Abstract
It has long been recognized that atomic emission of radiation is not an immutable property of an atom, but is instead dependent on the electromagnetic environment^{1} and, in the case of ensembles, also on the collective interactions between the atoms^{2,3,4,5,6}. In an open radiative environment, the hallmark of collective interactions is enhanced spontaneous emission—superradiance^{2}—with nondissipative dynamics largely obscured by rapid atomic decay^{7}. Here we observe the dynamical exchange of excitations between a single artificial atom and an entangled collective state of an atomic array^{9} through the precise positioning of artificial atoms realized as superconducting qubits^{8} along a onedimensional waveguide. This collective state is dark, trapping radiation and creating a cavitylike system with artificial atoms acting as resonant mirrors in the otherwise open waveguide. The emergent atom–cavity system is shown to have a large interactiontodissipation ratio (cooperativity exceeding 100), reaching the regime of strong coupling, in which coherent interactions dominate dissipative and decoherence effects. Achieving strong coupling with interacting qubits in an open waveguide provides a means of synthesizing multiphoton dark states with high efficiency and paves the way for exploiting correlated dissipation and decoherencefree subspaces of quantum emitter arrays at the manybody level^{10,11,12,13}.
Similar content being viewed by others
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 Dicke^{2}, the interaction of resonant atoms in such systems results in the formation of super and subradiant 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, pointlike 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) model^{14}: atoms coupled to a onedimensional (1D) continuum realized by an optical fibre or a microwave waveguide^{15,16}. Within this model, a diverse and rich set of phenomena await experimental study. For instance, one can synthesize an artificial cavity QED system^{9}, distill exotic manyexcitation dark states with fermionic spatial correlations^{10}, and use classical light sources to generate entangled and quantum manybody states of light^{11,12,13}.
A central technical hurdle common to these research avenues—reaching the socalled strong coupling regime, in which atom–atom interactions dominate decay—is experimentally difficult, especially in waveguide QED, because while the waveguide facilitates infiniterange interactions between atoms^{17,18}, it also provides a dissipative channel^{19}. Decoherence through this and other sources destroys the fragile manybody states of the system, which has limited the experimental state of the art to spectroscopic probes of waveguidemediated interactions^{20,21,22}. 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 reach^{9}. Additionally, if the timescale of singleatom 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 singleatom 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 cavitylike mode formed by the dark state of two singlequbit mirrors. Using waveguide transmission and individual addressing of the probe qubit, we observe spectroscopic and timedomain signatures of the collective dynamics of the qubit array, including vacuum Rabi oscillations between the probe qubit and the cavitylike 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 cavitylike mode is itself quantum nonlinear, as we show by characterizing the twoexcitation 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 equation^{9,23} 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
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 = λ_{0}/2 and a separate probe qubit centred in the middle of the array with onedimensional waveguide decay rate Γ_{1D,p}, and where λ_{0} = c/f_{0} is the wavelength of the field in the waveguide at the transition frequency of the qubits f_{0}. In this configuration, the effective Hamiltonian can be simplified in the singleexcitation manifold to
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 superradiantly emits into the waveguide at a rate of NΓ_{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 waveguide^{9}.
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 system^{9}. In this depiction, the probe qubit plays the part of a twolevel atom and the dark state mimics a highfinesse cavity, with the qubits in the λ_{0}/2spaced 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 singlequbit 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 (e〉_{p}G〉) and a single photon in the atomic cavity (g〉_{p}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 singlephoton precision.
The fabricated superconducting circuit used to realize the waveguide QED system is shown in Fig. 1b. The circuit consists of seven transmon qubits (Q_{j}, where j = 1–7), all of which are sidecoupled to the same coplanar waveguide (CPW). Each qubit’s transition frequency is tunable via an external flux bias port (Z_{1}–Z_{7}). We use the topcentre qubit in the circuit (Q_{4}) as a probe qubit. This qubit can be independently excited via a weakly coupled CPW drive line (XY_{4}), and is coupled to a lumpedelement microwave cavity (R_{4}) 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 (Γ_{1D,I}/2π = 20 MHz and Γ_{1D,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 onresonance 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 onresonance extinction. We find an onresonance 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 (Q_{2} and Q_{6}), which we tune to a frequency where their separation along the waveguide axis is d = λ_{0}/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 brightstate waveguide coupling rate of Γ_{1D,B} ≈ 2Γ_{1D} = 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 brightstate resonance (Fig. 2c). This pair of highly nonLorentzian 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 (XY_{4}) and the output port of the waveguide (see Fig. 2d). As the XY_{4} 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 cavityprobe qubit polariton spectrum signifies operation in the strong coupling regime.
To further investigate the signatures of strong coupling we perform timedomain measurements in which we prepare the system in the initial state g〉_{p}G〉 → e〉_{p}G〉 using a 10ns microwave π pulse applied at the XY_{4} 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 excitedstate 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 f_{p0} to eliminate mirror qubit interactions. From an exponential fit to the decay curve we find a decay rate of 1/T_{1} ≈ 2π × 1.19 MHz, in agreement with the result from waveguide spectroscopy at f_{p0}. 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 e〉_{p}G〉 undergoes vacuum Rabi oscillations with the dark state of the mirror qubits g〉_{p}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 photonmediated interactions dominating the decay and dephasing rates by roughly two orders of magnitude.
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 darkstate decay rate of T_{1,D} = 757 ns (274 ns) for type I (type II) mirror qubits, enhanced by approximately the cooperativity over the brightstate lifetime. In addition to the lifetime, we can measure the coherence time of the dark state with a Ramseylike sequence (see Fig. 3c), yielding \({T}_{2,{\rm{D}}}^{\ast }\) = 435 ns (191 ns) for type I (type II) mirror qubits. The collective darkstate 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 multiqubit 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 multiexcitation 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 longlived 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 darkstate lifetime. This is analogous to the electron shelving phenomenon, which leads to suppression of resonance fluorescence in threelevel atomic systems^{25}. 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 twoexcitation 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 subradiant 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 colocalized pairs of type I qubits (Q_{2} and Q_{3}, or Q_{5} and Q_{6}). 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 colocalized pair of mirror qubits Q_{2} and Q_{3}, as shown in Fig. 4a. This coupling results from nearfield components of the electromagnetic field that are excluded in the simple waveguide model. The nondegenerate 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 nearfield and waveguidemediated cooperative coupling. Probing the system with a weak continuous tone via the waveguide, we identify the two superradiant combinations of the compound atomic mirrors (Fig. 4c). Similar to the case of a twoqubit 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 waveguidemediated 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 \(\sqrt{4{g}^{2}+{\Delta }^{2}}\).
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 singlequbit 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 threedimensional architectures^{26}, but further improvement is theoretically possible. With better thermalization of the waveguide^{27} and coherence times in line with the best planar superconducting qubits^{28}, Purcell factors in excess of 10^{4} 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 decoherencefree subspaces^{29}. Even without improved Purcell factors, the control demonstrated here over the subradiant states of an atomic chain enables studies of the formation of fermionic correlations between excitations and the powerlaw decay dynamics associated with a critical open system in a modestly sized array^{10} (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 slowlight metamaterial waveguide^{30} would allow chipscale waveguide QED experiments with a much larger number of fixedfrequency qubits, in the range N = 10–100, where the full extent of the manybody dynamics of large quantum spin chains can be studied^{11,12,13}.
Methods
Fabrication
The device used in this work is fabricated on a 1 cm × 1 cm highresistivity (10 kΩ cm) silicon substrate. The ground plane, waveguides, resonator and qubit capacitors are patterned by electronbeam lithography followed by electronbeam evaporation of 120nm aluminium (Al) at a rate of 1 nm s^{−1}. A liftoff process is performed in Nmethyl2pyrrolidone at 80 °C for 1.5 h. The Josephson junctions are fabricated using doubleangle electronbeam evaporation on suspended Dolan bridges, following techniques similar to those in ref. ^{31}. The airbridges are patterned using greyscale electronbeam lithography and developed in a mixture of isopropyl alcohol and deionized water^{32}. After 2 h of resist reflow at 105 °C, electronbeam evaporation of 140nm 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 (Q_{2}, Q_{3}, Q_{5} and Q_{6}), type II mirror qubits (Q_{1} and Q_{7}), and the probe qubit (Q_{4}). The qubit frequency tuning range, waveguide coupling rate (Γ_{1D}), 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 Q_{4}, the Josephson energies of the junctions are determined to be E_{J1}/h = 18.4 GHz and E_{J2}/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 E_{J}/E_{C} = 81 at maximum frequency for Q_{4}.
Readout
We fabricate a lumpedelement resonator (shown in Extended Data Fig. 1b) to perform dispersive readout of the state of the central probe qubit (Q_{4}). The lumpedelement resonator consists of a capacitive claw and an inductive meander with a pitch of about 1 μm, effectively acting as a quarterwave resonator. The bare frequency of resonator and coupling to probe qubit are determined to be f_{r} = 5.156 GHz and g/2π = 116 MHz, respectively, giving a dispersive frequency shift of χ/2π = −2.05 MHz for Q_{4} 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 Q_{i} = 1.3 × 10^{5} and Q_{e} = 980 below the singlephoton level. It should be noted that the resonatorinduced Purcell decay rate of Q_{4} is \({\Gamma }_{1}^{{\rm{Purcell}}}/2{\rm{\pi }}\approx 70\;{\rm{kHz}}\), which is small compared to the decay rate into the waveguide Γ_{1D,p}/2π ≈ 1 MHz. The compact footprint of the lumpedelement 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 mode^{33}.
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 offresonance 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 propagation^{34}.
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 inuse qubits to relevant frequencies within 100 MHz of the working frequency and recorded the array of biases v_{0} and frequencies f_{0} of all qubits. Third, we varied the bias on only a single (jth) qubit and linearly interpolated the change in frequency (f_{i}) of the other (ith) qubits with respect to the bias voltage v_{j} 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:
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:
An example of such a crosstalk matrix between Q_{2}, Q_{4} and Q_{6} near f_{0} = (6.6, 6.6, 6.6) GHz used in the experiment is given by
This indicates that the crosstalk level between Q_{4} and either Q_{2} or Q_{6} is about 5%, while that between Q_{2} and Q_{6} 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 XY_{4} went through a directcurrent 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 (Z_{3}, Z_{4}) were attenuated by 20 dB and were filtered with a lowpass 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 (Z_{1}, Z_{2}, Z_{5}, Z_{6}, Z_{7}) are filtered by an additional lowpass 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 lowpass and bandpass filters on the output line suppressed noise sources outside the measurement spectrum.
A thinfilm ‘cold attenuator’, developed by Palmer and colleagues at the University of Maryland^{27}, 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 T_{1,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 halfperiod T_{SWAP} 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 T_{1,D} of the dark state, we excited the probe qubit with a resonant microwave πpulse, followed by applying an iSWAP gate. This prepared the collective dark state g〉_{p}D〉, offresonantly 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/2pulse. Next, application of an iSWAP gate maps this superposition to that of the dark state g〉_{p}(G〉 + D〉). After a varying delay time τ, another iSWAP gate was applied, followed by a detuned π/2pulse 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.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Purcell, E. M., Torrey, H. C. & Pound, R. V. Resonance absorption by nuclear magnetic moments in a solid. Phys. Rev. 69, 37–38 (1946).
Dicke, R. H. Coherence in spontaneous radiation processes. Phys. Rev. 93, 99–110 (1954).
Gross, M. & Haroche, S. Superradiance: an essay on the theory of collective spontaneous emission. Phys. Rep. 93, 301–396 (1982).
Osnaghi, S. et al. Coherent control of an atomic collision in a cavity. Phys. Rev. Lett. 87, 037902 (2001).
Majer, J. et al. Coupling superconducting qubits via a cavity bus. Nature 449, 443–447 (2007).
Röhlsberger, R., Schlage, K., Sahoo, B., Couet, S. & Rüffer, R. Collective Lamb shift in singlephoton superradiance. Science 328, 1248–1251 (2010).
Scully, M. O. Collective Lamb shift in single photon Dicke superradiance. Phys. Rev. Lett. 102, 143601 (2009).
Koch, J. et al. Chargeinsensitive qubit design derived from the Cooper pair box. Phys. Rev. A 76, 042319 (2007).
Chang, D. E., Jiang, L., Gorshkov, A. V. & Kimble, H. J. Cavity QED with atomic mirrors. New J. Phys. 14, 063003 (2012).
Albrecht, A. et al. Subradiant states of quantum bits coupled to a onedimensional waveguide. New J. Phys. 21, 025003 (2019).
Ramos, T., Pichler, H., Daley, A. J. & Zoller, P. Quantum spin dimers from chiral dissipation in coldatom chains. Phys. Rev. Lett. 113, 237203 (2014).
Mahmoodian, S. et al. Strongly correlated photon transport in waveguide quantum electrodynamics with weakly coupled emitters. Phys. Rev. Lett. 121, 143601 (2018).
GonzálezTudela, A., Paulisch, V., Chang, D. E., Kimble, H. J. & Cirac, J. I. Deterministic generation of arbitrary photonic states assisted by dissipation. Phys. Rev. Lett. 115, 163603 (2015).
Roy, D., Wilson, C. M. & Firstenberg, O. Strongly interacting photons in onedimensional continuum. Rev. Mod. Phys. 89, 021001 (2017).
Lodahl, P., Mahmoodian, S. & Stobbe, S. Interfacing single photons and single quantum dots with photonic nanostructures. Rev. Mod. Phys. 87, 347–400 (2015).
Gu, X., Kockum, A. F., Miranowicz, A., Liu, Y.x. & Nori, F. Microwave photonics with superconducting quantum circuits. Phys. Rep. 718/719, 1–102 (2017).
Dzsotjan, D., Sørensen, A. S. & Fleischhauer, M. Quantum emitters coupled to surface plasmons of a nanowire: a Green’s function approach. Phys. Rev. B 82, 075427 (2010).
AsenjoGarcia, A., Hood, J. D., Chang, D. E. & Kimble, H. J. Atomlight interactions in quasionedimensional nanostructures: a Green’sfunction perspective. Phys. Rev. A 95, 033818 (2017).
Kockum, A. F., Johansson, G. & Nori, F. Decoherencefree interaction between giant atoms in waveguide quantum electrodynamics. Phys. Rev. Lett. 120, 140404 (2018).
van Loo, A. F. et al. Photonmediated interactions between distant artificial atoms. Science 342, 1494–1496 (2013).
Hood, J. D. et al. Atom–atom interactions around the band edge of a photonic crystal waveguide. Proc. Natl Acad. Sci. USA 113, 10507–10512 (2016).
Sundaresan, N. M., Lundgren, R., Zhu, G., Gorshkov, A. V. & Houck, A. A. Interacting qubitphoton bound states with superconducting circuits. Phys. Rev. X 9, 011021 (2019).
Lalumière, K. et al. Inputoutput theory for waveguide QED with an ensemble of inhomogeneous atoms. Phys. Rev. A 88, 043806 (2013).
Astafiev, O. et al. Resonance fluorescence of a single artificial atom. Science 327, 840–843 (2010).
Cook, R. J. & Kimble, H. J. Possibility of direct observation of quantum jumps. Phys. Rev. Lett. 54, 1023–1026 (1985).
Rosario Hamann, A. et al. Nonreciprocity realized with quantum nonlinearity. Phys. Rev. Lett. 121, 123601 (2018).
Yeh, J.H., LeFebvre, J., Premaratne, S., Wellstood, F. C. & Palmer, B. S. Microwave attenuators for use with quantum devices below 100 mK. J. Appl. Phys. 121, 224501 (2017).
Bronn, N. T. et al. High coherence plane breaking packaging for superconducting qubits. Quantum Sci. Technol. 3, 024007 (2018).
Paulisch, V., Kimble, H. J. & GonzálezTudela, A. Universal quantum computation in waveguide QED using decoherence free subspaces. New J. Phys. 18, 043041 (2016).
Mirhosseini, M. et al. Superconducting metamaterials for waveguide quantum electrodynamics. Nat. Commun. 9, 3706 (2018).
Keller, A. J. et al. Al transmon qubits on silicononinsulator for quantum device integration. Appl. Phys. Lett. 111, 042603 (2017).
Rooks, M. J. et al. Low stress development of poly(methylmethacrylate) for high aspect ratio structures. J. Vacuum Sci. Technol. B 20, 2937–2941 (2002).
Chen, Z. et al. Fabrication and characterization of aluminum airbridges for superconducting microwave circuits. Appl. Phys. Lett. 104, 052602 (2014).
Wu, M.D., Deng, S.M., Wu, R.B. & Hsu, P. Fullwave characterization of the mode conversion in a coplanar waveguide rightangled bend. IEEE Trans. Microw. Theory Technol. 43, 2532–2538 (1995).
Acknowledgements
We thank J.H. Yeh and B. Palmer for the use of one of their cryogenic attenuators, which reduced thermal noise in the input waveguide line. This work was supported by the AFOSR MURI Quantum Photonic Matter (grant FA95501610323), the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (grant PHY1125565) with the support of the Gordon and Betty Moore Foundation, and the Kavli Nanoscience Institute at Caltech. D.E.C. acknowledges support from the ERC Starting Grant FOQAL, the MINECO Plan Nacional Grant CANS, the MINECO Severo Ochoa grant SEV20150522, the CERCA Programme/Generalitat de Catalunya and the Fundacio Privada Cellex. M.M. is supported through a KNI Postdoctoral Fellowship. X.Z. is supported by a Yariv/Blauvelt Fellowship. A.J.K. and A.S. are supported by IQIM Postdoctoral Scholarships. P.B.D. is supported by a Hertz Graduate Fellowship Award. A.A.G. is supported by the Global Marie Curie Fellowship under the LANTERN programme.
Reviewer information
Nature thanks Anton Kockum, Peter Lodahl and the other anonymous reviewer(s) for their contribution to the peer review of this work.
Author information
Authors and Affiliations
Contributions
M.M., E.K., P.B.D., A.A.G., D.E.C. and O.P. came up with the concept and planned the experiment. M.M., E.K., X.Z., P.B.D., A.S. and A.J.K. performed the device design and fabrication. E.K., X.Z., M.M., and A.S. performed the measurements and analysed the data. All authors contributed to the writing of the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data figures and tables
Extended Data Fig. 1 Scanning electron microscope image of the fabricated device (false colour highlights).
a, Type I (Q_{2}, Q_{3}) and type II (Q_{1}) mirror qubits coupled to the CPW. b, The central probe qubit (Q_{4}) and lumpedelement readout resonator (R_{4}) coupled to the CPW. The inset shows an inductive meander of the lumpedelement readout resonator. c, A superconducting quantum interference device (SQUID) loop with asymmetric Josephson junctions for the qubits. d, An airbridge placed across the waveguide to suppress the slotline mode.
Extended Data Fig. 2 Schematic of the measurement chain inside the dilution refrigerator.
The four types of input lines, the output line and their connection to the device inside a magnetic shield are illustrated. Attenuators are expressed as rectangles with labelled power attenuation and capacitor symbols correspond to directcurrent blocks. The thinfilm attenuator and a circulator (coloured red) are added to the waveguide input line and output line, respectively, in a second version of the setup and a second round of measurements to further protect the sample from thermal noise in the waveguide line. HEMT, highelectronmobility transistor.
Supplementary information
Supplementary Information
This file contains Supplementary Text and Data, Supplementary Figures 12, Supplementary Tables 12, and additional references.
Rights and permissions
About this article
Cite this article
Mirhosseini, M., Kim, E., Zhang, X. et al. Cavity quantum electrodynamics with atomlike mirrors. Nature 569, 692–697 (2019). https://doi.org/10.1038/s4158601911961
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s4158601911961
This article is cited by

Organic molecules pumped to resonance
Nature Physics (2024)

NonHermitian control between absorption and transparency in perfect zeroreflection magnonics
Nature Communications (2023)

Tunable directional photon scattering from a pair of superconducting qubits
Nature Communications (2023)

Hyperbolic whisperinggallery phonon polaritons in boron nitride nanotubes
Nature Nanotechnology (2023)

Manybody cavity quantum electrodynamics with driven inhomogeneous emitters
Nature (2023)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.