, \sqrt{\kappa}, 0, \sqrt{\Gamma} e^{i \phi_1}, \cdots, \sqrt{\Gamma} e^{i \phi_N}\right]^T a_\textrm{in} \end{equation} \tag{ 16 }$$

where E and V are the eigenvalues and the corresponding left eigenvectors of $H_\mathrm{eff}$. $a_\mathrm{in}$ is the amplitude of input field. The solution in the frequency domain is given by

$$\begin{equation} \mathbf{s}\left(\Delta\right) = i V^{-1}\left(\Delta I-E\right)^{-1} V P_\textrm{in} \end{equation} \tag{ 17 }$$

where $\Delta = \omega-\omega_c$ is the frequency detuning. The output fields of waveguide are given by

$$\begin{equation} a_{\text {out}}\left(\Delta\right) = R_{\text {out }} \mathbf{s}\left(\Delta\right) \end{equation} \tag{ 18 }$$

$$\begin{equation} b_{\text {out}}\left(\Delta\right) = a_\textrm{in}e^{i \phi_N}+T_{\text {out }} \mathbf{s}\left(\Delta\right) \end{equation} \tag{ 19 }$$

with

$$\begin{equation} R_{\text {out }} = \left[0,0, \sqrt{\kappa} e^{i \phi_N}, \sqrt{\Gamma} e^{i\left(N-1\right) \varphi}, \cdots, \sqrt{\Gamma} e^{i \varphi}, \sqrt{\Gamma}\right]^T \end{equation} \tag{ 20 }$$

$$\begin{equation} T_{\text {out }} = \left[0, \sqrt{\kappa} e^{i \phi_N}, 0, \sqrt{\Gamma} e^{i\left(N-1\right) \varphi}, \cdots, \sqrt{\Gamma} e^{i \varphi}, \sqrt{\Gamma}\right]^T. \end{equation} \tag{ 21 }$$

Subsequently, we can obtain the reflection and transmission spectra as $R(\Delta) = a_{\text {out}}^{\dagger}(\Delta) a_{\text {out}}(\Delta) /\left|a_{\text {in}}\right|^2$ and $T(\Delta) = b_{\text {out}}^{\dagger}(\Delta) b_{\text {out}}(\Delta) /\left|a_{\text {in}}\right|^2$, respectively. Equations (18)–(21) indicate that in this configuration, the pump photons can interfere with the scattering photons. While for the configuration of fluorescence, only the photons emitted by QE are detected. The emission spectrum of QE is defined as $S(\omega) = \lim _{t \rightarrow \infty} \operatorname{Re}$ $\left[\int_0^{\infty} \textrm{d} \tau\left\langle\sigma_{+}^{(0)}(t) \sigma_{-}^{(0)}(t+\tau)\right\rangle e^{i \omega \tau}\right]$, where the two-time correlation function $\left\langle\sigma_{+}^{(0)}(t) \sigma_{-}^{(0)}(t+\tau)\right\rangle$ can be obtained by using the quantum regression theorem [1, 20]. The equations for two-time correlation functions are as follows:

$$\begin{equation} \frac{\textrm{d} }{\textrm{d} \tau}\mathbf{c}\left(\tau\right) = -i V^{-1} E V \mathbf{c}\left(\tau\right) \end{equation} \tag{ 22 }$$

where $\mathbf{c}(\tau) = \left[\left\langle\sigma_{+}^{(0)}(0) \mathbf{s}(\tau)\right\rangle\right]^T$, with $\mathbf{s}(\tau)$ given in equation (15). With initial condition $c_0 = [1,0, \ldots, 0]^T$, we have $\mathbf{c}(\omega) = i V^{-1}(\omega I-E)^{-1} V c_0$, which yields the solution of $\left\langle\sigma_{+}^{(0)} \sigma_{-}^{(0)}(\omega)\right\rangle$ in the frequency domain. On the other hand, the emission spectrum of QE can be expressed as $S(\omega) = \operatorname{Re}\left[\left\langle\sigma_{+}^{(0)} \sigma_{-}^{(0)}(\omega)\right\rangle\right]$ by using the Fourier transform relations. We thus can obtain the emission spectrum of QE by substituting $\left\langle\sigma_{+}^{(0)} \sigma_{-}^{(0)}(\omega)\right\rangle$ into $S(\omega)$.

For the weak-coupling cavity QED system corresponding to figure 3(a), no spectral splitting is observed in both the transmission/reflection spectrum and the emission spectrum of QE for the bare cavity-QED without topological atom mirror, which are shown in the dashed curves in figure 3(d). However, we note that the emission spectrum of QE is obviously broaden, which is the superposition of two eigenmodes and it can be called as 'dark' strong coupling [79]. In contrast, the topological atom mirror brings the system into the strong-coupling regime, characterized by resolvable Rabi splitting with a width of ${\sim} 2 \sqrt{2} g$ seen in both the reflection spectrum and the emission spectrum of QE. In this case, the transmission is approximately zero since the edge state is localized at the left boundary of atom mirror (see figures 2(d) and 7(c) of appendix A.3). While for a strong-coupling cavity QED system, figure 3(e) shows that the Rabi peaks corresponding to topological cavity polaritons exhibit extremely sharp linewidth, which can be apparently observed in both the reflection spectrum and the emission spectrum of QE. Besides the Rabi splitting, multiple resonances located at the left and right sides of the reflection and transmission spectra are reminiscent of bulk states.

The topological bandgap separates the edge states and bulk states and plays a central role in determining the optical response of atom mirror. The gap of topological band can be controlled by the parameter φ. It is interesting to see the variation of reflection spectrum by tuning φ, which can reveal how the change of underlying band structure alters the decay of topological cavity polaritons. As the reflection spectrum of figure 3(f) shows, the locations of upper and lower bands can be clearly identified through the boundary where the reflectivity suddenly drops. Two bands are symmetrical with respect to $\phi = \pi/2$, but the linewidth of topological cavity polaritons is obviously increased as φ goes through $\pi/2$. In the parameter range of $\pi / 2\lt\phi \unicode{x2A7D} \pi$, the coupling of strong-coupling cavity QED system to topological edge state slightly broadens instead of narrowing the linewidth of polaritonic states, as the light gray line in the lower panel of figure 3(e) shows; meanwhile, there are two hybrid edge modes with a finite gap around the zero energy, similar to a SSH chain with even sites [41, 57]. The dramatic change of linewidth results from the opposite localization of edge states, which localize at the left (right) boundary of atom mirror for $0 \unicode{x2A7D} \phi \unicode{x2A7D} \pi / 2$ ($\pi / 2\lt\phi \unicode{x2A7D} \pi$), see the examples shown in figures 7(c) and (d) of appendix A.3. The results presented in figure 3(f) demonstrate the capacity of topological edge states in efficiently tuning the lifetime and linewidth of cavity polaritons at the single-quantum level. The emission spectrum of QE shown in figure 8(a) of appendix A.4 demonstrates the similar features observed in the reflection spectrum, but the signal of multiple resonances corresponding to bulk states are weak. Therefore, it is preferable to investigate the properties of topological cavity polaritons through the fluorescence of QE.

J₀ is another important parameter that determines the topological bandgap, and hence the lifetime $\tau_\mathrm{TO}$ exhibits strong dependence on J₀. For $J_0\gt0.5\Gamma$ and N > 3, the winding number is W = 1 for topological atom mirror with $d = 3\lambda_0/4$ [47]. As we see from the results shown in figure 4(a), the lifetime enhancement is still dependent on the specific choice of parameters due to the interplay between the dimerization and the long-range interaction mediated by waveguide. Therefore, it is necessary to obtain a substantial bandgap to produce significant lifetime enhancement even if two systems belong to homotopy group. Figure 4(a) plots the lifetime enhancement $\tau_\mathrm{TO}/\tau_0$ as the function of interaction strength J₀ and the number of atoms N, where it shows that $\tau_\mathrm{TO}\gt\gamma_0^{-1}$ can be achieved in a wide range of J₀ with a few tens of atoms in mirror. Remarkably, $\tau_\mathrm{TO}/\tau_0$ demonstrates an abrupt increase at a critical interaction strength $J_0^c$ regardless of N. This phenomenon can be understood by inspecting the emission spectrum of QE versus J₀, as shown in figure 4(b). We can see that the significant linewidth narrowing of Rabi peaks occurs at $J_0^c$ corresponding to the topological bandgap slightly larger than the width of Rabi splitting, i.e. $J_0^{\mathrm{c}} \gtrsim g / \sqrt{2} \cos (\varphi)$. The reason is that for $J_0\gt J_0^c$, the cavity polaritons are detuned from the superradiant bulk states and the corresponding dissipation is strongly suppressed, resulting in the significant enhancement of lifetime. However, a large J₀ is not always beneficial for enhancing the lifetime. As is seen in figure 4(a) and discussed earlier, there exists an optimal interaction strength $J_0^\mathrm{opt}$ for maximal $\tau_\mathrm{TO}/\tau_0$ as a consequence of the tradeoff between the delocalization of edge states and the dissipation induced by bulk states. We also notice that the anticrossing behavior can be observed at $J_0{\sim}4\Gamma$ (see figure 8(c) of appendix A.4 for a closeup and further discussion), implying the strong coupling between the topological cavity polaritons and the bulk states.

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Figure 4(c) shows that for a fixed J₀, there exists an optimal Q factor for the maximal enhancement of lifetime $\tau_\mathrm{TO}$. With continually increased Q factor, the lifetime enhancement decreases. It is because a high-Q cavity leads to the weak interaction between the cavity QED system and the topological atom mirror. It also explains why the optimal J₀ of maximal lifetime enhancement increases for high Q factor, since strong feedback from topological atoms is required in this case. Even so, a high-Q cavity is beneficial to produce the subradiant cavity polaritons (i.e. the lifetime $\tau_\mathrm{TO}\gt\gamma_0^{-1}$), as the white dashed line indicates, because τ₀ is too small for a low-Q cavity. We evaluate that over tenfold lifetime enhancement of cavity polaritons can be achieved for a cavity with Q factor of 10⁴ at 1550 nm and twofold lifetime enhancement is observed even for Q factor approaching 10⁵.

To shed insights into the dissipative properties of topological cavity polaritons, we diagonalize the Lindblad operator (equation (2)) to obtain the dissipative matrix $\gamma = \sum_m \chi_m\left|v_m\right\rangle\left\langle v_m\right|$, with χ_m being the dissipation spectrum, which is shown in figure 9(a) (see appendix A.5 for more details). Four dissipative modes are found to have large dissipation rate, which are two polarized radiating modes corresponding to the even and odd sites in topological atom mirror and other two are related to cavity modes, as we see from the wave function shown in figure 9(b) of appendix A.5. In our model, the dissipation of odd-polarized radiating mode is greater than that of even-polarized radiating mode due to the odd number of atoms. The dissipation rate $\Gamma_{\pm}$ from topological cavity polaritons to environment is given by the overlap between the polaritonic states and the radiating modes in the Lindblad operator, which is evaluated as [47]

$$\begin{equation} \Gamma_{\pm} = \left\langle\psi_{\pm}|\gamma| \psi_{\pm}\right\rangle = \sum_m \chi_m\left|\left\langle\psi_{\pm}| \chi_m\right\rangle\right|^2 \end{equation} \tag{ 23 }$$

where $\left|\psi_{\pm}\right\rangle$ is the eigenstate of $\operatorname{Re}\left[H_{\text {eff }}\right]$ corresponding to the cavity polaritons. The results are plotted in figure 4(d), where it shows that the dissipation rate of odd-polarized radiating mode approaches to zero around $J_0^\mathrm{opt}$, while the dissipation rate contributed by the even-polarized radiating mode (cavity modes) is monotonically decreasing (increasing) with increased J₀. We also find that the dissipation of radiating modes dramatically increases as J₀ approaches to $J_0^c$. In addition, a small φ that produces a large topological bandgap (see figure 3(f)) can reduce the dissipation rate of radiating modes. These results suggest that the system energy mainly dissipates from bulk states to environment in parameter range of $J_0\lt J_0^\mathrm{opt}$. While for $J_0\gt J_0^\mathrm{opt}$, the dissipation of cavity modes is dominated over the radiating modes and as a result, the minimum $\Gamma_{\pm}$ achieves around J₀ corresponding to zero dissipation rate (${\sim}10^{-3}\gamma_0$) of odd-polarized radiating mode.

In practice, the perturbations on system and imperfections of structure are inevitable. In this subsection, we investigate the impact of local disorder on the enhancement of lifetime for topological cavity polaritons. We introduce the random variation in atom position d_j , interaction strength J_j , and frequency ω_j in equations (10)–(13), then evaluate the lifetime of topological cavity polaritons in the same manner as figure 3(b).

In figure 5(a), we plot $\tau_\mathrm{TO}/\tau_0$ with disordered positions for topological atoms, where the position of the jth atom is $d_j+\Delta d_j$, with $-0.02 d \unicode{x2A7D} \Delta d_j \unicode{x2A7D} 0.02 d$. It shows that the disorder in atom positions has small impact on the lifetime for $J_0\lt J_0^\mathrm{opt}$, especially when $\tau_\mathrm{TO}/\tau_0\lt10$. Though it affects the lifetime in a negative manner, we find that $\tau_\mathrm{TO}/\tau_0\gt10$ can still be obtained around $J_0^\mathrm{opt}$. Different from the disorder in atom positions, the positive impact on lifetime is observed for disorder in interaction strengths with moderate disorder $-0.2 J_0 \unicode{x2A7D} \Delta J_j \unicode{x2A7D} 0.2 J_0$ and $J_0\gt J_0^\mathrm{opt}$, as figure 5(b) shows. In this case, the interaction strength between the jth and (j + 1)th atoms is given by $J_j+\Delta J_j$. The inset of figure 5(b) displays that $\tau_\mathrm{TO}/\tau_0$ at $J_0 = J_0^\mathrm{opt}$ manifests high robustness against the disorder in interaction strengths for various atom spacing. The results presented in figures 5(a) and (b) indicate that the disorder in atom positions has greater impact on the lifetime of topological cavity polaritons, as we can see that the lifetime variation of 2% disorder in atom positions is comparable and even slightly larger than that of 20% disorder in atom interactions around $J_0^\mathrm{opt}$. It is because the impact of disorder in atom positions is nonlocal, which affects the coupling between the disordered atom and all the other atoms through the waveguide-mediated long-range hoppings. On the contrary, the disorder in atom interactions is local perturbation, which only alters the coupling between neighboring sites. Therefore, the lifetime enhancement manifests higher robustness against disorder in atom interactions. As for disorder in atom frequencies, its main effect is on the energies of edge and bulk states, thus the impact on the lifetime of topological cavity polaritons is not obvious if the topological bandgap is sufficiently large. Figure 5(c) shows the lifetime enhancement in presence of disorder in atom frequencies, where the strong disorder strength is considered. The frequency of the jth atom is randomly distributed in range of $\omega_j \in [\omega_c-g / \sqrt{2}, \omega_c+g / \sqrt{2}]$, where the maximal disorder strength is a half of the width of Rabi splitting. We can see that similar to figure 5(b), disorder in atom frequencies also begins to have noticeable effect around $J_0^c$ (vertical dashed dotted line), but in this case the lifetime of topological cavity polaritons is less sensitive to disorder when $J_0\gt J_0^\mathrm{opt}$ (vertical dashed line) as expected. Therefore, it implies that disorder in atom interactions and frequencies mainly affect the edge states and bulk states of topological atom mirror, respectively. We conclude from figure 5 that the presence of moderate disorder in atom mirror will not severely spoil the enhanced lifetime of topological cavity polaritons.

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The proposed model can be implemented in various quantum systems. Particularly, it should be emphasized that the parameters used in this work are attainable in nanophotonic platform with semiconductor QEs. Taking InGaAs quantum dots for an example, the intrinsic decay is $\gamma_0 \approx 10\, \mu\mathrm{eV}$ at cryogenic temperatures [80]. For the strong-coupling regime under investigation, $\kappa = 20 \gamma_0$ corresponds to a cavity with Q factor of ${\sim} 6 \times 10^3$. The QE-cavity coupling strength $g = 20 \gamma_0 = 200\,\mu \mathrm{eV}$ can be obtained, for instance, in a WGM microdisk with 3µm radius [21, 34].

The dimerized interactions between the atoms in mirror can be achieved by additional structures or advanced technologies according to the kinds of quantum system. For quantum dot arrays, the switchable coherent coupling between two neighboring sites is usually provided by the tunnel barrier of electrostatic potential, which can be tuned through control gate [44, 74, 81]. In this respect, very recently the experimental realization of SSH chain based on ten semiconductor QEs with tunable interaction strengths has been reported [44]. With state-of-art technology, the precision of positioning a QE can reach ${\sim}15$ nm [82], which is less than 2% compared to the emission wavelength of QE. As the results of figure 5(a) indicates, such experimental uncertainties has limited impact on the lifetime of topological cavity polaritons. In addition, for the case of single-photon excitation as we study in this work, the topological atom mirror can be replaced by cavity counterpart [55], which is a more feasible experimental configuration to tune the system parameters. Besides the solid-state QEs, the technology of optical tweezers has already been applied to construct one- and two-dimensional atom arrays with the number of cold atoms up to 200 [83–85]. Alternatively, the cavity-magnon systems [13] and superconducting circuits [31, 43] are also promising candidates to implement topological atom mirror for the advance in realizing multi-atom interactions over extended distances. For instance, in superconducting qubit architecture the tunable coherent coupling between two qubits has been achieved with a Josephson junction and followed by a linear inductor to ground [86]. While in circuit quantum electrodynamics, the individual sites are formed with resonators while the tunnel couplings are implemented with coupling capacitors [74]. Therefore, the considerable enhancement of lifetime by over an order of magnitude predicted here is achievable for cavity polaritons in diverse quantum systems.

We derive the extended cascaded quantum master equation (equations (1) and (2)) by tracing out the degrees of freedom of the waveguide [65, 88]. The system Hamiltonian including the waveguide modes is given by ($\hbar = 1$)

$$\begin{equation} H_S = H+H_w+H_{s w} \end{equation} \tag{ A1 }$$

with $H = H_0+H_I+H_\mathrm{topo}$ given in equations (3)–(6). H_w is the free Hamiltonian of waveguide

$$\begin{equation} H_w = \sum_{\lambda = R, L} \int \textrm{d} \omega \omega b_\lambda^{\dagger} b_\lambda \end{equation} \tag{ A2 }$$

where b_L (b_R ) is the bosonic annihilation operator of the left-propagating (right-propagating) waveguide mode with frequency ω. The limits of integration are $(-\infty, 0]$ and $[0, \infty)$ for b_L and b_R , respectively, i.e. $H_w = \int_{-\infty}^0 \textrm{d} \omega \omega b_L^{\dagger} b_L+\int_0^{\infty} \textrm{d} \omega \omega b_R^{\dagger} b_R$. But the limits of integration are omitted in the following derivation for the sake of simplicity. H_sw is the interaction Hamiltonian that describes the cavity-waveguide and atom-waveguide interactions

$$\begin{equation} H_{s w} = i \sum_{\lambda = R, L} \int \textrm{d} \omega \left[\sqrt{\frac{\kappa_\lambda}{2 \pi}} b_\lambda^{\dagger} e^{-i k_\lambda x_0} c_\lambda+\sum_{j = 1}^N \sqrt{\frac{\gamma_\lambda}{2 \pi}} b_\lambda^{\dagger} e^{-i k_\lambda x_j} \sigma_{-}^{\left(j\right)}\right]+H . c . \end{equation} \tag{ A3 }$$

where the wave vector $k_\lambda = \omega_c/v_\lambda$, with $v_L\lt0$ ($v_R\gt0$) being the corresponding group velocity [65]. For the sake of convenience, we have used the notions c_R = $c_\textrm{ccw}$ and $c_L = c_\textrm{cw}$ since the CCW and CW modes are coupled to the right- and left-propagating guided modes, respectively. x₀ is the location of cavity-waveguide junction and x_j indicates the location of jth atom that couples to the waveguide. Applying the transformation $\widetilde{H} = U H U^{\dagger}+i \textrm{d} U / \textrm{d} t U^{\dagger}$ with $U = \exp \left[i\left(\omega_c \sum_{\lambda = R, L} c_\lambda^{\dagger} c_\lambda+\sum_{j = 1}^N \omega_j \sigma_{+}^{(j)} \sigma_{-}^{(j)}+\sum_{\lambda = R, L} \int \textrm{d} \omega \omega b_\lambda^{\dagger} b_\lambda\right)t\right]$, we have

$$\begin{equation} \begin{aligned} \widetilde{H}_{s w}\left(t\right) = i \sum_{\lambda = R, L}\int \textrm{d} \omega \left[\sqrt{\frac{\kappa_\lambda}{2 \pi}} b_\lambda^{\dagger} e^{i\Delta\omega_{c} t} e^{-i \omega x_0 / v_\lambda} c_\lambda +\sum_{j = 1}^N \sqrt{\frac{\gamma_\lambda}{2 \pi}} b_\lambda^{\dagger} e^{i\Delta\omega_{j} t} e^{-i \omega x_j / v_\lambda} \sigma_{-}^{\left(j\right)}\right]+H . c . \end{aligned} \end{equation} \tag{ A4 }$$

where $\Delta\omega_{c} = \omega-\omega_c$ and $\Delta\omega_{j} = \omega-\omega_j$. The equation of b_λ can be obtained from the Heisenberg equation

$$\begin{equation} \frac{\textrm{d}}{\textrm{d} t} b_\lambda\left(t\right) = \sqrt{\frac{\kappa_\lambda}{2 \pi}} c_\lambda\left(t\right) e^{i\Delta\omega_{c} t} e^{-i \omega x_0 / v_\lambda}+\sum_{j = 1}^N \sqrt{\frac{\gamma_\lambda}{2 \pi}} \sigma_{-}^{\left(j\right)}\left(t\right) e^{i\Delta\omega_{j} t} e^{-i \omega x_j / v_\lambda}. \end{equation} \tag{ A5 }$$

Formally integrating the above equation, we have

$$\begin{equation} b_\lambda\left(t\right) = \int_0^t \textrm{d} \tau \sqrt{\frac{\kappa_\lambda}{2 \pi}} c_\lambda\left(\tau\right) e^{i\Delta\omega_{c} \tau} e^{-i \omega x_0 / v_\lambda}+\sum_{j = 1}^N \int_0^t \textrm{d} \tau \sqrt{\frac{\gamma_\lambda}{2 \pi}} \sigma_{-}^{\left(j\right)}\left(\tau\right) e^{i\Delta\omega_{j} \tau} e^{-i \omega x_j / v_\lambda} \end{equation} \tag{ A6 }$$

where the initial condition $b_\lambda (0) = 0$ is imposed since the waveguide is initially in the vacuum state. On the other hand, the equation of motion of arbitrary operator O reads

$$\begin{equation} \begin{aligned} \frac{\textrm{d}}{\textrm{d} t} O\left(t\right) & = \sum_{\lambda = R, L} \int \textrm{d} \omega \sqrt{\frac{\kappa_\lambda}{2 \pi}}b_\lambda^{\dagger}\left(t\right) e^{i\Delta\omega_{c} t} e^{\frac{-i \omega x_0}{v_\lambda}}\left[O\left(t\right), c_\lambda\left(t\right)\right] \\ & \quad +\sum_{\lambda = R, L} \sum_{j = 1}^N \int \textrm{d} \omega \sqrt{\frac{\gamma_\lambda}{2 \pi}}b_\lambda^{\dagger}\left(t\right) e^{i\Delta\omega_{j} t} e^{\frac{-i \omega x_j}{v_\lambda}}\left[O\left(t\right), \sigma_{-}^{\left(j\right)}\left(t\right)\right]-H.c.. \end{aligned} \end{equation} \tag{ A7 }$$

Substituting $b_\lambda (t)$ into the above equation, we obtain

$$\begin{equation} \begin{aligned} \frac{\textrm{d}}{\textrm{d} t} O\left(t\right) & = \sum_{\lambda = R, L}\int_0^t \textrm{d} \tau \int \textrm{d} \omega\left[\frac{\kappa_\lambda}{2 \pi} c_\lambda^{\dagger}\left(\tau\right)+\sum_{j = 1}^N \frac{\sqrt{\kappa_\lambda \gamma_\lambda}}{2 \pi} \sigma_{+}^{\left(j\right)}\left(\tau\right) e^{\frac{i \omega x_{j0}}{v_\lambda}}\right] \times\left[O\left(t\right), c_\lambda\left(t\right)\right]e^{i\Delta\omega_{c}\left(t-\tau\right)} \\ & \quad +\sum_{\lambda = R, L} \sum_{j = 1}^N \int_0^t \textrm{d} \tau \int \textrm{d} \omega\left[\frac{\sqrt{\kappa_\lambda \gamma_\lambda}}{2 \pi} c_\lambda^{\dagger}\left(\tau\right) e^{\frac{-i \omega x_{j0}}{v_\lambda}}+\sum_{l = 1}^N \frac{\gamma_\lambda}{2 \pi} \sigma_{+}^{\left(l\right)}\left(\tau\right) e^{\frac{-i \omega x_{jl}}{v_\lambda}}\right] \\ & \quad \times\left[O\left(t\right), \sigma_{-}^{\left(j\right)}\left(t\right)\right] e^{i\Delta\omega_{c}\left(t-\tau\right)}-H.c. \end{aligned} \end{equation} \tag{ A8 }$$

where $x_{j0} = x_j-x_0$ and $x_{jl} = x_j-x_l$. We perform the Born–Markov approximation $O\left(t-x_{jl}/v_\lambda\right) \approx O\left(t\right)$ by assuming the time delay $\left|x_{jl}/v_\lambda\right|$ between the atoms and $\left|x_{j0}/v_\lambda\right|$ between the cavity modes and the atoms are sufficiently small and can be neglected. Therefore, we have

$$\begin{equation} \frac{\kappa_\lambda}{2 \pi} \int_0^t \textrm{d} \tau \int \textrm{d} \omega e^{i\Delta\omega_{c}\left(t-\tau\right)} c_\lambda^{\dagger}\left(\tau\right) = \kappa_\lambda \int_0^t \textrm{d} \tau \delta\left(t-\tau\right) c_\lambda^{\dagger}\left(\tau\right) = \frac{\kappa_\lambda}{2} c_\lambda^{\dagger}\left(t\right) \end{equation} \tag{ A9 }$$

$$\begin{align} \frac{\sqrt{\kappa_\lambda \gamma_\lambda}}{2 \pi} \int_0^t \textrm{d} \tau \int \textrm{d} \omega e^{i\Delta\omega_{c}\left(t-\tau\right)} e^{\frac{i \omega x_{j0}}{v_\lambda}} \sigma_{+}^{\left(j\right)}\left(\tau\right) & = \sqrt{\kappa_\lambda \gamma_\lambda} \int_0^t \textrm{d} \tau \delta\left(t+\frac{x_{j 0}}{v_\lambda}-\tau\right) e^{\frac{i \omega_c x_{j0}}{v_\lambda}} \sigma_{+}^{\left(j\right)}\left(\tau\right) \nonumber\\ & \approx \sqrt{\kappa_\lambda \gamma_\lambda} \Theta\left(-\frac{x_{j0}}{v_\lambda}\right) e^{i k_\lambda x_{j0}} \sigma_{+}^{\left(j\right)}\left(t\right). \end{align} \tag{ A10 }$$

Similarly,

$$\begin{equation} \begin{aligned} \frac{\sqrt{\kappa_\lambda \gamma_\lambda}}{2 \pi} \int_0^t \textrm{d} \tau \int \textrm{d} \omega e^{i\Delta\omega_{c}\left(t-\tau\right)} e^{\frac{-i \omega x_{j0}}{v_\lambda}} c_\lambda^{\dagger}\left(\tau\right) \approx \sqrt{\kappa_\lambda \gamma_\lambda} \Theta\left(\frac{x_{j 0}}{v_\lambda}\right) e^{-i k_\lambda x_{j 0}} c_\lambda^{\dagger}\left(t\right) \end{aligned} \end{equation} \tag{ A11 }$$

$$\begin{equation} \begin{aligned} \frac{\gamma_\lambda}{2 \pi} \sum_{l = 1}^N \int_0^t \textrm{d} \tau \int \textrm{d} \omega e^{i\Delta\omega_{c}\left(t-\tau\right)} e^{\frac{-i \omega x_{jl}}{v_\lambda}} \sigma_{+}^{\left(l\right)}\left(\tau\right) \approx \frac{\gamma_\lambda}{2} \sigma_{+}^{\left(j\right)}\left(t\right) +\gamma_\lambda \sum_{\substack{l = 1 \\ l \neq j}}^N \Theta\left(\frac{x_{j l}}{v_\lambda}\right) e^{-i k_\lambda x_{j l}} \sigma_{+}^{\left(l\right)}\left(t\right) \end{aligned} \end{equation} \tag{ A12 }$$

where $\Theta(t)$ is the step function, which accounts for the time order of the propagating direction of guided modes. For example, equation (A10) corresponds to left-propagating guided mode while equation (A11) only survives in right-propagating guided mode. Substituting equations (A9)–(A12) into equation (A8) and taking the averages, we obtain

$$\begin{equation} \begin{aligned} \frac{\textrm{d}}{\textrm{d} t}\langle O\left(t\right)\rangle & = \sum_{\lambda = R, L} \left\{\frac{\kappa_\lambda}{2}\left\langle c_\lambda^{\dagger}\left(t\right)\left[O\left(t\right), c_\lambda\left(t\right)\right]\right\rangle+\sum_{j = 1}^N \frac{\gamma_\lambda}{2}\left\langle\sigma_{+}^{\left(j\right)}\left(t\right)\left[O\left(t\right), \sigma_{-}^{\left(j\right)}\left(t\right)\right]\right\rangle \right. \\ & \quad \left.+\sum_{\substack{j, l = 1 \\ j \neq l}}^N \gamma_\lambda e^{-i k_\lambda x_{j l}}\left\langle\sigma_{+}^{\left(l\right)}\left(t\right)\left[O\left(t\right), \sigma_{-}^{\left(j\right)}\left(t\right)\right]\right\rangle \right\} \sum_{j = 1}^N \sqrt{\kappa_R \gamma_R}e^{-i k_R x_{j 0}}\left\langle c_R^{\dagger}\left(t\right)\left[O\left(t\right), \sigma_{-}^{\left(j\right)}\left(t\right)\right]\right\rangle \\ & \quad + \sum_{j = 1}^N \sqrt{\kappa_L \gamma_L}e^{i k_L x_{j 0}}\left\langle \sigma_{+}^{\left(j\right)}\left(t\right)\left[O\left(t\right), c_L\left(t\right)\right]\right\rangle -H.c.. \end{aligned} \end{equation} \tag{ A13 }$$

Since $\langle O(t)\rangle = \mathrm{Tr}\left[O(t)\rho(0)\right] = \mathrm{Tr}\left[O\rho(t)\right]$, we can simplify the average of operators in the above equation by using the cyclic property of trace. For example, the terms in the third line of equation (A13) can be written as

$$\begin{equation} \left\langle c_R^{\dagger}\left(t\right)\left[O\left(t\right), \sigma_{-}^{\left(j\right)}\left(t\right)\right]\right\rangle = \operatorname{Tr}\left[c_R^{\dagger} O \sigma_{-}^{\left(j\right)} \rho\left(t\right)-c_R^{\dagger} \sigma_{-}^{\left(j\right)} O \rho\left(t\right)\right] = \operatorname{Tr}\left\{O\left[\sigma_{-}^{\left(j\right)}, \rho\left(t\right) c_R^{\dagger}\right]\right\} \end{equation} \tag{ A14 }$$

$$\begin{equation} \left\langle\left[O\left(t\right), \sigma_{+}^{\left(j\right)}\left(t\right)\right] c_R\left(t\right)\right\rangle = \operatorname{Tr}\left[O \sigma_{+}^{\left(j\right)} c_R \rho\left(t\right)-\sigma_{+}^{\left(j\right)} O c_R \rho\left(t\right)\right] = \operatorname{Tr}\left\{O\left[\sigma_{+}^{\left(j\right)}, c_R \rho\left(t\right)\right]\right\} \end{equation} \tag{ A15 }$$

where the operators without time means to take the initial value, i.e. $O = O(0)$. Applying the cyclic property of trace to other terms of equation (A13), we obtain the following equation

$$\begin{equation} \begin{aligned} \frac{\textrm{d}}{\textrm{d} t}\langle O\left(t\right)\rangle & = \sum_{\lambda = R, L}\left\{\frac{\kappa_\lambda}{2}\left\langle O\left[c_\lambda, \rho\left(t\right) c_\lambda^{\dagger}\right]\right\rangle+\sum_{j = 1}^N \frac{\gamma_\lambda}{2}\left\langle O\left[\sigma_{-}^{\left(j\right)}, \rho\left(t\right) \sigma_{+}^{\left(j\right)}\right]\right\rangle \right.\\ &\quad \left. + \sum_{j, l = 1}^N \gamma_\lambda e^{-i k_\lambda x_{j l}}\left\langle O\left[\sigma_{-}^{\left(j\right)}, \rho\left(t\right) \sigma_{+}^{\left(l\right)}\right]\right\rangle\right\} + \sum_{j = 1}^N \sqrt{\kappa_R \gamma_R} e^{-i k_R x_{j 0}}\left\langle O\left[\sigma_{-}^{\left(j\right)}, \rho\left(t\right) c_R^{\dagger}\right]\right\rangle \\ & \quad+ \sum_{j = 1}^N \sqrt{\kappa_L \gamma_L} e^{i k_L x_{j 0}}\left\langle O\left[c_L, \rho\left(t\right) \sigma_{+}^{\left(j\right)}\right]\right\rangle-H . c .. \end{aligned} \end{equation} \tag{ A16 }$$

According to the relation $\langle O(t)\rangle = \mathrm{Tr}\left[O\rho(t)\right]$, we arrive at the extended cascaded quantum master equation

$$\begin{equation} \begin{aligned} \frac{\textrm{d}}{\textrm{d} t} \rho\left(t\right) & = -i\left[H, \rho\left(t\right)\right]+\sum_{\lambda = R, L} \frac{\kappa_\lambda}{2}\left\{\left[c_\lambda, \rho\left(t\right) c_\lambda^{\dagger}\right]-\left[c_\lambda^{\dagger}, c_\lambda \rho\left(t\right)\right]\right\} \\ & \quad +\sum_{\lambda = R, L} \sum_{j = 1}^N \frac{\gamma_\lambda}{2}\left\{\left[\sigma_{-}^{\left(j\right)}, \rho\left(t\right) \sigma_{+}^{\left(j\right)}\right]-\left[\sigma_{+}^{\left(j\right)}, \sigma_{-}^{\left(j\right)} \rho\left(t\right)\right]\right\} \\ & \quad +\sum_{\lambda = R, L} \sum_{\substack{j, l = 1 \\ j \neq l}}^N \gamma_\lambda\left\{e^{-i k_\lambda x_{j l}}\left[\sigma_{-}^{\left(j\right)}, \rho\left(t\right) \sigma_{+}^{\left(l\right)}\right]-e^{i k_\lambda x_{j l}}\left[\sigma_{+}^{\left(j\right)}, \sigma_{-}^{\left(l\right)} \rho\left(t\right)\right]\right\} \\ & \quad + \sum_{j = 1}^N \sqrt{\kappa_R \gamma_R}\left\{e^{-i k_R x_{j 0}}\left[\sigma_{-}^{\left(j\right)}, \rho\left(t\right) c_R^{\dagger}\right]-e^{i k_R x_{j 0}}\left[\sigma_{+}^{\left(j\right)}, c_R \rho\left(t\right)\right]\right\} \\ & \quad + \sum_{j = 1}^N \sqrt{\kappa_L \gamma_L}\left\{e^{i k_L x_{j 0}}\left[c_L, \rho\left(t\right) \sigma_{+}^{\left(j\right)}\right]-e^{-i k_L x_{j 0}}\left[c_L^{\dagger}, \sigma_{-}^{\left(j\right)}\rho\left(t\right)\right]\right\}. \end{aligned} \end{equation} \tag{ A17 }$$

Note that we define x₀ = 0 in the main text, thus $x_{j0} = x_j$ in the last two lines on the right-hand side. In addition, the second and third terms on the right-hand side can be expressed using the Liouvillian superoperator. Taking into account the free-space decay γ₀ of atoms, we arrive at the extended cascaded quantum master equation given in equations (1) and (2) in the main text.

To further verify the correctness of Lindblad operators derived in equation (A17), in figure 6 we compare the quantum dynamics obtained by numerically calculating the extended quantum master equation (equations (1) and (2)) using QuTip [89] with that given by equations (10)–(13), where good agreement can be seen.

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The extended cascaded quantum master equation in equations (1)–(6) in the main text does not account for the excitation of the system. In this subsection, we consider a concrete excitation configuration, i.e. planewave excitation through the right-propagating guided mode of waveguide (b_R ), and derive the corresponding quantum master equation. To include the excitation, we start by formally integrating equation (A5) and obtain

$$\begin{equation} \begin{aligned} b_R\left(t\right) = b_R\left(0\right) +\int_0^t \textrm{d} \tau \sqrt{\frac{\kappa_R}{2 \pi}} c_R\left(\tau\right) e^{i\left(\omega-\omega_c\right) \tau} e^{-i \omega x_0 / v} +\sum_{j = 1}^N \int_0^t \textrm{d} \tau \sqrt{\frac{\gamma_R}{2 \pi}} \sigma_{-}^{\left(j\right)}\left(\tau\right) e^{i\left(\omega-\omega_j\right) \tau} e^{-i \omega x_j / v} \end{aligned} \end{equation} \tag{ A18 }$$

where $v = v_R$ and $b_R (0)\neq0$ due to the existence of incident waveguide photons. Substituting the above equation into equation (A7), we find the additional terms compared to equation (A8)

$$\begin{equation} \sqrt{\frac{\kappa_R}{2 \pi}} \int_0^{\infty} \textrm{d} \omega b_R^{\dagger}\left(0\right) e^{i\left(\omega-\omega_c\right) t} e^{-i \omega x_0 / v}\left[O\left(t\right), c_R\left(t\right)\right]+H.c. \end{equation} \tag{ A19 }$$

and

$$\begin{equation} \sqrt{\frac{\gamma_R}{2 \pi}} \sum_{j = 1}^N \int_0^{\infty} \textrm{d} \omega b_R^{\dagger}\left(0\right) e^{i\left(\omega-\omega_j\right) t} e^{-i \omega x_j / v}\left[O\left(t\right), \sigma_{-}^{\left(j\right)}\left(t\right)\right]+H.c. \end{equation} \tag{ A20 }$$

which yields the following Lindblad operator for excitation

$$\begin{equation} \mathcal{D}_p\left[\rho\right] = \left[c_R, \rho_c\left(t\right)\right]-\left[c_R^{\dagger}, \rho_c^{\dagger}\left(t\right)\right]+\sum_{j = 1}^N\left\{\left[\sigma_{-}^{\left(j\right)}, \rho_e\left(t\right)\right]-\left[\sigma_{+}^{\left(j\right)}, \rho_e^{\dagger}\left(t\right)\right]\right\} \end{equation} \tag{ A21 }$$

where $\rho_c (t) = \rho(t) p_c^{\dagger}$ and $\rho_e (t) = \rho(t) p_e^{\dagger}$ is the driving for CCW mode and atoms, respectively, with $p_c(t) = \sqrt{\frac{\kappa_R}{2 \pi}} \int \textrm{d} \omega b_R(0) e^{-i\left(\omega-\omega_c\right) t} e^{i \omega x_0 / v}$ and $p_e(t) = \sqrt{\frac{\gamma_R}{2 \pi}} \int \textrm{d} \omega b_R(0) e^{-i\left(\omega-\omega_j\right) t} e^{i \omega x_j / v}$ accounting for the absorption of the incident waveguide photons. In this work, we consider the single-photon planewave excitation, the multiphoton excitation can be referred to, e.g. [91–93]. We can see that for a monochromatic planewave, p_c and p_e reduce to a complex number. In this case, we have

$$\begin{equation} \mathcal{D}_p\left[\rho\right] = \left[c_R, \rho\left(t\right)\right] p_c^*-\left[c_R^{\dagger}, \rho\left(t\right)\right] p_c+\sum_{j = 1}^N\left\{\left[\sigma_{-}^{\left(j\right)}, \rho\left(t\right)\right] p_e^*-\left[\sigma_{+}^{\left(j\right)}, \rho\left(t\right)\right] p_e\right\}. \end{equation} \tag{ A22 }$$

Accordingly, the extended cascaded quantum master equation is given by

$$\begin{equation} \frac{\textrm{d} }{\textrm{d}t}{\rho} = -i\left[H, \rho\right]+\mathcal{D}\left[\rho\right]+\mathcal{D}_p\left[\rho\right] \end{equation} \tag{ A23 }$$

which yields equations (14)–(16) in the main text.

To derive the input–output relations, we integrate equation (A5) from t to t_f (i.e. $t_f\gt t$) and obtain

$$\begin{equation} \begin{aligned} b_R\left(t\right) = b_R\left(t_f\right) +\int_t^{t_f} \textrm{d} \tau \sqrt{\frac{\kappa_R}{2 \pi}} c_R\left(\tau\right) e^{i\left(\omega-\omega_c\right) \tau} e^{-i \omega x_0 / v} +\sum_{j = 1}^N \int_t^{t_f} \textrm{d} \tau \sqrt{\frac{\gamma_R}{2 \pi}} \sigma_{-}^{\left(j\right)}\left(\tau\right) e^{i\left(\omega-\omega_j\right) \tau} e^{-i \omega x_j / v}. \end{aligned} \end{equation} \tag{ A24 }$$

By comparing with equation (A18), we have

$$\begin{equation} \begin{aligned} b_R\left(t\right) = b_R\left(0\right)+\int_0^{t_f} \textrm{d} \tau \sqrt{\frac{\kappa_R}{2 \pi}} c_R\left(\tau\right) e^{i\left(\omega-\omega_c\right) \tau} e^{-i \omega x_0 / v} +\sum_{j = 1}^N \int_0^{t_f} \textrm{d} \tau \sqrt{\frac{\gamma_R}{2 \pi}} \sigma_{-}^{\left(j\right)}\left(\tau\right) e^{i\left(\omega-\omega_j\right) \tau} e^{-i \omega x_j / v}. \end{aligned} \end{equation} \tag{ A25 }$$

We use the following definition of input–output operators [1, 78]

$$\begin{equation} b_{\text {out }}\left(t\right) = \frac{1}{\sqrt{2 \pi}} \int_0^{\infty} \textrm{d} \omega b_R\left(t_f\right) e^{i\left(\omega-\omega_c\right) x_N/v} e^{-i\left(\omega-\omega_c\right) t} \end{equation} \tag{ A26 }$$

$$\begin{equation} b_{\text {in }}\left(t\right) = \frac{1}{\sqrt{2 \pi}} \int_0^{\infty} \textrm{d} \omega b_R\left(0\right) e^{-i\left(\omega-\omega_c\right) t} \end{equation} \tag{ A27 }$$

where a phase factor corresponding to the light propagating from the waveguide-cavity junction to the rightmost atom appears in equation (A26). It means that the right output field propagates freely after scattered by the rightmost atom. Using equations (A26) and (A27), we can obtain the input–output relation from equation (A25)

$$\begin{equation} \begin{aligned} \frac{1}{\sqrt{2 \pi}} & \int_0^{\infty} \textrm{d} \omega b_R\left(t_f\right) e^{i\left(\omega-\omega_c\right) x_N/v} e^{-i\left(\omega-\omega_c\right) t} \\ & = \frac{1}{\sqrt{2 \pi}} \int_0^{\infty} \textrm{d} \omega b_R\left(0\right) e^{i\left(\omega-\omega_c\right) x_N/v} e^{-i\left(\omega-\omega_c\right) t} \\ &\quad +\frac{1}{\sqrt{2 \pi}} e^{i\left(\omega-\omega_c\right) x_N/v} \int_0^{t_f} \textrm{d} \tau \int_0^{\infty} \textrm{d} \omega \sqrt{\frac{\kappa_R}{2 \pi}} c_R\left(\tau\right) e^{-i\left(\omega-\omega_c\right)\left(t-\tau\right)} e^{-i \omega x_0/v} \\ &\quad +\frac{1}{\sqrt{2 \pi}} e^{i\left(\omega-\omega_c\right) x_N/v} \sum_{j = 1}^N \int_0^t \textrm{d} \tau \int_0^{\infty} \textrm{d} \omega \sqrt{\frac{\gamma_R}{2 \pi}} \sigma_{-}^{\left(j\right)}\left(\tau\right) e^{-i\left(\omega-\omega_c\right)\left(t-\tau\right)} e^{-i \omega x_j/v} \end{aligned} \end{equation} \tag{ A28 }$$

$$\begin{equation} \begin{aligned} b_{\text{out}}\left(t\right) & = b_{\text{in}} \left(t-\frac{x_N}{v}\right)+\frac{1}{\sqrt{2 \pi}} \int_0^{t_f} \textrm{d} \tau \int_0^{\infty} \textrm{d} \omega \sqrt{\frac{\kappa_R}{2 \pi}} c_R\left(\tau\right) e^{-i\left(\omega-\omega_c\right)\left(t-\tau-\frac{x_{N 0}}{v}\right)} e^{-i k_R x_0} \\ &\quad +\frac{1}{\sqrt{2 \pi}} \sum_{j = 1}^N \int_0^{t_f} \textrm{d} \tau \int_0^{\infty} \textrm{d} \omega \sqrt{\frac{\gamma_R}{2 \pi}} \sigma_{-}^{\left(j\right)}\left(\tau\right) e^{-i\left(\omega-\omega_c\right)\left(t-\tau-\frac{x_{N j}}{v}\right)} e^{-i k_R x_j}. \end{aligned} \end{equation} \tag{ A29 }$$

Since $\kappa_R\gg0$, we can extend the lower limit of integration to $-\infty$

$$\begin{equation} \begin{aligned} b_{\text {out }}\left(t\right) & = b_{\text {in }}\left(t -\frac{x_N}{v}\right)+\frac{1}{\sqrt{2 \pi}} \int_0^{t_f} \textrm{d} \tau \int_{-\infty}^{\infty} \textrm{d} \omega \sqrt{\frac{\kappa_R}{2 \pi}} c_R\left(\tau\right) e^{-i\left(\omega-\omega_c\right)\left(t-\tau-x_{N 0} / v\right)} e^{-\frac{i \omega_c x_0}{v}} \\ &\quad +\frac{1}{\sqrt{2 \pi}} \sum_{j = 1}^N \int_0^{t_f} \textrm{d} \tau \int_{-\infty}^{\infty} \textrm{d} \omega \sqrt{\frac{\gamma_R}{2 \pi}} \sigma_{-}^{\left(j\right)}\left(\tau\right) e^{-i\left(\omega-\omega_c\right)\left(t-\tau-x_N / v\right)} e^{-\frac{i \omega_c x_j}{v}}. \end{aligned} \end{equation} \tag{ A30 }$$

Using the identity $\int_{-\infty}^{\infty} e^{i k x} \textrm{d} k = 2 \pi \delta(k)$, we obtain

$$\begin{equation} \begin{aligned} b_{\text{out}}\left(t\right)& = b_{\text{in}} \left(t-\frac{x_N}{v}\right)+\sqrt{\kappa_R} \int_0^{t f} \textrm{d} \tau c_R\left(\tau\right) \delta\left(t-\frac{x_{N 0}}{v}-\tau\right) e^{-i k_R x_0} \\ & \quad +\sqrt{\gamma_R} \sum_{j = 1}^N \int_0^{t_f} \textrm{d} \tau \sigma_{-}^{\left(j\right)}\left(\tau\right) \delta\left(t-\frac{x_{N j}}{v}-\tau\right) e^{-i k_R x_j}. \end{aligned} \end{equation} \tag{ A31 }$$

Applying the Born–Markovian approximation, the above equation becomes

$$\begin{equation} b_{\text{out}}\left(t\right) \approx b_{\text {in}}\left(t\right)+\sqrt{\kappa_R} c_R\left(t\right) e^{-i k_R x_0}+\sqrt{\gamma_R} \sum_{j = 1}^N \sigma_{-}^{\left(j\right)}\left(t\right) e^{-i k_R x_j}. \end{equation} \tag{ A32 }$$

With a fashion similar to equations (A25)–(A32), we can obtain the input–output relation for left-propagating guided mode

$$\begin{equation} a_{\text{out}}\left(t\right) \approx a_{\text{in}}\left(t\right)+\sqrt{\kappa_L} c_L\left(t\right) e^{i k_L x_0}+\sqrt{\gamma_L} \sum_{j = 1}^N \sigma_{-}^{\left(j\right)}\left(t\right) e^{i k_L x_j} \end{equation} \tag{ A33 }$$

with

$$\begin{equation} a_{\text{out}}\left(t\right) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^0 \textrm{d} \omega b_L\left(t_f\right) e^{-i\left(\omega-\omega_c\right) x_N/v} e^{-i\left(\omega-\omega_c\right) t} \end{equation} \tag{ A34 }$$

and

$$\begin{equation} a_{\text{in}}\left(t\right) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^0 \textrm{d} \omega b_L\left(0\right) e^{-i\left(\omega-\omega_c\right) t} \end{equation} \tag{ A35 }$$

where a phase factor is added in $a_\text{out} \left(t\right)$ since the left output field propagates freely after scattered by the waveguide-cavity junction. Since $a_\text{in} \left(t\right) = 0$ for left-incident planewave, the input–output relations equations (A32) and (A33) yield equations (18)–(21) in the main text. Note that $a_\text{in} \left(t\right)$ (equation (A35)) is different from the field amplitude a_in in equation (16).

To demonstrate the dissipationless feature of edge states, the spontaneous emission rate of atoms is omitted, i.e. γ₀ = 0. In figure 7(a), we plot the decay rates $\Gamma_\mathrm{edge} = -2\operatorname{Im}\left[E_\mathrm{edge} \right]$ of edge states with atom spacings $d = \lambda_0/4$ and $3\lambda_0/4$ versus intensity strength J₀. It shows that $\Gamma_\mathrm{edge}$ of edge states with $d = 3\lambda_0/4$ suddenly drops to zero at a critical $J_0 \approx \Gamma/2$ regardless of φ, demonstrating the dissipationless feature; while the edge states in dissipative topological phase ($d = \lambda_0/4$) manifests the distinct feature of slowly decreased $\Gamma_\mathrm{edge}$ without transition as J₀ increases. Different from the original SSH model, in our setup the interplay between the dimerized interaction and the waveguide-mediated long-range interaction gives rise to topological phase transition. Without considering the dissipation interaction, the zero-energy states are found at atomic spacing $d = \lambda_0/4$ and $3\lambda_0/4$, and the corresponding winding number is W = 1 for $J_0/\gamma_0\gt0$ and $J_0/\gamma_0\gt0.5$, respectively [47]. However, as we see from figure 7(a), the edge state at $d = \lambda_0/4$ is not protected from the dissipation, while at $d = 3\lambda_0/4$ the edge state becomes dissipationless for $J_0/\Gamma\gt0.5$. Appendix A.5 is devoted to study the dissipation spectrum, where we show that how the Lindblad operator protects the dissipationless edge state.

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In figures 7(c) and (d), we investigate the probability distributions of edge states for a bare topological atom mirror, i.e. without coupling to the cavity QED system. We can see that the delocalization populates all unit cells for a sufficient large interaction strength ($J_0\gt10\Gamma$), thus $\Gamma_\mathrm{edge} \neq 0$ for dissipationless edge states. Furthermore, the topological edge states localize at the left (right) boundary when $0\unicode{x2A7D}\phi\unicode{x2A7D}\pi/2$ ($\pi/2\lt\phi\unicode{x2A7D}\pi$). The long-range hoppings between topological atoms induced by waveguide destroys the exponential localization of edge states: the probability decreases non-monotonically from the boundary and extends to the opposite boundary, exhibiting strong delocalization with a large intensity strength J₀. The delocalization means the increased dissipation of atom mirror and therefore, we can see in figure 7(b) that it requires more atoms to achieve high reflectivity for a large J₀. But on the other hand, a large J₀ can produce a wide topological bandgap, which is beneficial to suppress the dissipation from edge states to bulk states and achieve high reflectivity. The balance between the delocalization and the dissipation induced by bulk states yields an optimal J₀ for the maximal reflectivity of topological atom mirror with a fixed number of atoms.

We stress that for γ₀ = 0, both topological and trivial atom mirrors have unity reflectivity, while the cavity polaritons of latter are dissipative. Therefore, the formation of bound cavity polaritons cannot understood as a result of classical destructive interference between the cavity field and the reflected field through atom mirror. The evidence supporting our claim is that the lifetime of topological cavity polaritons is prominently greater than the non-topological cavity polaritons, as we see in figure 7(e), while the maximal reflectivity of topological atom mirror is slightly smaller than the trivial atom mirror with Bragg reflection condition (i.e. $d = \lambda_0/2$), as figure 7(f) shows. It is because there is a superradiant state with decay rate $N\Gamma$ in trivial atom mirror, which has zero energy at $d = \lambda_0/2$, see the eigenenergies shown in the inset of figure 1, and hence resonantly couples to the QE and cavity. As a result, the bound cavity polaritons cannot form with a trivial atom mirror. In contrast, the zero-energy edge state at $d = 3\lambda_0/4$ has zero decay when γ₀ = 0. Combined with a large bandgap that separates the cavity polaritons from bulk states, the topological cavity polaritons can be bound.

In order to clarify the effects of direct interaction on lifetime enhancement of cavity polaritons, we include the direct interaction between atoms in trivial mirror with the same interaction strength J₀ as that of topological atom mirror, but the interaction is not staggered, i.e. equal to the case of $\mathrm{cos}(\phi) = 0$ in topological atom mirror. The resultant lifetime of cavity polaritons are shown as the green line in figure 7(e), where we can see that the introduction of such coherent coupling cannot achieve significant enhancement of lifetime for trivial atom mirror. The results shown in figures 7(e) and (f) indicate the significance of topological atom mirror in enhancing the lifetime of cavity polaritons.

Figure 8(a) presents the logarithmic plot of the emission spectrum of QE versus φ. By comparing with the reflection spectrum shown in figure 3(f), we can see that they demonstrate similar pattern and features, such as the variation of topological bandgap and the distinct linewidth of cavity polaritons for the left and right edge states. However, for observing the linewidth narrowing of polaritonic states (Rabi peaks), the emission spectrum of QE is preferred since the signal of bulk states is weak. In addition, we find that the composed system shows a feature of SSH model with even sites, where two hybrid edge modes with a finite gap can be seen in the region of $\pi/2\lt\phi\unicode{x2A7D}\pi$, which is hard to recognize in the reflection spectrum. The probability distributions shown in figure 8(b) indicate the formation of hybrid edge modes with even and odd parities, distinguishing from the left or right edge states of bare topological atom mirror that populate either the odd or even sites.

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On the other hand, we find an inconspicuous anticrossing at $\phi{\sim}0.4\pi$ in figure 8(a), indicating the strong coupling between bulk states and polaritonic states. This anticrossing behaviour is more evident in the emission spectrum of QE versus J₀, as figure 8(c) shows. In stark contrast to the conventional strong-coupling anticrossing that is observed by tuning the frequency detuning, here J₀ plays the role of frequency detuning in the strong-coupling anticrossing. It is because the variation of J₀ does not alter the energy of polaritonic states, but the width of topological bandgap linearly enlarges with increased J₀. Therefore, J₀ can change the energies of bulk states.

It is important to note that in the reflection spectrum of figure 3(e), two dips corresponding to cavity polaritons are nonvanishing around $\phi = 0.33\pi$, while this phenomenon is not found in the emission spectrum of QE, as figure 8(d) shows, where the black arrows indicate the polaritonic states. The results reveal that the formation of dark cavity polaritons is related to specific excitation method. For the configuration of planewave excitation, these dark cavity polaritons stem from destructive interference between the incident and scattering photons.

The dissipative interaction is described by the Lindblad operator $\mathcal{D}[\rho]$ (equation (2)), which can be recast in terms of jump operators v_m , the (N + 3) eigenvectors of $\mathcal{D}[\rho]$. The decay rates χ_m are found as the corresponding eigenvalues. $\mathcal{D}[\rho]$ can be rewritten as [47, 94]

$$\begin{equation} \mathcal{D}\left[\rho\right] = \sum_m \chi_m\left(v_m \rho v_m^{\dagger}-\frac{1}{2} \rho v_m^{\dagger} v_m-\frac{1}{2} v_m^{\dagger} v_m \rho\right). \end{equation} \tag{ A36 }$$

The diagonalization of $\mathcal{D}[\rho]$ yields a dissipation matrix γ, which can be expressed as follows

$$\begin{equation} \gamma = \sum_m \chi_m\left|v_m\right\rangle\left\langle v_m\right| \end{equation} \tag{ A37 }$$

where χ_m is also called the dissipation spectrum. Figure 9(a) shows χ_m for the composed system in the strong-coupling regime with 31 atoms in mirror. We can see that the dissipation of modes m = 1 and 34 is $\chi_{1,34}{\sim}\kappa/2$, thus they are related to two cavity modes. Two radiating modes indexed by m = 2 and 3 can be found in the dissipation spectrum, whose dissipation is much greater than other modes. Figure 9(b) plots the wave function of two radiating modes versus atoms index, where we can identify the odd and even polarization for m = 2 and 3, respectively. With the eigenstates $\left|\psi_n\right\rangle$ of Hamiltonian $\operatorname{Re}\left[H_\mathrm{eff} \right]$, we can evaluate the dissipation rate $\Gamma_n$ of the nth eigenstate to the environment

$$\begin{equation} \Gamma_n = \left\langle\psi_n|\gamma| \psi_n\right\rangle = \sum_m \Gamma_n^m \end{equation} \tag{ A38 }$$

with $\Gamma_n^m$ being the contribution of the mth mode in dissipation spectrum

$$\begin{equation} \Gamma_n^m = \chi_m\left|\left\langle\psi_n \mid v_m\right\rangle\right|^2. \end{equation} \tag{ A39 }$$

Particularly, the eigenstates corresponding to cavity polaritons is indicated by $n = \pm$. Figure 4(d) shows the contributions of cavity modes ($\Gamma_\pm^{1}$ and $\Gamma_\pm^{34}$) and radiating modes ($\Gamma_\pm^{2}$ and $\Gamma_\pm^{3}$) to the dissipation of cavity polaritons.

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