Quantum electrodynamics in a topological waveguide

Harnessing the topological properties of photonic bands Haldane and Raghu (2008); Lu et al. (2014); Ozawa et al. (2019) is a burgeoning paradigm in the study of periodic electromagnetic structures. Topological concepts discovered in electronic systems von Klitzing (1986); Hasan and Kane (2010) have now been translated and studied as photonic analogs in various microwave and optical systems Lu et al. (2014); Ozawa et al. (2019). In particular, symmetry-protected topological phases Chen et al. (2013) which do not require time-reversal-symmetry breaking, have received significant attention in experimental studies of photonic topological phenomena, both in the linear and nonlinear regime Smirnova et al. (2019). One of the simplest canonical models is the Su-Schrieffer-Heeger (SSH) model Su et al. (1979); Asbóth et al. (2016), which was initially used to describe electrons hopping along a one-dimensional dimerized chain with a staggered set of hopping amplitudes between nearest-neighbor elements. The chiral symmetry of the SSH model, corresponding to a symmetry of the electron amplitudes found on the two types of sites in the dimer chain, gives rise to two topologically distinct phases of electron propagation. The SSH model, and its various extensions, have been used in photonics to explore a variety of optical phenomena, from robust lasing in arrays of microcavities St-Jean et al. (2017); Zhao et al. (2018) and photonic crystals Ota et al. (2018), to disorder-insensitive 3rd harmonic generation in zigzag nanoparticle arrays Kruk et al. (2019).

Utilization of quantum emitters brings new opportunities in the study of topological physics with strongly interacting photons Carusotto and Ciuti (2013), where single-excitation dynamics Cai et al. (2019) and topological protection of quantum many-body states de Léséleuc et al. (2019) in the SSH model have recently been investigated. In a similar vain, a topological photonic bath can also be used as an effective substrate for endowing special properties to quantum matter. For example, a photonic waveguide which localizes and transports electromagnetic waves over large distances, can form a highly effective quantum light-matter interface Haroche and Raimond (2006); Lodahl et al. (2015); Chang et al. (2018) for introducing non-trivial interactions between quantum emitters. Several systems utilizing highly dispersive electromagnetic waveguide structures have been proposed for realizing quantum photonic matter exhibiting tailorable, long-range interactions between quantum emitters Douglas et al. (2015); Shi et al. (2018); Liu and Houck (2016); Hung et al. (2016). With the addition of non-trivial topology to such a photonic bath, exotic classes of quantum entanglement can be generated through photon-mediated interactions of a chiral Barik et al. (2018); Lodahl et al. (2017) or directional nature Bello et al. (2019); García-Elcano et al. (2019).

With this motivation, here we investigate the properties of quantum emitters coupled to a topological waveguide which is a photonic analog of the SSH model Bello et al. (2019). Our setup is realized by coupling superconducting transmon qubits Koch et al. (2007) to an engineered superconducting metamaterial waveguide Mirhosseini et al. (2018); Ferreira et al. (2020), consisting of an array of sub-wavelength microwave resonators with SSH topology. Combining the notions from waveguide quantum electrodynamics (QED) Lodahl et al. (2015); Chang et al. (2018); Gu et al. (2017); Roy et al. (2017) and topological photonics Lu et al. (2014); Ozawa et al. (2019), we observe qubit-photon bound states with directional photonic envelopes inside a bandgap and cooperative radiative emission from qubits inside a passband dependent on the topological configuration of the waveguide. Coupling of qubits to the waveguide also allows for quantum control over topological edge states, enabling quantum state transfer between distant qubits via a topological channel.

The SSH model describing the topological waveguide studied here is illustrated in Fig. 1a. Each unit cell of the waveguide consists of two photonic sites, A and B, each containing a resonator with resonant frequency $\omega_{0}$. The intra-cell coupling between A and B sites is $J(1+\delta)$ and the inter-cell coupling between unit cells is $J(1-\delta)$. The discrete translational symmetry (lattice constant $d$) of this system allows us to write the Hamiltonian in terms of momentum-space operators, $\hat{H}/\hbar=\sum_{k}(\hat{\mathbf{v}}_{k})^{\dagger}\>\mathbf{h}(k)\>\hat{\mathbf{v}}_{k}$, where $\hat{\mathbf{v}}_{k}=(\hat{a}_{k},\hat{b}_{k})^{T}$ is a vector operator consisting of a pair of A and B sublattice photonic mode operators, and the $k$-dependent kernel of the Hamiltonian is given by,

Here, $f(k)\equiv-J[(1+\delta)+(1-\delta)e^{-ikd}]$ is the momentum-space coupling between modes on different sublattice, which carries information about the topology of the system. The eigenstates of this Hamiltonian form two symmetric bands centered about the reference frequency $\omega_{0}$ with dispersion relation

We construct a circuit analog of this canonical model using an array of inductor-capacitor (LC) resonators with alternating coupling capacitance and mutual inductance as shown in Fig. 1a. The topological phase of the circuit model is determined by the relative size of intra- and inter-cell coupling between neighboring resonators, including both the capacitive and inductive contributions. Strictly speaking, this circuit model breaks chiral symmetry of the original SSH Hamiltonian Asbóth et al. (2016); Ozawa et al. (2019), which ensures the band spectrum to be symmetric with respect to $\omega=\omega_{0}$. Nevertheless, the topological protection of the edge states under perturbation in the intra- and inter-cell coupling strengths remains valid under certain conditions, and the existence of edge states still persists due to the presence of inversion symmetry within the unit cell of the circuit analog, leading to a quantized Zak phase Zak (1989). For detailed analysis of the modeling, symmetry, and robustness of the circuit topological waveguide see Apps. A and B.

The circuit model is realized using standard fabrication techniques for superconducting metamaterials discussed in Refs. Mirhosseini et al. (2018); Ferreira et al. (2020), where the coupling between sites is controlled by the physical distance between neighboring resonators. Due to the near-field nature, the coupling strength is larger (smaller) for smaller (larger) distance between resonators on a device. An example unit cell of a fabricated device in the topological phase is shown in Fig. 1b (the values of intra- and inter-cell distances are interchanged in the trivial phase). We find a good agreement between the measured transmission spectrum and a theoretical curve calculated from a LC lumped-element model of the test structures with 8 unit cells in both trivial and topological configurations (Fig. 1c,d). For the topological configuration, the observed peak in the waveguide transmission spectrum at 6.636 GHz inside the MBG signifies the associated edge state physics in our system.

The non-trivial properties of the topological waveguide can be accessed by coupling quantum emitters to the engineered structure. To this end, we prepare Device I consisting of a topological waveguide in the trivial phase with 9 unit cells, whose boundary is tapered with specially designed resonators before connection to external ports (see Fig. 2a). The tapering sections at both ends of the array are designed to reduce the impedance mismatch to the external ports ($Z_{0}=50\>\Omega$) at frequencies in the upper passband (UPB). This is crucial for reducing ripples in the waveguide transmission spectrum in the passbands Ferreira et al. (2020). The device contains 14 frequency-tunable transmon qubits Koch et al. (2007) coupled to every site on the 7 unit cells in the middle of the array (labeled Q${}_{i}^{\alpha}$, where $i=$1-7 and $\alpha$=A,B are the cell and sublattice indices, respectively). Properties of Device I and the tapering section are discussed in further detail in Apps. C and D, respectively.

For qubits lying within the middle bandgap, the topology of the waveguide manifests itself in the spatial profile of the resulting qubit-photon bound states. When the qubit transition frequency is inside the bandgap, the emission of a propagating photon from the qubit is forbidden due to the absence of photonic modes at the qubit resonant frequency. In this scenario, a stable bound state excitation forms, consisting of a qubit in its excited state and a waveguide photon with exponentially localized photonic envelope John and Wang (1990); Kurizki (1990). Generally, bound states with a symmetric photonic envelope emerge due to the inversion symmetry of the photonic bath with respect to the qubit location Liu and Houck (2016). In the case of the SSH photonic bath, however, a directional envelope can be realized Bello et al. (2019) for a qubit at the centre of the MBG ($\omega_{0}$), where the presence of a qubit creates a domain wall in the SSH chain and the induced photonic bound state is akin to an edge state (refer to App. E for a detailed description). For example, in the trivial phase, a qubit coupled to site A (B) acts as the last site of a topological array extended to the right (left) while the subsystem consisting of the remaining sites extended to the left (right) is interpreted as a trivial array. Mimicking the topological edge state, the induced photonic envelope of the bound state faces right (left) with photon occupation only on B (A) sites (Fig. 2b), while across the trivial boundary on the left (right) there is no photon occupation. The opposite directional character is expected in the case of the topological phase of the waveguide. The directionality reduces away from the center of the MBG, and is effectively absent inside the upper or lower bandgaps.

We experimentally probe the directionality of qubit-photon bound states by utilizing the coupling of bound states to the external ports in the finite-length waveguide of Device I (see Fig. 2c). The external coupling rate $\kappa_{\text{e},p}$ ($p=1,2$) is governed by the overlap of modes in the external port $p$ with the tail of the exponentially attenuated envelope of the bound state, and therefore serves as a useful measure to characterize the localization Biondi et al. (2014); Liu and Houck (2016); Mirhosseini et al. (2018). To find the reference frequency $\omega_{0}$ where the bound state becomes most directional, we measure the external linewidth of the bound state seen from each port as a function of qubit tuning. For Q${}_{4}^{\text{B}}$, which is located near the center of the array, we find $\kappa_{\text{e},1}$ to be much larger than $\kappa_{\text{e},2}$ at all frequencies inside MBG. At $\omega_{0}/2\pi=6.621\>\text{GHz}$, $\kappa_{\text{e},2}$ completely vanishes, indicating a directionality of the Q${}_{4}^{\text{B}}$ bound state to the left. Plotting the external coupling at this frequency to both ports against qubit index, we observe a decaying envelope on every other site, signifying the directionality of photonic bound states is correlated with the type of sublattice site a qubit is coupled to. Similar measurements when qubits are tuned to other frequencies near the edge of the MBG, or inside the upper bandgap (UBG), show the loss of directionality away from $\omega=\omega_{0}$ (App. F).

A remarkable consequence of the distinctive shape of bound states is direction-dependent photon-mediated interactions between qubits (Fig. 2d,e). Due to the site-dependent shapes of qubit-photon bound states, the interaction between qubits becomes substantial only when a qubit on sublattice A is on the left of the other qubit on sublattice B, i.e., $j>i$ for a qubit pair (Q${}_{i}^{\text{A}}$,Q${}_{j}^{\text{B}}$). From the avoided crossing experiments centered at $\omega=\omega_{0}$, we extract the qubit-qubit coupling as a function of cell displacement $i-j$. An exponential fit of the data gives the localization length of $\xi=1.7$ (in units of lattice constant), close to the estimated value from the circuit model of our system (see App. C). While theory predicts the coupling between qubits in the remaining combinations to be zero, we report that coupling of $|g_{ij}^{\text{AA,BB}}|/2\pi\lesssim 0.66\>\text{MHz}$ and $|g_{ij}^{\text{AB}}|/2\pi\lesssim 0.48\>\text{MHz}$ (for $i>j$) are observed, much smaller than the bound-state-induced coupling, e.g., $|g_{45}^{\text{AB}}|/2\pi=32.9\>\text{MHz}$. We attribute such spurious couplings to the unintended near-field interaction between qubits. Note that we find consistent coupling strength of qubit pairs dependent only on their relative displacement, not on the actual location in the array, suggesting that physics inside MBG remains intact with the introduced waveguide boundaries. In total, the avoided crossing and external linewidth experiments at $\omega=\omega_{0}$ provide strong evidence of the shape of qubit-photon bound states, compatible with the theoretical photon occupation illustrated in Fig. 2b.

In the passband regime, i.e., when the qubit frequencies lie within the upper or lower passbands, the topology of the waveguide is imprinted on cooperative interaction between qubits and the single-photon scattering response of the system. The topology of the SSH model can be visualized by plotting the complex-valued $f(k)$ for $k$ values in the first Brillouin zone (Fig. 3a). In the topological (trivial) phase, the contour of $f(k)$ encloses (excludes) the origin of the complex plane, resulting in the winding number of 1 (0) and the corresponding Zak phase of $\pi$ (0) Zak (1989). This is consistent with the earlier definition based on the sign of $\delta$. It is known that for a regular waveguide with linear dispersion, the coherent exchange interaction $J_{ij}$ and correlated decay $\Gamma_{ij}$ between qubits at positions $x_{i}$ and $x_{j}$ along the waveguide take the forms $J_{ij}\propto\sin{\varphi_{ij}}$ and $\Gamma_{ij}\propto\cos{\varphi_{ij}}$ Chang et al. (2012); Lalumière et al. (2013), where $\varphi_{ij}=k|x_{i}-x_{j}|$ is the phase length. In the case of our topological waveguide, considering a pair of qubits coupled to A/B sublattice on $i$/$j$-th unit cell, this argument additionally collects the phase $\phi(k)\equiv\arg{f(k)}$ Bello et al. (2019). This is an important difference compared to the regular waveguide case, because the zeros of equation

determine wavevectors (and corresponding frequencies) where perfect Dicke super-radiance Dicke (1954) occurs. Due to the properties of $f(k)$ introduced above, for a fixed cell-distance $\Delta n\equiv|i-j|\geq 1$ between qubits there exists exactly $\Delta n-1$ ($\Delta n$) frequency points inside the passband where perfect super-radiance occurs in the trivial (topological) phase. An example for the $\Delta n=2$ case is shown in Fig. 3b. Note that although Eq. (2) is satisfied at the band-edge frequencies $\omega_{\text{min}}$ and $\omega_{\text{max}}$ ($kd=\{0,\pi\}$), they are excluded from the above counting due to breakdown of the Born-Markov approximation that is assumed in obtaining the particular form of cooperative interaction in this picture.

To experimentally probe signatures of perfect super-radiance, we tune the frequency of a pair of qubits across the UPB of Device I while keeping the two qubits resonant with each other. We measure the waveguide transmission spectrum $S_{21}$ during this tuning, keeping track of the lineshape of the two-qubit resonance as $J_{ij}$ and $\Gamma_{ij}$ varies over the tuning. Drastic changes in the waveguide transmission spectrum occur whenever the two-qubit resonance passes through the perfectly super-radiant points, resulting in a swirl pattern in $|S_{21}|$. Such patterns arise from the disappearance of the peak in transmission associated with interference between photons scattered by imperfect super- and sub-radiant states, resembling the electromagnetically-induced transparency in a V-type atomic level structure Witthaut and Sorensen (2010). As an example, we discuss the cases with qubit pairs (Q${}_{2}^{\text{A}}$,Q${}_{4}^{\text{B}}$) and (Q${}_{2}^{\text{B}}$,Q${}_{5}^{\text{A}}$), which are shown in Fig. 3c. Each qubit pair configuration encloses a three-unit-cell section of the waveguide; however for the (Q${}_{2}^{\text{A}}$,Q${}_{4}^{\text{B}}$) pair the waveguide section is in the trivial phase, whereas for (Q${}_{2}^{\text{A}}$,Q${}_{4}^{\text{B}}$) the waveguide section is in the topological phase. Both theory and measurement indicate that the qubit pair (Q${}_{2}^{\text{A}}$,Q${}_{4}^{\text{B}}$) has exactly one perfectly super-radiant frequency point in the UPB. For the other qubit pair (Q${}_{2}^{\text{B}}$,Q${}_{5}^{\text{A}}$), with waveguide section in the topological phase, two such points occur (corresponding to $\Delta n=2$). This observation highlights the fact that while the topological phase of the bulk in the SSH model is ambiguous, a finite section of the array can still be interpreted to have a definite topological phase. Apart from the unintended ripples near the band-edges, the observed lineshapes are in good qualitative agreement with the theoretical expectation in Ref. Bello et al. (2019). Detailed description of the swirl pattern and similar measurement results for other qubit combinations with varying $\Delta n$ are reported in App. G.

Finally, to explore the physics associated with topological edge modes, we fabricated a second device, Device II, which realizes a closed quantum system with 7 unit cells in the topological phase (Fig. 4a). We denote the photonic sites in the array by ($i$,$\alpha$), where $i=$1-7 is the cell index and $\alpha=$A,B is the sublattice index. Due to reflection at the boundary, the passbands on this device appear as sets of discrete resonances. The system supports topological edge modes localized near the sites (1,A) and (7,B) at the boundary, labeled E${}_{\text{L}}$ and E${}_{\text{R}}$. The edge modes are spatially distributed with exponentially attenuated tails directed toward the bulk. In a finite system, the non-vanishing overlap between the envelopes of edge states generates a coupling which depends on the localization length $\xi$ and the system size $L$ as $G\sim e^{-L/\xi}$. In Device II, two qubits denoted Q${}_{\text{L}}$ and Q${}_{\text{R}}$ are coupled to the topological waveguide at sites (2,A) and (6,B), respectively. Each qubit has a local drive line and a flux-bias line, which are connected to room-temperature electronics for control. The qubits are dispersively coupled to readout resonators, which are loaded to a coplanar waveguide for time-domain measurement. The edge mode E${}_{\text{L}}$ (E${}_{\text{R}}$) has photon occupation on sublattice A (B), inducing interaction $g_{\text{L}}$ ($g_{\text{R}}$) with Q${}_{\text{L}}$ (Q${}_{\text{R}}$). Due to the directional properties discussed earlier, bound states arising from Q${}_{\text{L}}$ and Q${}_{\text{R}}$ have photonic envelopes facing away from each other inside the MBG, and hence have no direct coupling to each other. For additional details on Device II and qubit control, refer to App. H.

We probe the topological edge modes by utilizing the interaction with the qubits. While parking Q${}_{\text{L}}$ at frequency $f_{\text{q}}=6.835\>\text{GHz}$ inside MBG, we initialize the qubit into its excited state by applying a microwave $\pi$-pulse to the local drive line. Then, the frequency of the qubit is parametrically modulated Naik et al. (2017) such that the first-order sideband of the qubit transition frequency is nearly resonant with E${}_{\text{L}}$. After a variable duration of the frequency modulation pulse, the state of the qubit is read out. From this measurement, we find a chevron-shaped oscillation of the qubit population in time centered at modulation frequency $242.5\>\text{MHz}$ (Fig. 4b). We find the population oscillation at this modulation frequency to contain two harmonic components as shown in Fig. 4c, a general feature of a system consisting of three states with two exchange-type interactions $g_{1}$ and $g_{2}$. In such cases, three single-excitation eigenstates exist at $0$, $\pm{g}$ with respect to the bare resonant frequency of the emitters (${g}\equiv\sqrt{g_{1}^{2}+g_{2}^{2}}$), and since the only possible spacing between the eigenstates in this case is ${g}$ and $2{g}$, the dynamics of the qubit population exhibits two frequency components with a ratio of two. From fitting the Q${}_{\text{L}}$ population oscillation data in Fig. 4c, the coupling between E${}_{\text{L}}$ and E${}_{\text{R}}$ is extracted to be $G/2\pi=5.05\>\text{MHz}$. Parking Q${}_{\text{R}}$ at the bare resonant frequency $\omega_{\text{E}}/2\pi=6.601\>\text{GHz}$ of the edge modes, E${}_{\text{R}}$ strongly hybridizes with Q${}_{\text{R}}$ and is spectrally distributed at $\pm g_{\text{R}}$ with respect to the original frequency ($g_{\text{R}}/2\pi=57.3\>\text{MHz}$). As this splitting is much larger than the coupling of E${}_{\text{R}}$ to E${}_{\text{L}}$, the interaction channel E${}_{\text{L}}$$\leftrightarrow$E${}_{\text{R}}$ is effectively suppressed and the vacuum Rabi oscillation only involving Q${}_{\text{L}}$ and E${}_{\text{L}}$ is recovered (Fig. 4d) by applying the above-mentioned pulse sequence on Q${}_{\text{L}}$. A similar result was achieved by applying a simultaneous modulation pulse on Q${}_{\text{R}}$ to put its first-order sideband near-resonance with the bare edge modes (instead of parking it near resonance), which we call the double-modulation scheme. From the vacuum Rabi oscillation Q${}_{\text{L}}$$\leftrightarrow$E${}_{\text{L}}$ (Q${}_{\text{R}}$$\leftrightarrow$E${}_{\text{R}}$) using the double-modulation scheme, we find the effective qubit-edge mode coupling to be $\tilde{g}_{\text{L}}/2\pi=23.8\>\text{MHz}$ ($\tilde{g}_{\text{R}}/2\pi=22.5\>\text{MHz}$).

The half-period of vacuum Rabi oscillation corresponds to an iSWAP gate between Q${}_{\text{L}}$ and E${}_{\text{L}}$ (or Q${}_{\text{R}}$ and E${}_{\text{R}}$), which enables control over the edge modes with single-photon precision. As a demonstration of this tool, we perform remote population transfer between Q${}_{\text{L}}$ and Q${}_{\text{R}}$through the non-local coupling of topological edge modes E${}_{\text{L}}$ and E${}_{\text{R}}$. The qubit Q${}_{\text{L}}$ (Q${}_{\text{R}}$) is parked at frequency 6.829 GHz (6.835 GHz) and prepared in its excited (ground) state. The transfer protocol, consisting of three steps, is implemented as follows: i) an iSWAP gate between Q${}_{\text{L}}$ and E${}_{\text{L}}$ is applied by utilizing the vacuum Rabi oscillation during the double-modulation scheme mentioned above, ii) the frequency modulation is turned off and population is exchanged from E${}_{\text{L}}$ to E${}_{\text{R}}$ using the interaction $G$, iii) another iSWAP gate between Q${}_{\text{R}}$ and E${}_{\text{R}}$ is applied to map the population from E${}_{\text{R}}$ to Q${}_{\text{R}}$. The population of both qubits at any time within the transfer process is measured using multiplexed readout Chen et al. (2012) (Fig. 4e). We find the final population in Q${}_{\text{R}}$ after the transfer process to be 87 %. Numerical simulations suggest that (App. H) the infidelity in preparing the initial excited state accounts for 1.6 % of the population decrease, the leakage to the unintended edge mode due to ever-present interaction $G$ contributes 4.9 %, and the remaining 6.5 % is ascribed to the short coherence time of qubits away from the flux-insensitive point [$T_{2}^{*}=344\>(539)$ ns for Q${}_{\text{L}}$ (Q${}_{\text{R}}$) at working point].

We expect that a moderate improvement on the demonstrated population transfer protocol could be achieved by careful enhancement of the excited state preparation and the iSWAP gates, i.e. optimizing the shapes of the control pulses Khaneja et al. (2005); Motzoi et al. (2009); Didier et al. (2018); Hong et al. (2020). The coherence-limited infidelity can be mitigated by utilizing a less flux-sensitive qubit design Sete et al. (2017); Hutchings et al. (2017) or by reducing the generic noise level of the experimental setup Ott (2009). Further, incorporating tunable couplers Chen et al. (2014) into the existing metamaterial architecture to control the localization length of edge states in situ will fully address the population leakage into unintended interaction channels, and more importantly, enable robust quantum state transfer over long distances Lang and Büchler (2017). Together with many-body protection to enhance the robustness of topological states de Léséleuc et al. (2019), building blocks of quantum communication Kimble (2008) under topological protection are also conceivable.

Looking forward, we envision several research directions to be explored beyond the work presented here. First, the topology-dependent photon scattering in photonic bands that is imprinted in the cooperative interaction of qubits can lead to new ways of measuring topological invariants in photonic systems Tran et al. (2017). The directional and long-range photon-mediated interactions between qubits demonstrated in our work also opens avenues to synthesize non-trivial quantum many-body states of qubits, such as the double Néel state Bello et al. (2019). Even without technical advances in fabrication Dunsworth et al. (2018); Rosenberg et al. (2017); Foxen et al. (2017), a natural scale-up of the current system will allow for the construction of moderate to large-scale quantum many-body systems. Specifically, due to the on-chip wiring efficiency of a linear waveguide QED architecture, with realistic refinements involving placement of local control lines on qubits and compact readout resonators coupled to the tapered passband (intrinsically acting as Purcell filters Jeffrey et al. (2014)), we expect that a fully controlled quantum many-body system consisting of 100 qubits is realizable in the near future. In such systems, protocols for preparing and stabilizing de Léséleuc et al. (2019); Ma et al. (2017, 2019) quantum many-body states could be utilized and tested. Additionally, the flexibility of superconducting metamaterial architectures Mirhosseini et al. (2018); Ferreira et al. (2020) can be further exploited to realize other novel types of topological photonic baths Lodahl et al. (2017); Bello et al. (2019); García-Elcano et al. (2019). While the present work was limited to a one-dimensional system, the state-of-the-art technologies in superconducting quantum circuits Krantz et al. (2019) utilizing flip-chip methods Rosenberg et al. (2017); Foxen et al. (2017) will enable integration of qubits into two-dimensional metamaterial surfaces. It also remains to be explored whether topological models with broken time-reversal symmetry, an actively pursued approach in systems consisting of arrays of three-dimensional microwave cavities Anderson et al. (2016); Owens et al. (2018), could be realized in compact chip-based architectures. Altogether, our work sheds light on opportunities in superconducting circuits to explore quantum many-body physics originating from novel types of photon-mediated interactions in topological waveguide QED, and paves the way for creating synthetic quantum matter and performing quantum simulation Blatt and Roos (2012); Gross and Bloch (2017); Lewis-Swan et al. (2019); Georgescu et al. (2014); Carusotto et al. (2020).