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

Cavity polaritons derived from strong light–matter interaction provide a basis for efficient manipulation of quantum states via cavity field. Polaritons with narrow linewidth and long lifetime are appealing in applications, such as quantum sensing and storage. Here, we propose a prototypical arrangement to implement a whispering-gallery-mode resonator with one-dimensional topological atom mirror, which allows to boost the lifetime of cavity polaritons over an order of magnitude. This considerable enhancement attributes to the coupling of polaritonic states to dissipationless edge states protected by the topological bandgap of atom mirror that suppresses the leakage of cavity modes. When exceeding the width of Rabi splitting, topological bandgap can further reduce the dissipation from polaritonic states to bulk states, giving arise to subradiant cavity polaritons with extremely sharp linewidth. The resultant Rabi oscillation experiences decay rate lower than the free-space decay of a single quantum emitter. Inheriting from the topologically protected properties of edge states, the subradiance of cavity polaritons can be preserved in disordered atom mirror with moderate perturbations involving the atomic frequency, interaction strengths and location fluctuations. Our work opens up a new paradigm of topology-engineered quantum states with robust quantum coherence for future applications in quantum computing and network.


I. INTRODUCTION
Cavity quantum electrodynamics (QED) constitutes one of the cornerstones of quantum optics, where the coherent exchange of single photon between the quantum emitter (QE) and the cavity mode, known as Rabi oscillation, can take place in the strong-coupling regime and results in the formation of polaritonic states consisting of entangled atom and photon components [1,2].The corresponding bosonic quasiparticle, termed cavity polaritons, offers a scheme for controllable storage and transfer of quantum states and a rich variety of technologies and applications, such as on-chip quantum light source [3,4], quantum sensing [5,6], scalable quantum computing and quantum information processing [2,[7][8][9].Great effort has been devoted into achieving strong coupling in various QED platforms [10][11][12][13], while less attention has been paid to reduce the linewidth of cavity polaritons [14][15][16][17], which is beneficial for diverse quantum-optics applications [18][19][20][21][22][23][24].For instance, reducing the linewidth of resonant systems enables to detect weak signals and achieves better measurement sensitivity for precision sensing in experiments [22][23][24][25][26][27].Moreover, linewidth represents de-cay rate, thereby quantum states with narrower linewidth means longer lifetime, a feature highly desirable for quantum storage and quantum memory [28][29][30][31][32].The lifetime of cavity polaritons is often limited by the quality (Q) factor of cavity, since the linewidth of QE is usually smaller than that of cavity in many cavity-QED systems [11,13,33,34].However, a high-Q cavity in general features a large volume [35][36][37] or requires sophisticated design [38,39] that is demanding for nanofabrication.
Beyond the conventional quantum optics, recently topological quantum optics appeared as a rapidly growing field for controlling light-matter interaction in manybody quantum systems by exploiting the concept of topology [40][41][42][43][44][45][46][47][48][49][50].In analogy to photonic topological insulators, the emergence of exotic topological states in quantum systems, characterized by localized edge states and interface states, demonstrates intriguing optical response and has motivated the development of functional quantum devices with robustness against the structural disorder and impurities, such as topological single-photon circulator [51], topologically protected qubits [42,45,52], unconventional photon transport [47,49,53], faulttolerant topological quantum computing [54][55][56], to mention a few.Among these topological quantum systems, atom arrays can serve as a versatile platform for topological light manipulation, with functionalities beyond the classical mirror that reflects the light [42,44,57,58].In particular, topological quantum states can become sub- radiant through the collective interference [42,44,52], whose radiative loss can be strongly suppressed and significantly smaller than the free-space decay rate of a single atom.This unique feature of atom arrays combined with topological protection provides extra degrees of freedom to manipulate the quantum states.
Triggered by the prospect of manipulating cavity polaritons through topological effects, we propose a topological edge states-engineered cavity QED system consisting of a whispering-gallery-mode (WGM) resonator coupled to a one-dimensional (1D) topological atom mirror with long-range hoppings mediated by waveguide.With sufficiently strong atom-waveguide coupling, edge states become dissipationless through topological phase transition [46].By virtue of the exponential localization and topological protection of dissipationless edge states, this simple configuration enables unprecedented linewidth narrowing and decay suppression of polaritonic states with small atom array.By analyzing the energy spectrum and spectral properties of the composed system, we predict that typically a dozen atoms are adequate to produce subradiance for cavity polaritons and the resultant subnatural linewidth can be experimentally evidenced from either the reflection spectrum of waveguide or the fluorescence of QE.Our scheme can provide a viable approach to realize the long-time storage of quantum states in QED systems with cavities of moderate Q factor and explore the topological manipulation of quantum states on integrated optoelectronic platform.

A. Model and Theory
The system under investigation is depicted in Fig. 1 and comprises of a hybrid cavity QED system based on a WGM ring resonator and a waveguide QED system [59].The resonator supports a series of WGM resonances, but only a pair of degenerate clockwise (CW) and counterclockwise (CCW) modes with the same WGM order is considered.This simplification is reasonable for a realistic WGM resonator operated in the visible and near-infrared ranges, where the linewidth of QE can be much smaller than the frequency spacing between adjacent WGM resonances [20,33,60,61].A QE is embedded inside the resonator, which couples to a waveguide with a topological atom mirror at the right end.The nearestneighbor interactions between topological atoms change alternatively to form 1D diatomic chain by mimicking the Su-Schrieffer-Heeger (SSH) model [40,62], in addition to the long-range interactions mediated by waveguide.An extended cascaded quantum master equation is derived in Appendix A and employed to describe the quantum dynamics of the composed system (see also Refs.[19,59] for details) with Lindblad operator 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 waveguide.
is the Liouvillian superoperator for the dissipation of operator O. c ccw (c 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.σ (j) − is the lowering operator of the jth atom located at x j and particularly, σ (0) − represents the atom inside the cavity, which we refer to QE hereafter to distinguish from the atoms in the mirror.N and x 0 denote the number of atoms and the location of waveguide-cavity junction, respectively.γ 0 and γ λ (λ = 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 (γ R = γ L = Γ) and cavity modes (κ R = κ L = κ) to two chiral guided modes of waveguide.Meanwhile, the coherent interaction between atoms can be tailored by adjusting the atom spacing [63,64].Without loss of generality, we focus on the case of equal atom spacing, i.e., x j+1 −x j = d for j ≥ 1.The total Hamiltonian reads with the free Hamiltonian (4) and the interaction Hamiltonian for cavity QED system and the Hamiltonian describing the coherent coupling between adjacent atoms 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 atoms and we assume ω j = ω 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 Fig. 1).Explicitly, the staggered hoppings can be written as J ± = J 0 [1 ± cos(ϕ)], with J 0 and ϕ being the interaction strength and the tunable parameter of dimerization strength that control the bandgap and localization of edge states, respectively.In absence of dimerized interactions (J 0 = 0), the band structure of atom mirror is topologically trivial, which is centrosymmetric with respect to d = λ 0 /2 and plotted in the inset of Fig. 1.The band structure is modified by the dimerized interactions and gives rise to localized edge states in the strong topological regime with J 0 ≫ γ 0 , which exhibits the periodicity of λ 0 = 2π/k 0 in d.The inset of Fig. 1 also plots the band structure of topological atom mirror with an odd number of sites and J 0 = 8Γ, where 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 atom separations, d = λ 0 /4 and 3λ 0 /4 (indicated by the vertical dashed line in the inset of Fig. 1), where the coupling between topological atoms is fully dispersive [63,64] but no energy shift is observed.This protection stems from the chiral symmetry of SSH chain; however, the topological phases of d = λ 0 /4 and 3λ 0 /4 are distinct [46]: the former is dissipative while the latter is dissipationless.A brief discussion on topological phase transition can be found in Appendix C. Hereafter, atom mirrors with and without the dimerized interactions are called the topological and trivial atom mirrors, respectively, their interac- tion with the polaritonic states of strong-coupling cavity QED system yields the topological and non-topological cavity polaritons.In the following study, we focus on the case of d = 3λ 0 /4, where the dissipationless edge state can produce prominent anisotropic scattering of photon [47].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.
We consider 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 [28,30].To better understand how the topological edge state affects the quantum dynamics and photon transport, we derive the effective Hamiltonian from Eqs. ( 1)-( 6) under the open boundary condition for atom mirror, which is given by with the non-Hermitian free Hamiltonian and the virtual-photon interaction Hamiltonian accounting for the waveguide-mediated long-range hoppings where the first and second lines characterize the nonlocal interactions between atoms and between the cavity modes and the atoms, respectively.ϕ j = k 0 x + (j − 1)φ is the effective phase of light propagating from the waveguide-cavity junction to the jth atom, where φ = k 0 d.Due to the open boundaries of the system, we directly diagonalize H eff to obtain the single-photon band structure and the corresponding eigenstates.For the case of N = 31 atoms, Fig. 2(a) displays the probability distributions of all eigenstates, which is indexed as m = 1, 2, . . ., 34 by increasingly sorting the decay rates (i.e., −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 Fig. 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 freespace decay rate γ 0 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.Fig. 2(c) compares the decay rates of topological and non-topological cavity polaritons versus the number of atom N , where 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 = 8Γ, 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.Fig. 2(c) also indicates that there exists an optimal interaction strength J 0 ≈ 8Γ, denoted as J opt 0 , corresponding to the smallest decay rate of topological cavity polaritons, which is about γ 0 /2, a half of the decay rate of a bare atom.It implies that the reduction of 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., γ 0 = 0).This claim is confirmed by the results presented in the inset of Fig. 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.
Besides the decay rate, the probability distributions of non-topological and topological cavity polaritons are also distinct, as Fig. 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 = 8Γ 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 is the foundation to realize the topologyengineered cavity polaritons.Similar phenomena are also observed for J 0 = 10Γ, 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 J 0 is a consequence of waveguide-mediated long-range hoppings between topological atoms [47,53,65].The delocalization tends to populate all topological atoms and leads to the significant decline of reflection when J 0 exceeds a critical value where the probability is extended to the last cell of topological atom mirror, as illustrated in Fig. 6(b) of Appendix C. 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, Fig. 2(c) also shows the increased decay of topological cavity polaritons for a small J 0 .It is attributed to the coupling of topological cavity polaritons to bulk states and yields J opt 0 for the minimum decay, which we will discuss in the later part of the work.

B. Linewidth narrowing and the enhanced lifetime of subradiant cavity polaritons
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 [Eqs.(1)-( 6)] 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 Fig. 3(a).While for a bare cavity QED system already in the strong-coupling regime, Fig. 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 τ TO /τ 0 shown in Figs.3(b) and (c), where τ TO and τ 0 are the lifetimes of topological and non-topological cavity polaritons, respectively.We find that for topological cavity polaritons, τ TO allows more than 11 cycles of energy exchange between the QE and the cavity, while the nontopological cavity polaritons cannot accomplish a complete period of Rabi oscillation within τ 0 .We also find that τ TO /τ 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 found at φ = 3π/2.
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 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 B and also Ref. [67] for detailed derivation) with and where E and V are the eigenvalues and the corresponding left eigenvectors of H eff .a in is the amplitude of input field.The solution in the frequency domain is given by where ∆ = ω − ω c is the frequency detuning.The output fields of waveguide are given by b out = a in e iϕ N + T out s(∆) with     − (t + τ ) can be obtained by using the quantum regression theorem [1,19].The equation for two-time correlation functions is as follows: where c(τ ) = σ + (0)s(τ ) T , with s(τ ) given in Eq. (15).With initial condition − (ω) in the frequency domain.On the other hand, the emission spectrum of QE can be expressed as by using the Fourier transform relations.We thus can obtain the emission spectrum of QE by substituting σ For the weak-coupling cavity QED system corresponding to Fig. 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 Fig. 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 [68].In contrast, the topological atom mirror brings the system into the strong-coupling regime, characterized by resolvable Rabi splitting with a width of ∼ 2 √ 2g 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 Fig. 2(c) and Fig. 6(c) in Appendix C].While for a strong-coupling cavity QED system, Fig. 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 Fig. 3(f) shows, the locations of upper and lower bands can be clearly identified through the boundary where the reflection suddenly drops.Two bands are symmetrical with respect to ϕ = π/2, but the linewidth of topological cavity polaritons is obviously increased as ϕ goes though π/2.In the parameter range of π/2 < ϕ ≤ π, 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 Fig. 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 [40,53].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 ≤ ϕ ≤ π/2 (π/2 < ϕ ≤ π), see the examples shown in Figs.6(c) and (d) in Appendix C. The results presented in Fig. 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 Fig. 7(a) of Appendix D 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.
J 0 is another important parameter that determines the topological bandgap, and hence the lifetime τ TO exhibits strong dependence on J 0 .Fig. 4(a) plots the lifetime enhancement τ TO /τ 0 as the function of interaction strength J 0 and the number of atoms N , where it shows that τ TO > γ −1 0 can be achieved in a wide range of J 0 with a few tens of atoms in mirror.Remarkably, τ TO /τ 0 demonstrates an abrupt increase at a critical interaction strength J c 0 regardless of N .This phenomenon can be understood by inspecting the emission spectrum of QE versus J 0 , as shown in Fig. 4(b).We can see that the sig- nificant linewidth narrowing of Rabi peaks occurs at J c 0 corresponding to the topological bandgap slightly larger than the width of Rabi splitting, i.e., J c 0 ≳ g/ √ 2 cos(φ).The reason is that for J 0 > J c 0 , 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 J 0 is not always beneficial for enhancing the lifetime.As is seen in Fig. 4(a) and discussed earlier, there exists an optimal interaction strength J opt 0 for maximal τ TO /τ 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 ∼ 4Γ [see Fig. 7(c) in Appendix D for a closeup and further discussion], implying the strong coupling between the topological cavity polaritons and the bulk states.
To shed insights into the dissipative properties of topological cavity polaritons, we diagonalize the Lindblad operator [Eq.(2)] to obtain the dissipative matrix γ = m χ m |v m ⟩ ⟨v m |, with χ m being the dissipation spectrum, which is shown in Fig. 8(a) (see Appendix E 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 Fig. 8(b) of Appendix E. 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 Γ ± 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 [46] where |ψ ± ⟩ is the eigenstate of Re [H eff ] corresponding to the cavity polaritons.The results are plotted in Fig. 4(c), where it shows that the dissipation rate of odd-polarized radiating mode approaches to zero around J opt 0 , while the dissipation rate contributed by the even-polarized radiating mode (cavity modes) is monotonically decreasing (increasing) with increased J 0 .We also find that the dissipation of radiating modes dramatically increases as J 0 approaches to J c 0 .In addition, a small ϕ that produces a large topological bandgap [see Fig. 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 < J opt 0 .While for J 0 > J opt 0 , the dissipation of cavity modes is dominated over the radiating modes and as a result, the minimum Γ ± achieves around J 0 corresponding to zero dissipation rate (∼ 10 −3 γ 0 ) of odd-polarized radiating mode.

C. Robustness against the disorder in topological atom mirror
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.In Fig. 5(a), we plot τ TO /τ 0 with disordered positions for topological atoms, where the position of the jth atom is d j + ∆d j , with −0.02d ≤ ∆d j ≤ 0.02d.It shows that the disorder in atom positions has small impact on the lifetime for J 0 < J opt 0 , especially when τ TO /τ 0 < 10.Though it affects the lifetime in a negative manner, we find that τ TO /τ 0 > 10 can still be obtained around J opt 0 .Different from the disorder in atom positions, the positive impact on lifetime is observed for disorder in interaction Disorder in atom positions strengths with moderate disorder −0.2J 0 ≤ ∆J j ≤ 0.2J 0 and J 0 > J opt 0 , as Fig. 5(b) shows.In this case, the interaction strength between the jth and (j + 1)th atoms is given by J j + ∆J j .The inset of Fig. 5(b) displays that τ TO /τ 0 at J 0 = J opt 0 manifests high robustness against the disorder in interaction strengths for various atom spacing.The results presented in Figs.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 opt 0 .It is because the impact of disorder in atom positions is nonlocal, which affects the coupling between the disordered atom and all 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.Fig. 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 , where the maximal disorder strength is a half of the width of Rabi splitting.We can see that similar to Fig. 5(b), disorder in atom frequencies also begins to have noticeable effect around J c 0 (vertical dashed dotted line), but in this case the lifetime of topological cavity polaritons is less sensitive to disorder when J 0 > J opt 0 (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 Fig. 5 that the presence of moderate disorder in atom mirror will not severely spoil the enhanced lifetime of topological cavity polaritons.
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 γ 0 ≈ 10µeV at cryogenic temperatures [69].For the strong-coupling regime under investigation, κ = 20γ 0 corresponds to a cavity with Q factor of ∼ 6 × 10 3 .The QE-cavity coupling strength g = 20γ 0 = 200µeV can be obtained, for instance, in a WGM microdisk with 3µm radius [20,33].For quantum dot arrays, the switchable coupling between two neighboring sites is usually provided by the tunnel barrier of electrostatic potential, which can be tuned through control gate [43,70,71].Very recently, the experimental realization of SSH chain based on ten semiconductor QEs with tunable interaction strengths has been reported [43].With state-of-art technology, the precision of positioning a QE can reach ∼ 15nm [72], which is less than 2% compared to the emission wavelength of QE.As the results of Fig. 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 [51], 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 upto 200 [73][74][75].Alternatively, the cavity-magnon systems [13] and superconducting circuits [30,42] are also promising candidates to implement topological atom mirror for the advance in realizing multiatom interactions over extended distances.Therefore, the considerable enhancement of lifetime by over an order of magnitude predicted here is achievable for cavity polaritons in diverse quantum systems.

III. CONCLUSION
In summary, we propose a scheme for narrowing the linewidth of cavity polaritons combined with robustness by coupling a one-dimensional topological atom mirror to the cavity QED system based on WGM resonator.The cavity polaritons become subradiant, with a linewidth smaller than that of a single QE through the coupling of cavity mode to edge states in dissipationless topological phase.Accordingly, the lifetime can be improved by over an order of magnitude.The subradiance of cavity polaritons are protected by the topological bandgap and hence can survive in the disordered atom mirror.
Our architecture exhibits prominent advantages in at least three aspects.Firstly, the maximal enhancement of lifetime is achieved in a cavity with moderate Q factor of 10 3 − 10 4 , which gets rid of the drawback of poor excitation and collection efficiencies in conventional approach that reduces the linewidth by the use of a high-Q cavity.This feature combined with the openness of semiinfinite waveguide benefits the practical applications.Importantly, several unit cells, typically 10 − 20 atoms, are sufficient to narrow the linewidth of cavity polaritons to a value comparable to a single QE in the free space.Topological atom mirror of this scale has been demonstrated with state-of-art technology of nanofabrication.Last but not least, the property of topological protection empowers the subradiant cavity polaritons to have high tolerance for fabrication imperfections and experimental uncertainties.Moving forward, future endeavors can devote to explore the effects of coherent time-delayed feedback on lifetime enhancement [76], or conceive the scheme of in situ and dynamical topological manipulation of quantum states [70].Therefore, our scheme offers a promising platform for exploring topological quantum optics and may be potentially used for long-time storage of quantum states in experiments, which is crucial to push quantum technologies toward practical applications.
The equation of b λ can be obtained from the Heisenberg equation Formally integrating the above equation, we have where the initial condition b λ (0) = 0 is imposed since the waveguide is in the vacuum state.On the other hand, the equation of motion of arbitrary operator O reads Substituting b λ (t) into the above equation, we obtain − (τ )e iωx jl /v e −i(ω−ωc)(t−τ ) (A8) where x j0 = x j − x 0 and x jl = x j − x l .We perform the Markov approximation by assuming the time delay x jl /v between the atoms and x j0 between the cavity modes and the atoms are sufficiently small and can be neglected.Therefore, we have   where ρ c (t) = ρ(t)p † c and ρ e (t) = ρ(t)p † e describe the driving from the incident source for CCW mode and atoms, respectively, with p c (t) = κ R 2π dωb R (0)e −i(ω−ωc)t e iωx0/v and p e (t) = γ R 2π dωb R (0)e −i(ω−ωj )t e iωxj /v accounting for the absorption of the incident waveguide photons.We can see that for a monochromatic planewave, p c and p e reduce to a complex number.In this case, we have where a phase factor corresponding to the light propagating from the waveguide-cavity junction to the rightmost atom appears in Eq. (B9).It means that the right output field propagates freely after scattered by the rightmost atom.
where χ m is called the dissipation spectrum and |v m ⟩ is the corresponding wave function.Fig. 8(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 χ 1,34 ∼ κ/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.

FIG. 1 .
FIG. 1. Schematic of a whispering-gallery-mode (WGM) ring cavity coupled to a quantum emitter (QE) and a waveguide with a one-dimensional topological atom mirror at the right end.xj indicates the location of the jth element.The staggered hoppings between nearest-neighbor sites in topological atom mirror simulate the Su-Schrieffer-Heeger (SSH) chain that supports the topological edge states.The pair of sites with stronger coupling defines a unit cell, as the pink translucent box indicates (J+ > J−).The inset shows the real energy spectrum of topological atom mirror versus atom spacing d for nine atoms.Vertical dashed line indicates the dissipationless topological edge state at d = 3λ0/4 under investigation.Other parameters are J0 = 8Γ, ϕ1 = 0 and ϕ = 0.3π.ain and aout, bout stand for the input and output fields for planewave excitation, respectively.

3 FIG. 2 .
FIG. 2. (a) Probability distributions of the composed system.Color of each eigenstate gives the decay rate (imaginary eigenenergy).(b) Complex eigenenergies for cavity QED system with topological (blue circles) and trivial atom mirrors (blue dots).The horizontal dashed line indicates the subradiant region and the vertical green thick lines label the polaritonic states.Complex eigenenergies of bare cavity QED system is also shown for comparison (pink squares), which corresponds to the right axis.(c) Decay rate of topological cavity polaritons Γp versus odd N for various J0.The results of cavity polaritons with trivial atom mirror (Γ 0 p ) and perfect classical mirror (i.e., with unity reflectivity, see Ref. [19] for the model in detail) are shown for comparison.The inset shows Γp/Γ 0 p for various κ with J0 = 8Γ.(d) Probability distributions of topological edge state-engineered cavity polaritons [m = 1, 2 in (a)] and trivial cavity polaritons.Dashed black curve is exponential fits to the probability of cavity polaritons versus cells index.Parameters not mentioned are g = 20γ0, κ = 20γ0, J0 = 8Γ, Γ = 5γ0, N = 31, ϕ1 = 0 and ϕ = 0.3π.

FIG. 3 .
FIG. 3. (a) and (b) Population dynamics of QE and cavity mode with and without the topological atom mirror in the weakand strong-coupling regimes, respectively.The inset in (b) shows the short-time dynamics.The lifetime of cavity polaritons is defined as the time that the population decays from 1 to e −1 .The black dashed line shows the dynamics of initially excited bare QE in the free space.(c) Topology-enhanced lifetime τTO/τ0 versus φ = k0d in the strong coupling.(d) and (e) Reflection and transmission for left-incident planewave (upper panel) and normalized emission spectrum (lower panel) corresponding to the parameters of (a) and (b), respectively.The non-shaded region indicates the topological bandgap.The light gray line in (e) shows the emission spectrum of QE with ϕ = 0.85π.(f) Reflection with topological atom mirror versus ϕ.Parameters for weak coupling are g = 5γ0, κ = 20γ0, J0 = 5Γ, Γ = 5γ0, N = 31, ϕ1 = 0 and ϕ = 0.3π.While the strong coupling is g = 20γ0, J0 = 8Γ and other parameters remain unchanged.The critical coupling strength for strong coupling is gc = (κ + γ0) /2 √ 2 [66].The subscripts 'TO' and 'bare' indicate the spectra with and without the topological atom mirror, respectively.
) Subsequently, we can obtain the reflection and transmission spectra asR(∆) = a † out (∆)a out (∆)/ |a in | 2andT (∆) = b † out (∆)b out (∆)/ |a in | 2, respectively.Eqs.(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(ω) = lim t→∞ Re ∞ 0 dτ σ

FIG. 4 . 2 √
FIG. 4. (a) Topology-enhanced lifetime of cavity polaritons τTO/τ0 versus the number of atoms N and the interaction strength J0.Orange dashed dotted line indicates the critical J0 (i.e., J c 0 ) for topological bandgap with a width of Egap = 2 √ 2g.Dashed white line surrounds the parameters range of τTO > γ −1 0 .Orange star denotes τTO/τ0 for dynamics shown in Fig. 3(b).(b) Emission spectrum of QE versus J0.The horizontal and vertical white dashed lines indicate the locations of cavity polaritons and J c 0 , respectively.(c) Dissipation from topological cavity polaritons to two cavity modes (blue lines) and the even-and odd-polarized radiating modes (light gray and red lines) for ϕ = 0.3π (solid lines with dots) and 0.2π (dashed lines with circles).Parameters not mentioned are the same as Fig. 3(b).

FIG. 5 .
FIG. 5. Topologically robust enhancement of lifetime τTO/τ0 versus interaction strength J0 with disorder in atom positions (a), atom interactions (b), and atom frequencies (c).The disorder ranges are ∆dj ∈ [−0.02d, 0.02d] for atom positions, ∆Jj ∈ [−0.2J0, 0.2J0] for atom interactions, and ∆ωj ∈ [−g/ √ 2, g/ √ 2] for atom frequencies.The pink solid line represents the mean value averaged over 100 random realizations of disordered topological atom mirror, with pink shaded area indicating the standard deviation.The green shaded region indicates the parameter range of J c 0 ≤ J0 ≤ J opt 0 , where the vertical dashed dotted line and dashed line label J c 0 and J opt 0 , respectively.The insets show τTO/τ0 at J0 = J opt 0 versus atom spacing d.Other parameters are the same as Fig. 3(b).

Fig. 8 ( 2 (
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 |ψ n ⟩ of Hamiltonian Re [H eff ], we can evaluate the dissipation rate Γ n of the nth eigenstate to the environment Γ n = ⟨ψ n |γ|ψ n ⟩ = n being the contribution of the mth mode in dissipation spectrumΓ m n = χ m ⟨ψ n | v m ⟩ E3)Particularly, the eigenstates corresponding to cavity polaritons is indicated by n = ±.Fig.4(d) shows the contributions of cavity modes (Γ 1 ± and Γ 34 ± ) and radiating modes (Γ 2 ± and Γ 3 ± ) to the dissipation of cavity polaritons.

FIG. 8 .
FIG. 8. Dissipation spectrum (a) and the wave function of two radiating modes (b) of topological atom mirror.The parameters are the same as Fig. 4(d).