Main

The collective interaction of atoms in the presence of a radiation field has been studied since the early days of quantum physics. As first studied by Dicke2, the interaction of resonant atoms in such systems results in the formation of super- and sub-radiant states in the spontaneous emission. While Dicke’s central insight—that atoms interact coherently even through an open environment—was used to understand the radiation properties of an idealized, point-like atomic gas, the dynamical properties of ordered, distant atoms coupled to open environments also exhibit novel physics. In their most essential form, such systems can be studied within the canonical waveguide quantum electrodynamics (QED) model14: atoms coupled to a one-dimensional (1D) continuum realized by an optical fibre or a microwave waveguide15,16. Within this model, a diverse and rich set of phenomena await experimental study. For instance, one can synthesize an artificial cavity QED system9, distill exotic many-excitation dark states with fermionic spatial correlations10, and use classical light sources to generate entangled and quantum many-body states of light11,12,13.

A central technical hurdle common to these research avenues—reaching the so-called strong coupling regime, in which atom–atom interactions dominate decay—is experimentally difficult, especially in waveguide QED, because while the waveguide facilitates infinite-range interactions between atoms17,18, it also provides a dissipative channel19. Decoherence through this and other sources destroys the fragile many-body states of the system, which has limited the experimental state of the art to spectroscopic probes of waveguide-mediated interactions20,21,22. However, by using collective dark states, where the precise positioning of atoms protects them from substantial waveguide emission decoherence, the strong coupling limit is predicted to be within reach9. Additionally, if the timescale of single-atom emission into the waveguide is long enough to permit measurement and manipulation of the system, the coherent dynamics can be driven and probed at the single-atom level. Here we overcome these hurdles with a waveguide QED system consisting of transmon qubits coupled to a common microwave waveguide, strengthening opportunities for a range of waveguide QED physics.

As a demonstration of this tool, we construct such an emergent cavity QED system and probe its linear and nonlinear dynamics. The system features an ancillary probe qubit and a cavity-like mode formed by the dark state of two single-qubit mirrors. Using waveguide transmission and individual addressing of the probe qubit, we observe spectroscopic and time-domain signatures of the collective dynamics of the qubit array, including vacuum Rabi oscillations between the probe qubit and the cavity-like mode. These oscillations provide direct evidence of strong coupling between these modes as well as a natural method of efficiently creating and measuring dark states that are inaccessible through the waveguide. Unlike traditional cavity QED, our cavity-like mode is itself quantum nonlinear, as we show by characterizing the two-excitation dynamics of the array.

The collective evolution of an array of resonant qubits coupled to a 1D waveguide can be formally described by a master equation9,23 of the form \(\mathop{\hat{\rho }}\limits^{.}=-i/\hslash [{\hat{H}}_{{\rm{e}}{\rm{f}}{\rm{f}}},\hat{\rho }]+\sum _{m,n}{\Gamma }_{m,n}{\hat{\sigma }}_{{\rm{g}}{\rm{e}}}^{m}\hat{\rho }{\hat{\sigma }}_{{\rm{e}}{\rm{g}}}^{n}\), where \({\hat{\sigma }}_{{\rm{g}}{\rm{e}}}^{m}=|{g}_{m}\rangle \langle {e}_{m}|\), \(|g\rangle \,{\rm{a}}{\rm{n}}{\rm{d}}\,\langle e|\) are a qubit’s ground and excited states, respectively, and m and n represent the indices of the qubit array. Within the Born–Markov approximation, the effective Hamiltonian can be written in the interaction picture as

$${ {\hat{H}} }_{{\rm{eff}}}=\hbar \sum _{m,n}\left({J}_{m,n}-i\frac{{\Gamma }_{m,n}}{2}\right){\hat{\sigma }}_{{\rm{eg}}}^{m}{\hat{\sigma }}_{{\rm{ge}}}^{n}$$
(1)

where ħ = h/2π is the reduced Planck constant. Figure 1a depicts the waveguide QED system considered in this work. The system consists of an array of N qubits separated by distance d = λ0/2 and a separate probe qubit centred in the middle of the array with one-dimensional waveguide decay rate Γ1D,p, and where λ0 = c/f0 is the wavelength of the field in the waveguide at the transition frequency of the qubits f0. In this configuration, the effective Hamiltonian can be simplified in the single-excitation manifold to

$${\hat{H}}_{{\rm{e}}{\rm{f}}{\rm{f}}}=-\frac{i\hslash N{\Gamma }_{1{\rm{D}}}}{2}{\hat{S}}_{{\rm{B}}}^{\dagger }{\hat{S}}_{{\rm{B}}}-\frac{i\hslash {\Gamma }_{1{\rm{D}},{\rm{p}}}}{2}{\hat{\sigma }}_{{\rm{e}}{\rm{e}}}^{({\rm{p}})}+\hslash J({\hat{\sigma }}_{{\rm{g}}{\rm{e}}}^{({\rm{p}})}{\hat{S}}_{{\rm{D}}}^{\dagger }+{\rm{H}}.{\rm{c}}.)$$
(2)

where \({\hat{S}}_{{\rm{B}}},\,{\hat{S}}_{{\rm{D}}}=(1/\sqrt{N})\sum _{m > 0}({\hat{\sigma }}_{{\rm{g}}{\rm{e}}}^{m}\mp {\hat{\sigma }}_{{\rm{g}}{\rm{e}}}^{-m}){(-1)}^{m}\) are the lowering operators of the bright collective state (B) and the fully symmetric dark collective state (D) of the qubit array, as shown in Fig. 1a, and where m > 0 and m < 0 denote qubits to the right and left of the probe qubit, respectively). As shown by the last term in the Hamiltonian, the probe qubit is coupled to this dark state at a cooperatively enhanced rate \(2J=\sqrt{N}\sqrt{{\Gamma }_{1{\rm{D}}}{\Gamma }_{1{\rm{D}},{\rm{p}}}}\). (H.c. is the Hermitian conjugate.) The bright state super-radiantly emits into the waveguide at a rate of 1D. The collective dark state has no coupling to the waveguide, and a decoherence rate \({\Gamma }_{{\rm{D}}}^{{\rm{^{\prime} }}}\) that is set by parasitic damping and dephasing not captured in the simple waveguide QED model (see Supplementary Notes 2 and 3). In addition to the bright and dark collective states described above, there exist an additional N − 2 collective states of the qubit array with no coupling to either the probe qubit or the waveguide9.

Fig. 1: Waveguide QED setup.
figure 1

a, The top schematic shows the cavity configuration of the waveguide QED system consisting of an array of N mirror qubits (N = 2 shown; green) coupled to the waveguide with an inter-qubit separation of λ0/2, with a probe qubit (red) at the centre of the mirror array. The middle schematic shows the analogous cavity QED system with correspondence to waveguide parameters. The bottom panel shows the energy-level diagram of the system of three qubits (two mirror, one probe). The mirror dark state |D〉 is coupled to the excited state of the probe qubit |ep at a cooperatively enhanced rate of \(2J=\sqrt{2{\Gamma }_{1{\rm{D}}}{\Gamma }_{1{\rm{D}},{\rm{p}}}}\). The bright state |B〉 is decoupled from the probe qubit. b, Optical image of the fabricated waveguide QED circuit . Tunable transmon qubits interact via microwave photons in a superconducting CPW (false-colour orange trace). The CPW is used for externally exciting the system and is terminated in a 50-Ω load. The insets show scanning electron microscope images of the different qubit designs used in our experiment. The probe qubit, designed to have Γ1D,p/2π = 1 MHz, is accessible via a separate CPW (XY4; false-colour blue trace) for state preparation, and is also coupled to a compact microwave resonator (R4; false-colour cyan) for dispersive readout. The mirror qubits come in two types: type I, with Γ1D/2π = 20 MHz and type II, with Γ1D/2π = 100 MHz. c, Waveguide transmission spectrum across individual qubit resonances. The top panel shows the probe qubit (Q4); the bottom panel shows type I (Q6, green curve) and type II (Q1, blue curve) mirror qubits. The inset shows a zoomed-in view of the centre of the curves with the same axes. From a Lorentzian lineshape fit of the measured waveguide transmission spectra we infer Purcell factors of P1D = 11 for the probe qubit and P1D = 98 (219) for the type I (type II) mirror qubit.

The subsystem consisting of a coupled probe qubit and the symmetric dark state of the mirror qubit array can be described as an analogue to a cavity QED system9. In this depiction, the probe qubit plays the part of a two-level atom and the dark state mimics a high-finesse cavity, with the qubits in the λ0/2-spaced array acting as atomic mirrors (see Fig. 1a). In general, provided that the fraction of excited array qubits remains small as N increases, \({ {\hat{S}} }_{{\rm{D}}}\) stays nearly bosonic and the analogy to the Jaynes–Cummings model remains valid. By mapping the waveguide parameters to those of a cavity QED system, the cooperativity between probe qubit and atomic cavity can be written as \({\mathscr{C}}={(2J)}^{2}/({\Gamma }_{1{\rm{D}},{\rm{p}}}+{\Gamma }_{{\rm{p}}}^{{\rm{^{\prime} }}}){\Gamma }_{{\rm{D}}}^{{\rm{^{\prime} }}}\approx N{P}_{1{\rm{D}}}\)   . Here \({P}_{1{\rm{D}}}={\Gamma }_{1{\rm{D}}}/{\Gamma }_{{\rm{D}}}^{{\rm{^{\prime} }}}\)is the single-qubit Purcell factor, which quantifies the ratio of the waveguide emission rate to the parasitic damping and dephasing rates. Attaining \({\mathscr{C}} > 1\) is a prerequisite for observing coherent quantum effects. Referring to the energy level diagram of Fig. 1a, by sufficiently reducing the waveguide coupling rate of the probe qubit one can also realize a situation in which \(J > max\{({\Gamma }_{1{\rm{D}},{\rm{p}}}+{\Gamma }_{{\rm{p}}}^{{\rm{^{\prime} }}}),\,{\Gamma }_{{\rm{D}}}^{{\rm{^{\prime} }}}\}\)  , corresponding to the strong coupling regime of cavity QED between the excited state of the probe qubit (|ep|G〉) and a single photon in the atomic cavity (|gp|D〉) (see Supplementary Note 2). This mapping of a waveguide QED system onto a cavity QED analogue therefore allows us to use cavity QED techniques to efficiently probe the dark states of the qubit array with single-photon precision.

The fabricated superconducting circuit used to realize the waveguide QED system is shown in Fig. 1b. The circuit consists of seven transmon qubits (Qj, where j = 1–7), all of which are side-coupled to the same coplanar waveguide (CPW). Each qubit’s transition frequency is tunable via an external flux bias port (Z1–Z7). We use the top-centre qubit in the circuit (Q4) as a probe qubit. This qubit can be independently excited via a weakly coupled CPW drive line (XY4), and is coupled to a lumped-element microwave cavity (R4) to enable dispersive readout of its state. The other six qubits are mirror qubits. The mirror qubits come in two different types (I and II), which are designed with different waveguide coupling rates (Γ1D,I/2π = 20 MHz and Γ1D,II/2π = 100 MHz) in order to provide access to a range of Purcell factors. Type I mirror qubits also lie in pairs across the CPW and have large (about 50 MHz) direct coupling. We characterize the waveguide and parasitic coupling rates of each individual qubit by measuring the phase and amplitude of microwave transmission through the waveguide (see Fig. 1c)24. Measurements are performed in a dilution refrigerator at a base temperature of 8 mK (see Methods). For a sufficiently weak coherent drive the effects of qubit saturation can be neglected and the on-resonance extinction of the coherent waveguide tone relates to a lower bound on the Purcell factor of the individual qubit. Any residual waveguide thermal photons, however, can result in weak saturation of the qubit and a reduction in the on-resonance extinction. We find an on-resonance intensity transmittance as low as 2 × 10−5 for the type II mirror qubits, corresponding to an upper bound on the CPW mode temperature of 43 mK and a lower bound for the Purcell factor of 200. Further details of the design, fabrication and measured parameters of the probe qubit and each mirror qubit are provided in the Methods.

The transmission through the waveguide, in the presence of the probe qubit, can also be used to measure spectroscopic signatures of the collective dark state of the qubit array. As an example of this, we use a single pair of type I mirror qubits (Q2 and Q6), which we tune to a frequency where their separation along the waveguide axis is d = λ0/2. The remaining qubits on the CPW are decoupled from the waveguide input by tuning their frequency away from the measurement point. Figure 2a shows the waveguide transmission spectrum for a weak coherent tone in which a broad resonance dip is evident, corresponding to the bright state of the mirror qubit pair. We find a bright-state waveguide coupling rate of Γ1D,B ≈ 2Γ1D = 2π × 26.8 MHz by fitting a Lorentzian lineshape to the spectrum. The dark state of the mirror qubits is not observable in this waveguide spectrum, but it becomes observable when measuring the waveguide transmission with the probe qubit tuned to resonance with the mirror qubits (see Fig. 2b). In addition to the broad response from the bright state, in this case two spectral peaks appear near the centre of the bright-state resonance (Fig. 2c). This pair of highly non-Lorentzian spectral features result from the Fano interference between the broad bright state and the hybridized polariton resonances formed between the dark state of the mirror qubits (the atomic cavity photon) and the probe qubit. This hybridized probe qubit and the atomic cavity eigenstates can be more clearly observed by measuring the transmission between the probe qubit drive line (XY4) and the output port of the waveguide (see Fig. 2d). As the XY4 line does not couple to the bright state owing to the symmetry of its positioning along the waveguide, we observe two well resolved resonances (Fig. 2e) with mode splitting 2J/2π ≈ 6 MHz, when the probe qubit is nearly resonant with the dark state. Observation of vacuum Rabi splitting in the hybridized atomic cavity-probe qubit polariton spectrum signifies operation in the strong coupling regime.

Fig. 2: Vacuum Rabi splitting.
figure 2

a, Transmission through the waveguide for two type I mirror qubits (M1; Q2 and Q6) on resonance, with the remaining qubits tuned away from the measurement frequency range. b, Transmission through the waveguide as a function of the flux bias tuning voltage of the probe qubit. c, Waveguide transmission spectrum for the probe qubit (P; Q4) and the mirror qubits tuned to resonance. d, Transmission spectrum measured between the probe qubit drive line XY4 and the waveguide output as a function of flux bias tuning of the probe qubit. e, XY4-to-waveguide transmission spectrum for the three qubits tuned to resonance. Transmittance here has been scaled. The dashed red lines in d and solid black line in e are the predictions of a numerical model with experimentally measured qubit parameters. The prediction in e includes slight power broadening effects.

To further investigate the signatures of strong coupling we perform time-domain measurements in which we prepare the system in the initial state |gp|G〉 → |ep|G〉 using a 10-ns microwave π pulse applied at the XY4 drive line. Following excitation of the probe qubit we use a fast (5 ns) flux bias pulse to tune the probe qubit into resonance with the collective dark state of the mirror qubits (the atomic cavity) for a desired interaction time τ. Upon returning to its initial frequency after the flux bias pulse, the excited-state population of the probe qubit is measured via the dispersively coupled readout resonator. In Fig. 3a we show a timing diagram and plot three measured curves of the probe qubit’s population dynamics versus τ. The top red curve corresponds to the measured probe qubit’s free decay, where the probe qubit is shifted to a detuned frequency fp0 to eliminate mirror qubit interactions. From an exponential fit to the decay curve we find a decay rate of 1/T1 ≈ 2π × 1.19 MHz, in agreement with the result from waveguide spectroscopy at fp0. In the middle green and bottom blue curves we plot the measured probe qubit’s population dynamics while interacting with an atomic cavity formed from type I and type II mirror qubit pairs, respectively. In both cases the initially prepared state |ep|G〉 undergoes vacuum Rabi oscillations with the dark state of the mirror qubits |gp|D〉. Along with the measured data we plot a theoretical model where the waveguide coupling, parasitic damping, and dephasing rate parameters of the probe qubit and dark state are taken from independent measurements, and the detuning between probe qubit and dark state is left as a free parameter (see Supplementary Note 2). From the excellent agreement between measurement and model we infer an interaction rate of 2J/2π = 5.64 MHz (13.0 MHz) and a cooperativity of \({\mathscr{C}}=94\,(172)\) (172) for the type I (type II) mirror system. For both mirror types we find that the system is well within the strong coupling regime \(J\gg max\{({\Gamma }_{1{\rm{D}},{\rm{p}}}+{\Gamma }_{{\rm{p}}}^{{\rm{^{\prime} }}}),{\Gamma }_{{\rm{D}}}^{{\rm{^{\prime} }}}\}\), with the photon-mediated interactions dominating the decay and dephasing rates by roughly two orders of magnitude.

Fig. 3: Vacuum Rabi oscillations.
figure 3

a, Measured population of the excited state of the probe qubit for three different scenarios. The top, red curve represents the probe qubit (P) tuned to fp0 = 6.55 GHz, with all mirror qubits tuned away, corresponding to free population decay. The middle, green curve represents the probe qubit tuned into resonance with a pair of type I mirror qubits (M1; Q2 and Q6) at frequency fM1 = 6.6 GHz corresponding to dI = λ0/2. The bottom, blue curve represents the probe qubit tuned to resonance with type II mirror qubits (M2; Q1 and Q7) at frequency fM2 = 5.826 GHz corresponding to dII = λ0/2. Inset, the sequence of pulses applied during the measurement. T1 and TSWAP are the population decay time and half of the oscillation time period for the spontaneous decay curve and the vacuum Rabi oscillations, respectively. b, Measurement of the population decay time (T1,D) of the dark state of the type I (green curve) and type II (blue curve) mirror qubits. c, Corresponding Ramsey coherence time \(\left({T}_{2,{\rm{D}}}^{\ast }\right)\) of the type I (top, green) and type II (bottom, blue) dark states. d, Waveguide transmission spectrum through the atomic cavity without (brown data points) and with (orange data points) pre-population of the cavity. Here the atomic cavity is initialized in a single photon state using an iSWAP gate acting on the probe qubit followed by detuning of the probe qubit away from resonance. In both cases the transmission measurement is performed using coherent rectangular pulses with a duration of 260 ns and a peak power of P ≈ 0.03 (ħω0Γ1D). Solid lines show theory fits from numerical modelling of the system. e, Energy level diagram of the 0 (|G〉), 1 (|D〉 and |B〉), and 2 (|E〉) excitation manifolds of the atomic cavity indicating waveguide-induced decay and excitation pathways. f, Rabi oscillation with two excitations in the system of the probe qubit and atomic cavity. The shaded region shows the first iSWAP gate in which an initial probe qubit excitation is transferred to the atomic cavity. Populating the probe qubit with an additional excitation at this point results in strong damping of subsequent Rabi oscillations due to the rapid decay of state |E〉. The dashed brown curve is the predicted result for interaction of the probe qubit with an equivalent linear cavity. In df the atomic cavity is formed from the type I mirror qubits Q2 and Q6.

The tunable interaction time in our measurement sequence also permits state transfer between the probe qubit and the dark state of the mirror qubits using an iSWAP gate. We measure the dark state’s population decay in a protocol where we excite the probe qubit and transfer the excitation into the dark state (see Fig. 3b). From an exponential fit to the data we find a dark-state decay rate of T1,D = 757 ns (274 ns) for type I (type II) mirror qubits, enhanced by approximately the cooperativity over the bright-state lifetime. In addition to the lifetime, we can measure the coherence time of the dark state with a Ramsey-like sequence (see Fig. 3c), yielding \({T}_{2,{\rm{D}}}^{\ast }\) = 435 ns (191 ns) for type I (type II) mirror qubits. The collective dark-state coherence time is slightly shorter than its population decay time, hinting at correlated sources of noise in the distantly entangled qubits forming the dark state (see discussion, Supplementary Note 3).

These experiments have so far probed the waveguide and the multi-qubit array with a single excitation, where the cavity QED analogy is helpful for understanding the response. However, this analogy is not fully suitable for understanding multi-excitation dynamics, where the quantum nonlinear response of the qubits leads to a number of interesting phenomena. To observe this, we populate the atomic cavity with a single photon via an iSWAP gate and then measure the transmission of weak coherent pulses through the waveguide. Figure 3d shows transmission through the atomic cavity formed from type I mirror qubits before and after adding a single photon. The sharp change in the transmissivity of the atomic cavity is a result of trapping in the long-lived dark state of the mirror qubits. The dark state has no transition dipole to the waveguide channel (see Fig. 3e), and thus cannot participate in absorption or emission of photons when probed via the waveguide. As a result, populating the atomic cavity with a single photon makes the cavity nearly transparent to incoming waveguide signals for the duration of the dark-state lifetime. This is analogous to the electron shelving phenomenon, which leads to suppression of resonance fluorescence in three-level atomic systems25. As a further example, we use the probe qubit to prepare the cavity in the doubly excited state via two consecutive iSWAP gates. In this case, with only two mirror qubits and the rapid decay via the bright state of the two-excitation state |E〉 of the mirror qubits (refer to Fig. 3e), the resulting probe qubit population dynamics shown in Fig. 3f have a strongly damped response \(\left({\mathscr{C}} < 1\right)\) with weak oscillations occurring at the vacuum Rabi oscillation frequency. This is in contrast to the behaviour of a linear cavity (shown as the dashed brown curve in Fig. 3f), where driving the second photon transition leads to persistent Rabi oscillations with a frequency that is \(\sqrt{2}\) times larger than the vacuum Rabi oscillations. Further analysis of the nonlinear behaviour of the atomic cavity is provided in Supplementary Note 4.

The waveguide QED superconducting circuit of Fig. 1b can also be used to investigate the spectrum of sub-radiant states that emerge when N > 2 and there is direct interaction between mirror qubits. This situation is realized by taking advantage of the capacitive coupling between co-localized pairs of type I qubits (Q2 and Q3, or Q5 and Q6). Although in an idealized 1D waveguide model there is no cooperative interaction term between qubits with zero separation along the waveguide, we observe a strong coupling (with the measured interaction rate, g/2π = 46 MHz) between the co-localized pair of mirror qubits Q2 and Q3, as shown in Fig. 4a. This coupling results from near-field components of the electromagnetic field that are excluded in the simple waveguide model. The non-degenerate hybridized eigenstates of the qubit pair effectively behave as a compound atomic mirror. The emission rate of each compound mirror to the waveguide can be adjusted by setting the detuning, Δ, between the pair. As illustrated in Fig. 4b, resonantly aligning the compound atomic mirrors on both ends of the waveguide results in a hierarchy of bright and dark states involving both near-field and waveguide-mediated cooperative coupling. Probing the system with a weak continuous tone via the waveguide, we identify the two super-radiant combinations of the compound atomic mirrors (Fig. 4c). Similar to the case of a two-qubit cavity, we identified the collective dark states via the probe qubit. As evidenced by the measured Rabi oscillations shown in Fig. 4d, the combination of direct and waveguide-mediated interactions of mirror qubits in this geometry results in the emergence of a pair of collective entangled states of the four qubits acting as strongly coupled atomic cavities with a frequency separation \(\sqrt{4{g}^{2}+{\Delta }^{2}}\).

Fig. 4: Compound atomic mirrors, N = 4.
figure 4

a, Avoided mode crossing of a pair of type I mirror qubits positioned on opposite sides of the CPW. Near the degeneracy point, the qubits form a pair of compound eigenstates consisting of symmetric (|S〉) and antisymmetric (|A〉) states with respect to the waveguide axis. b, Measured transmission through the waveguide with the pair of compound atomic mirrors aligned in frequency. The two broad resonances correspond to super-radiant states |B1〉 and |B2〉 as indicated. As we tune the frequency of the probe qubit, we observe the (avoided-crossing-like) signature of the interaction of the probe qubit with each dark state. c, Illustration of the single-excitation manifold of the collective states of N = 4 mirror qubits forming a pair of compound atomic cavities. The bright (super-radiant) and dark (sub-radiant) states can be identified by comparing the symmetry of the compound qubit states with the resonant radiation field pattern in the waveguide. d, Probe qubit measurements of the two dark states, |D1〉 and |D2〉. In these measurements the frequency of each dark state is shifted to ensure λ0/2 separation between the two compound atomic mirrors.

In conclusion, we have realized a synthetic cavity QED system in which to observe and drive the coherent dynamics that emerge from correlated dissipation in an open waveguide, paving the way for several research avenues beyond the work presented here. Our current work has reached single-qubit Purcell factors exceeding 200, which is an order of magnitude larger than the experimental state of the art in planar superconducting quantum circuits and on par with values achieved in less scalable three-dimensional architectures26, but further improvement is theoretically possible. With better thermalization of the waveguide27 and coherence times in line with the best planar superconducting qubits28, Purcell factors in excess of 104 should be achievable. In this regime, with an already realized system size of N = 4, a universal set of quantum gates with fidelity above 0.99 could theoretically be realized by encoding information in decoherence-free subspaces29. Even without improved Purcell factors, the control demonstrated here over the sub-radiant states of an atomic chain enables studies of the formation of fermionic correlations between excitations and the power-law decay dynamics associated with a critical open system in a modestly sized array10 (N = 10). Further, the demonstrated ability to measure the population decay time and coherence time for the entangled states of multiple distant qubits provides a valuable experimental tool with which to examine the sources of correlated decoherence in circuit QED. Finally, reducing the frequency disorder of transmon qubits beyond the values measured in our system (δf ≈ 60 MHz) and using a slow-light metamaterial waveguide30 would allow chip-scale waveguide QED experiments with a much larger number of fixed-frequency qubits, in the range N = 10–100, where the full extent of the many-body dynamics of large quantum spin chains can be studied11,12,13.

Methods

Fabrication

The device used in this work is fabricated on a 1 cm × 1 cm high-resistivity (10 kΩ cm) silicon substrate. The ground plane, waveguides, resonator and qubit capacitors are patterned by electron-beam lithography followed by electron-beam evaporation of 120-nm aluminium (Al) at a rate of 1 nm s−1. A liftoff process is performed in N-methyl-2-pyrrolidone at 80 °C for 1.5 h. The Josephson junctions are fabricated using double-angle electron-beam evaporation on suspended Dolan bridges, following techniques similar to those in ref. 31. The airbridges are patterned using greyscale electron-beam lithography and developed in a mixture of isopropyl alcohol and deionized water32. After 2 h of resist reflow at 105 °C, electron-beam evaporation of 140-nm Al is performed at a rate of 1 nm s−1 following 5 min of Ar ion milling. Liftoff is done in the same fashion as before.

Qubits

We design and fabricate transmon qubits in three different variants for the experiment (see Extended Data Fig. 1a, b): type I mirror qubits (Q2, Q3, Q5 and Q6), type II mirror qubits (Q1 and Q7), and the probe qubit (Q4). The qubit frequency tuning range, waveguide coupling rate (Γ1D), and parasitic decoherence rate (Γ') can be extracted from waveguide spectroscopy measurements of the individual qubits. The values for all the qubits inferred in this manner are listed in Extended Data Table 1. Note that Γ' is defined as damping and dephasing from channels other than the waveguide at zero temperature. The inferred value of Γ' from waveguide spectroscopy measurements is consistent with this definition in the zero temperature waveguide limit (effects of finite waveguide temperature are considered in Supplementary Note 1). The standard deviation in maximum frequencies of the four identically designed qubits (type I) is found to be 61 MHz, equivalent to about 1% qubit frequency disorder in our fabrication process. Asymmetric Josephson junctions are used in the superconducting quantum interference device (SQUID) loops of all qubits (Extended Data Fig. 1c) to reduce dephasing from flux noise, limiting the tuning range of qubits to around 1.3 GHz. For Q4, the Josephson energies of the junctions are determined to be EJ1/h = 18.4 GHz and EJ2/h = 3.5 GHz, giving a junction asymmetry of \(d\equiv \frac{{E}_{{\rm{J}}1}-{E}_{{\rm{J}}2}}{{E}_{{\rm{J}}1}+{E}_{{\rm{J}}2}}=0.68\). The anharmonicity is measured to be η/2π = −272 MHz and EJ/EC = 81 at maximum frequency for Q4.

Readout

We fabricate a lumped-element resonator (shown in Extended Data Fig. 1b) to perform dispersive readout of the state of the central probe qubit (Q4). The lumped-element resonator consists of a capacitive claw and an inductive meander with a pitch of about 1 μm, effectively acting as a quarter-wave resonator. The bare frequency of resonator and coupling to probe qubit are determined to be fr = 5.156 GHz and g/2π = 116 MHz, respectively, giving a dispersive frequency shift of χ/2π = −2.05 MHz for Q4 at maximum frequency. The resonator is loaded to the common waveguide in the experiment, and its internal and external quality factors are measured to be Qi = 1.3 × 105 and Qe = 980 below the single-photon level. It should be noted that the resonator-induced Purcell decay rate of Q4 is \({\Gamma }_{1}^{{\rm{Purcell}}}/2{\rm{\pi }}\approx 70\;{\rm{kHz}}\), which is small compared to the decay rate into the waveguide Γ1D,p/2π ≈ 1 MHz. The compact footprint of the lumped-element resonator is critical for minimizing the distributed coupling effects that may arise from interference between direct qubit decay to the waveguide and the Purcell decay of the qubit via the resonator path.

Suppression of spurious modes

In our experiment we use a coplanar transmission line for realizing a microwave waveguide. In addition to the fundamental propagating mode of the waveguide, which has even symmetry with respect to the waveguide axis, these structures also support a set of modes with odd symmetry, known as slotline modes. The propagation of a slotline mode can be completely suppressed in a waveguide with perfectly symmetric boundary conditions. However, in practice perfect symmetry cannot be maintained over the full waveguide length, which unavoidably leads to the presence of the slotline modes as a spurious loss channel for the qubits. Crossovers connecting ground planes across the waveguide are known to suppress propagation of slotline modes, and to this effect Al airbridges have been used in superconducting circuits with negligible impedance mismatch for the desired CPW mode33.

In this experiment, we place airbridges (Extended Data Fig. 1d) along the waveguide and control lines at specific distances set with the following considerations. Airbridges create a reflecting boundary for slotline modes, and if placed at a distance d a discrete resonance corresponding to a wavelength 2d is formed. Therefore, placing airbridges over distances less than λ/4 apart, where λ is the wavelength of the mode resonant with the qubits, pushes the slotline resonances of the waveguide sections between the airbridges to substantially higher frequencies. In this situation, the dissipation rate of qubits via the spurious channel is substantially suppressed by the off-resonance Purcell factor \({\Gamma }_{1}^{{\rm{P}}{\rm{u}}{\rm{r}}{\rm{c}}{\rm{e}}{\rm{l}}{\rm{l}}}\approx {(g/\Delta )}^{2}\kappa \), where Δ denotes the detuning between the qubit transition frequency and the frequency of the odd mode in the waveguide section between the two airbridges, and where the parameters g and κ are the interaction rate of the qubit and the decay rate of the slotline cavity modes, respectively. In addition, we place the airbridges before and after bends in the waveguide to ensure that the fundamental waveguide mode is not converted to the slotline mode upon propagation34.

Crosstalk in flux biasing

We tune the frequency of each qubit by supplying a bias current to individual flux control lines (Z lines), which control the magnetic flux in the qubit’s SQUID loop. In our system, the Z lines are attached to external wires in two forms with different configurations, which allows the qubit frequency to be tuned in ‘slow’ and ‘fast’ timescales (See Extended Data Fig. 2). The bias currents were generated via independent bias voltages generated by seven arbitrary waveform generator channels, allowing for simultaneous tuning of all qubits. In practice, independent frequency tuning of each qubit needs to be accompanied by small changes in the flux bias of the qubits in the near physical vicinity of the qubit of interest, owing to crosstalk between adjacent Z control lines.

In this experiment, we characterized the crosstalk between bias voltage channels of the qubits in the following way. First, we tuned the qubits not in use to frequencies more than 800 MHz away from the working frequency (which is set as either 5.83 GHz or 6.6 GHz). These qubits were controlled by fixed biases such that their frequencies, even in the presence of crosstalk from other qubits, remained far enough from the working frequency and hence were not considered for the rest of the analysis. Second, we tuned the remaining in-use qubits to relevant frequencies within 100 MHz of the working frequency and recorded the array of biases v0 and frequencies f0 of all qubits. Third, we varied the bias on only a single (jth) qubit and linearly interpolated the change in frequency (fi) of the other (ith) qubits with respect to the bias voltage vj on the jth qubit, finding the crosstalk matrix component \({M}_{ij}={({\rm{\partial }}{f}_{i}/{\rm{\partial }}{v}_{j})}_{{\boldsymbol{v}}={{\boldsymbol{v}}}_{0}}\). Repeating this step, we obtained the following (approximately linearized) relation between the frequencies f and bias voltages v of the qubits:

$${\boldsymbol{f}}\approx {{\boldsymbol{f}}}_{0}+M({\boldsymbol{v}}-{{\boldsymbol{v}}}_{0})$$

Finally, we took the inverse of the above relation to find the bias voltages v that are required for tuning the qubits to frequencies f:

$${\boldsymbol{v}}\approx {{\boldsymbol{v}}}_{0}+{M}^{-1}({\boldsymbol{f}}-{{\boldsymbol{f}}}_{0}).$$

An example of such a crosstalk matrix between Q2, Q4 and Q6 near f0 = (6.6, 6.6, 6.6) GHz used in the experiment is given by

$$M=(\begin{array}{ccc}0.2683 & -0.0245 & -0.0033\\ -0.0141 & -0.5310 & 0.0170\\ 0.0016 & 0.0245 & 0.4933\end{array}){{\rm{G}}{\rm{H}}{\rm{z}}{\rm{V}}}^{-1}$$

This indicates that the crosstalk level between Q4 and either Q2 or Q6 is about 5%, while that between Q2 and Q6 is less than 1%. We repeated similar steps for other configurations in the experiment.

Measurement setup

Extended Data Fig. 2 illustrates the outline of the measurement chain in our dilution refrigerator. The sample was enclosed in a magnetic shield that was mounted at the mixing chamber. We have outlined four different types of input lines used in our experiment. Input lines to the waveguide and XY4 went through a direct-current block at room temperature and then were attenuated by 20 dB at the 4 K stage, followed by additional 40 dB of attenuation at the mixing chamber. The fast flux tuning lines (Z3, Z4) were attenuated by 20 dB and were filtered with a low-pass filter with corner frequency at 225 MHz to minimize thermal noise photons while maintaining short rise and fall time of pulses for fast flux control. The slow flux tuning lines (Z1, Z2, Z5, Z6, Z7) are filtered by an additional low-pass filter with 64 kHz corner frequency at the 4 K stage to further suppress noise photons. In addition, the waveguide signal output path contained a high electron mobility transistor amplifier at the 4 K plate. Three circulators were placed between the high electron mobility transistor and the sample to ensure (>70 dB) isolation of the sample from the amplifier noise. In addition, a series of low-pass and band-pass filters on the output line suppressed noise sources outside the measurement spectrum.

A thin-film ‘cold attenuator’, developed by Palmer and colleagues at the University of Maryland27, was added to the measurement path in order to achieve better thermalization between the microwave coaxial line and its thermal environment. Similarly, an additional circulator was added to the waveguide measurement chain in later setups to further protect the device against thermal photons (both the attenuator and circulator are highlighted in red in the schematic in Extended Data Fig. 2). The effect of this change is discussed in Supplementary Note 1.

Dark state characterization

We characterized the collective dark state of mirror qubits with population decay time T1,D and Ramsey coherence time \({T}_{2,{\rm{D}}}^{\ast }\) using the cooperative interaction with the probe qubit. For each configuration of mirror qubits, we obtained the Rabi oscillation curve (see Figs. 3a, 4d) using a fast flux bias pulse on the probe qubit as explained in the main text. The half-period TSWAP of Rabi oscillation results in a complete transfer of probe qubit population to the collective dark state and vice versa, hence defining an iSWAP gate.

To measure the population decay time T1,D of the dark state, we excited the probe qubit with a resonant microwave π-pulse, followed by applying an iSWAP gate. This prepared the collective dark state |gp|D〉, off-resonantly decoupled from the probe qubit. After free evolution of the dark state for a variable duration τ, another iSWAP gate was applied to transfer the remaining dark state population back to the probe qubit. Finally, we measured the state of the probe qubit and performed an exponential fitting to the resulting decay curve.

Likewise, we measured the Ramsey coherence time \({T}_{2,{\rm{D}}}^{\ast }\) of the dark state as follows. First, we excited the probe qubit to a superposition (|g〉 + |e〉)p|G〉 of the ground and excited states by applying a detuned microwave π/2-pulse. Next, application of an iSWAP gate maps this superposition to that of the dark state |gp(|G〉 + |D〉). After a varying delay time τ, another iSWAP gate was applied, followed by a detuned π/2-pulse on the probe qubit. Measurement of the state of the probe qubit resulted in a damped oscillation curve the decay envelope of which gave the Ramsey coherence time of the dark state involved in the experiment. Note that the fast oscillation frequency in this curve is determined by the detuning of the dark state with respect to the frequency of the microwave pulses applied to the probe qubit.