Abstract
Waveguide quantum electrodynamics offers a wide range of possibilities to effectively engineer interactions between artificial atoms via a onedimensional open waveguide. While these interactions have been experimentally studied in the few qubit limit, the collective properties of such systems for larger arrays of qubits in a metamaterial configuration has so far not been addressed. Here, we experimentally study a metamaterial made of eight superconducting transmon qubits with local frequency control coupled to the mode continuum of a waveguide. By consecutively tuning the qubits to a common resonance frequency we observe the formation of super and subradiant states, as well as the emergence of a polaritonic bandgap. Making use of the qubits quantum nonlinearity, we demonstrate control over the latter by inducing a transparency window in the bandgap region of the ensemble. The circuit of this work extends experiments with one and two qubits toward a fullblown quantum metamaterial, thus paving the way for largescale applications in superconducting waveguide quantum electrodynamics.
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Introduction
The recent advances in the field of quantum information processing has led to a rising demand to explore new systems beyond cavity quantum electrodynamics (QED). One promising candidate is waveguide QED, where quantum systems interact coherently with the mode continuum of a waveguide instead of a cavity. After the pioneering works with single qubits, including the demonstration of resonance flourescence^{1}, quantum correlations of light and single photon routers^{2}, attention shifted to the realization of multiple qubits coupled to a common waveguide. It was derived^{3} and experimentally verified^{4} that multiple qubits obtain an infinite range photon mediated effective interaction, which can be tuned with the interqubit distance d. Furthermore, the shared collective excitations are of polaritonic nature with lifetimes ranging from extremely sub to superradiant relative to the radiative lifetime of the individual qubits^{5,6}. The strong intrinsic nonlinearity of the qubits was recently shown to give rise to partially localized polaritons^{7}, topological edge states^{8,9}, and quantum correlations in the scattered light of the array^{10}. The collective quantum properties are exploited in the field of quantum metamaterials^{11,12}. Here, the quantum coherence of the constituting qubits is used to engineer a global optical response which depends on their quantum state^{13,14,15,16}. With respect to quantum information processing, multiqubit waveguide QED systems could be harnessed in numerous applications such as on demand, highly efficient creation of multiphoton and entangled states^{17,18,19}, storage devices for microwave pulses^{20}, atomic mirrors^{21}, numberresolved photon detection^{22}, slow and even stopped light^{23}. Experimentally, waveguide QED systems have been realized on several platforms including atoms^{24}, quantum dots coupled to nanophotonic waveguides^{25} and defect centers in diamonds^{26}. Even though superconducting qubits feature advantages such as frequency control, high coherence, near perfect extinction and absence of position and number disorder, superconducting multiqubit waveguide QED systems have been studied only recently to some extend^{27,28}.
Here, we investigate the mode spectrum of a metamaterial formed by a densely spaced array of eight superconducting transmon qubits coupled to a coplanar waveguide. By employing dedicated fluxbias lines for each qubit, we establish control over their transition frequencies. Thus we are able to alter the number of resonant qubits N at will, allowing us to observe super and subradiant modes as well as the gradual formation of a bandgap. Our spectroscopic measurements show, that through this control the global optical susceptibility of the metamaterial can be tuned. A demonstration for the collective AutlerTownes splitting (ATS) of eight qubits is presented, which marks an important step toward the implementation of quantum memories.
Results and discussion
Circuit design and properties
The sample under investigation is depicted in Fig. 1(a). The spacing d between adjacent qubits is 400 μm, which is smaller than the corresponding wavelength λ at all accessible frequencies (\(\varphi =\frac{2\pi }{\lambda }d=0.05\!\!0.16\)). The dense spacing is chosen in order to increase the width of the expected bandgap of Δω = Γ_{10}/φ ≫ Γ_{10} and to fulfil the metamaterial limit of subwavelength dimensions^{29}. Here Γ_{10} is the radiative decay rate of the individual qubits into the waveguide. The qubits are overcoupled, ensuring a multimode Purcelllimited rate Γ_{10}/2π ≈ 6.4 MHz for high extinction and, simultaneously, better subradiant state visibility. All qubits are individually frequency controllable between 3 and 8 GHz by changing the critical currents of the qubit SQUIDs with local fluxbias lines. We compensate unwanted magnetic crosstalk between the fluxbias lines and neighbouring qubits by extracting and diagonalising the full mutual inductance matrix M (see Methods). This allows us to counteract the parasitic crosstalk flux by sending appropriate currents to all qubits, which are not actively tuned. With this calibration scheme we achieve precise control over the individual qubit frequencies. We estimate the residual crosstalk to be smaller than 10^{−3}. The effective Hamiltonian of this system, after formally tracing out photonic degrees of freedom and applying the Markov approximation, is described by^{6},
with the bosonic creation operator \({b}_{j}^{\dagger }\left0\right\rangle =\left{{\rm{e}}}_{j}\right\rangle\), exciting the jth qubit at frequency ω_{j}; where we assume a ∝ e^{+iωt} time dependence of the excitations. Here, χ_{j}/2π is the qubit anharmonicity for which we find spectroscopically weakly varying values around −275 MHz. The last term of H_{eff} describes the effective qubit–qubit coupling. Due to the specific choice of small d its imaginary part dominates over the real part, leading to a suppressed exchangetype interaction between the qubits. The expected eigenfrequencies ω_{ξ} of H_{eff} in the single excitation limit and χ → 0 are shown in Fig. 2(a).
Bandstructure and collective metamaterial excitations
First, we characterize the mode spectrum of the metamaterial in dependence on the number of resonant qubits N by measuring the transmission coefficient S_{21}(ω) while the qubits are consecutively tuned to a common resonance frequency at ω_{r}/2π = 7.898 GHz, compare Fig. 1 (b) and Fig. 2 (a). The incident photon power P_{inc} is kept below the single photon level (P_{inc} ≪ \({\hbar}\)ωΓ_{10}) to avoid saturation of the qubits. For a single qubit, the well known resonancefluorescence was observed as a single dip in transmission^{1}. By fitting the complex transmission data to the expected transmission function S_{21}(ω) (see Methods) the individual coherence properties of all eight qubits at ω_{r} can be extracted (see Supplementary Note 1). We find the average radiative rates Γ_{10}/2π = 6.4 MHz and the intrinsic nonradiative rates Γ_{nr}/2π = 240 – 560 kHz. For N ≥ 2 resonant qubits the system obtains multiple eigenmodes and the super and subradiant polariton modes start to emerge. The superradiant mode is manifested as a wide transmission dip above ω_{r}, the subradiant modes can be identified as transmission peaks below ω_{r}. We note that the peak shape is created by Fanointerference of the sub and the superradiant modes (see Supplementary Note 5), which also causes the calculated eigenfrequencies of H_{eff} to not exactly coincide with the maximum of the peaks in Fig. 1 (b) and Fig. 2(a). For N ≥ 6 qubits a second darker subradiant mode is visible between ω_{r} and the brightest subradiant mode. The limiting factor for the observation of the subradiant states is the intrinsic qubit coherence as given by Γ_{nr}, setting an upper threshold for the maximum observable lifetime. Darker subradiant states with Γ_{ξ} < Γ_{nr} decay in the qubits into dielectric channels or dephase due to flux noise in the SQUIDs before they are remitted into the waveguide. The observed transmission coefficient ∣S_{21}∣ is in good agreement with the calculated transmission based on a transfer matrix approach^{30}. The asymmetric lineshape of the resonances is a parasitic effect caused by interference of the signal with lowQ standing waves in the cryostat^{31}. We account for this effect in the transfer matrix calculation by adding semireflective inductances in front and after the qubit array. As shown in Fig. 2(a), a frequency region of strongly suppressed transmission with ∣S_{21}∣^{2} <−25 dB is opening up above ω_{r} with increasing N. This effect is associated with the emergence of a polaritonic bandgap, where the effective refractive index becomes purely imaginary, as expected for any kind of resonant periodic structures^{29,32}. For N = 8 qubits we extract a bandgap bandwidth of Δω ≈ 1.9Γ_{10}. Compared to the expected bandgap width of Δω = Γ_{10}/φ ≈ 6.3Γ_{10} of the structure for N → ∞ this places our system size in the transitioning regime between a single atom and a fully extended metamaterial with a continuous mode spectrum. The radiative decay rates Γ_{ξ} of the observed subradiant states shown in Fig. 2(b) are extracted by fitting Lorentzians to the corresponding modes in the reflection data (not shown). It can be generally shown, that ∣S_{21}∣^{2} indeed obtains a Lorentzian shape in the vicinity of each ω_{ξ}^{30,33,34}. From a fit to a power law ∝ N^{b} with exponent b = 2.93 ± 0.13 we find that the rate of the brightest subradiant states scales with Γ_{ξ} ∝ N^{3}, which we also find analytically for densely packed qubit structures with φ ≪ 1 from H_{eff} (see Supplementary Note 4). The found scaling is the complementary asymptotic of the theoretically predicted Γ_{ξ} ∝ N^{−3} law for the darkest subradiant modes in references^{5,6,32}. Small deviations between calculated and measured values of Γ_{ξ} are caused by imperfect qubit tuning and distortions of the observed reflection coefficient due to the microwave background.
The control over the mode spectrum can be further elaborated with the offresonant situation, where one qubit has a finite detuning Δ from the common resonance frequency ω_{r} of the residual qubits by sweeping it through the common resonance as shown in Fig. 3(a). Here, qubit 8 is tuned through the collective resonance of qubits 1–7. For large detunings ∣Δ∣ ≫ Γ_{10} the modes of the ensemble and qubit 8 are not hybridized (not shown). For smaller detunings an additional partially hybridized subradiant mode appears, which becomes for Δ ≈ 0 the brightest subradiant state of the fully hybridized 8qubit system. The result is in good agreement with the transfer matrix calculation and direct diagonalization of H_{eff} as shown in Fig. 3(b). The level repulsion between the eigenstates is caused by the residual exchangetype interaction between the qubits due to the finite interqubit distance d. In Fig. 3 there are several blind spots where the subradiant states turn completely dark, occurring when the frequency of the detuned qubit matches the frequency of a dark mode. This is explained by the Fanolike interferences^{35} between the detuned qubit resonance and the modes of the resonant qubits, which are analyzed in more detail in the Supplementary Note 5.
Collective AutlerTownes splitting
The intrinsic quantum nonlinearity of the system due to the anharmonic nature of the qubits can be probed by increasing the microwave power beyond the single photon regime (P_{inc} > \({\hbar}\)ωΓ_{10}). At higher power the 0 → 1 transition of a single qubit will start to saturate and transmission at resonance will increase back to unity^{1}, see Fig. 4(b). We could experimentally verify the prediction of the saturation of an ensemble of resonant qubits at higher drive rates of refs. ^{1,3}, which can be observed in Fig. 4(a). We point out that all spectroscopic features of the metamaterial, such as super and subradiant modes as well as the bandgap, are saturable. The power P_{50%} needed to saturate the transmission at ω_{r} to 50% (∣S_{21}∣^{2} = 0.5) grows with an approximate scaling \(\propto \mathrm{ln}\,(N)\). The observed quantum nonlinear behavior establishes a border to previously studied metamaterials consisting of harmonic resonators rather than qubits^{36}. Furthermore the anharmonic level structure of the transmon can be used to electromagnetically open a transparency window around the frequency of the 0 → 1 transition, by employing the ATS^{37}. As indicated in Fig. 4(c), a coherent control tone with Rabi strength Ω_{c} and frequency ω_{c} drives the 1 → 2 transition, while a weak microwave tone with Ω_{p} ≪ Ω_{c} is probing the transmission at frequencies ω_{p} around the 0 → 1 transition. The control tone is dressing the first qubit level, creating two hybridized levels, which are separated proportionally to its amplitude, and is therefore creating a transparency window with respect to the probe. To the best of our knowledge this effect was so far only demonstrated for a single qubit in superconducting waveguide QED, leading to applications like single photon routers^{2}. Here, we observe the collective ATS for up to N = 8 resonant qubits (Fig. 4(d)). Analogously to the single qubit case, the observed level splitting is proportional to Ω_{c}. At control tone powers >−110 dBm the bandgap is rendered fully transparent and transmission close to unity is restored. In the experiment we find slightly smaller than unity values of ∣S_{21}∣ ≈ 0.75 due to the interference of the signal with the microwave background. In the bandstructure picture the collective ATS can be understood as the dressed states of the individual qubits giving rise to two independent Bloch bands^{38}. Therefore, the two branches of the collective ATS with supressed transmission are collectively broadened bandgaps and have a larger linewidth than the dressed states of the single qubit ATS. As pointed out in reference^{38} the collective ATS demonstrates active control over the bandstructure of the metamaterial via the parameters ω_{c} and Ω_{c}. The observed splittings are in agreement with a transfer matrix calculation and a full master equation simulation (see Supplementary Note 6). Minor deviations are caused by imperfect qubit tuning and differing qubit anharmonicities of about σ_{χ} ≈ 10 MHz. We argue here that the single photon router concept with a single qubit can significantly be improved with multiple qubits, which form a much wider stopband with higher saturation power, thus permitting to route also multiple photons.
In conclusion, we demonstrated a fully controllable quantum metamaterial consisting of eight densely packed transmon qubits coupled to a waveguide. Such intermediate system size combined with individual qubit control allowed us to explore the transition from a single mode regime to a continuous band spectrum. By tuning the qubits consecutively to resonance, we observed the emergence of a polaritonic bandgap, and confirmed the scaling of the brightest dark mode decay rate with the qubit number. Active control over the band structure of the ensemble was demonstraded by inducing a transparency window in the bandgap region, using the AutlerTownes effect. Our work promotes further research with higher qubit numbers to realize a longliving quantum memory.
Methods
Fabrication
The sample is fabricated with two consecutive lithography steps from thermally evaporated aluminum in a Plassys MEB550s shadow evaporator on a 500 μm sapphire substrate. In a first step solely the qubits are patterned with 50 keV electron beam lithography. The Josephson junctions are patterned with a bridgefree fabrication technique^{39} using a PMMA/PMMAMAA double resist stack. Before doubleangle evaporation, the developed resist stack is cleaned for 6 min with an oxygenplasma to remove resist residuals in the junction area to reduce the impact of junction aging^{39}. In a second optical lithography step we pattern the CPWwaveguide and the groundplane in a liftoffprocess on S1805 photo resist. The SQUIDs are formed by two Josephson junctions, enclosing an area of 560 μm^{2}. By design, the junction areas are 0.12 and 0.17 μm^{2} with a designed asymmetry of 17%.
Calibration of magnetic crosstalk
We extract the full 8 × 8 mutual inductance matrix M between the qubits and the bias coils in 28 consecutive measurements. For that, the transmission through the chip is observed at a fixed frequency while tuning the currents of two qubits I_{x} and I_{y} such that they get tuned through the observation frequency. Fitting the slope of the observed qubit traces gives access to M_{xy}/M_{xx} and M_{yx}/M_{yy}. When M is known, compensation currents which are send to all qubits which are not actively tuned, can be calculated to compensate unwanted crosstalk. We estimate the residual crosstalk to be smaller than 0.1%. Further information is provided in the Supplementary Note 2.
Normalization of spectroscopic data
We normalize the transmission data by dividing the raw data \({S}_{21}^{\,\text{raw}\,}\) by the transmission data at high powers \({S}_{21}^{\text{sat}}{:}{S}_{21}^{\,\text{calib}\,}(\omega )={S}_{21}^{\,\text{raw}\,}(\omega )/(a{S}_{21}^{\,\text{sat}\,}(\omega ))\), where a ≈ 1 is a constant factor accounting for weak fluctuations of the amplifier gain. Reflection data is normalized by dividing the raw data \({S}_{22}^{\,\text{raw}\,}\) with its maximum value at the qubit resonance frequency f_{r}: \({S}_{22}^{\,\text{calib}}={S}_{22}^{\text{raw}}/{S}_{22}^{\text{raw}\,}({f}_{\text{r}})\). This approximation is justified for the sample under investigation since the extinction of single and multiple qubits is very close to 1. A more rigorous normalization of the data based on energy conservation ∣S_{21}∣^{2} + ∣S_{11}∣^{2} ≈ 1 is not applicable here, due to differing signal paths for reflection and transmission measurements, as pointed out in the main text. In addition, in any experimental system the insertion losses due to impedance mismatches of the feedline can not be avoided. Since they are in general not symmetric on both sides of the sample, ∣S_{11}∣ ≠ ∣S_{22}∣ and therefore ∣S_{21}∣^{2} + ∣S_{22}∣^{2} ≠ 1.
Characterization of individual qubits
The amplitude transmission coefficient of a driven twolevel system sidecoupled to a waveguide S_{21} is given by^{2}:
The decoherence rate γ_{10} = Γ_{10}/2 + Γ_{nr} is the sum of radiative decay Γ_{10} and nonradiative decay rates Γ_{nr} = Γ_{Φ} + Γ_{l}/2. Here Γ_{Φ} accounts for pure dephasing of the qubit and Γ_{l} for all non radiative relaxation channels. We define the extinction coefficient as \(1{(1{{{\Gamma }}}_{10}/2{\gamma }_{10})}^{2}\), measuring the suppression of powertransmission at very low drive powers. A circle fitting procedure^{40} is used to fit equation (2) to the measured complex transmitted signal S_{21} in the limit of weak driving \({{{\Omega }}}_{{\rm{p}}}^{2}\ll ({{{\Gamma }}}_{10}+{{{\Gamma }}}_{l}){\gamma }_{10}\). The decoherence rates of the individual qubits and further details on the fitting procedure are provided in the Supplementary Note 1.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work has received funding from the Deutsche Forschungsgemeinschaft (DFG) by the Grant No. US 18/151, the European Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement No. 863313 (SUPERGALAX), and by the Initiative and Networking Fund of the Helmholtz Association, within the Helmholtz Future Project ‘Scalable solid state quantum computing’. We acknowledge financial support from Studienstiftung des Deutschen Volkes (J.D.B.), LandesgraduiertenförderungKarlsruhe (A.S.) and Helmholtz International Research School for Teratronics (T.W.). Analysis of subradiant mode lifetimes, performed by A.N.P., has been supported by the Russian Science Foundation Grant 201200194. Basic concepts for this work were developed with the financial support from the Russian Science Foundation (contract No. 161200095). A.V.U. acknowledges partial support from the Ministry of Education and Science of the Russian Federation in the framework of the Increase Competitiveness Program of the National University of Science and Technology MISIS (contract No. K22020022).
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J.D.B. fabricated the devices supported by H.R. J.D.B. performed the measurements with support of A.S. and T.W. A.S. developed concepts for the crosstalk calibration. A.V.U. and H.R. setup the measurement facility. J.D.B. analyzed the data. A.N.P. performed the calculations on Fanointerference and linewidthscaling. J.D.B. and A.N.P. wrote the paper. All authors contributed to the discussion. The project was supervised by H.R. and A.V.U.
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Brehm, J.D., Poddubny, A.N., Stehli, A. et al. Waveguide bandgap engineering with an array of superconducting qubits. npj Quantum Mater. 6, 10 (2021). https://doi.org/10.1038/s4153502100310z
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DOI: https://doi.org/10.1038/s4153502100310z
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