Slowing down light in a qubit metamaterial

In the light of quantum information processing achieving control over the speed of light in artificially structured media, slowing it down and eventually stopping the light, has recently regained attention. This functionality is of vital importance for the realization of long-living quantum memories¹ and for controlling and synchronizing the flow of information.² One of the most prominent techniques to realize slow light is based on electromagnetically induced transparency (EIT)—a quantum interference effect between different excitation pathways, rendering an otherwise opaque medium transparent and creating a steep dispersion profile.³ Early demonstrations of slow light in EIT media involved vapors of sodium⁴ and rubidium atoms,⁵ followed by further experiments with cold atoms coupled to nanofibers,⁶ and eventually by the demonstration of a highly efficient quantum memory.⁷ A promising candidate beyond atoms to observe these effects is superconducting qubits in the context of cavity-free waveguide quantum electrodynamics (wQED). The observation of resonance florescence revealed that even single qubits can have strong coupling to propagate electromagnetic waves and show extinction coefficients close to unity.^8,9 First demonstrations of the single-qubit Autler–Townes splitting (ATS) with ladder-type three-level systems,^9,10 showcased their potential suitability for the EIT-related applications. However, it was subsequently argued that in these experiments, the EIT regime was not unambiguously reached.¹¹ 

More recently, focus shifted toward multi-qubit systems with infinite range interactions,^12,13 super- and subradiant polaritonic excitations,^13–16 collective ATS,¹⁴ topological edge states,^17,18 creation of non-classical light,¹⁹ proposals for superconducting quantum memories,²⁰ and quantum metameterials.^21,22 Even though superconducting wQED systems can match the requirements of large optical depth and high coherence,²³ slow light has been so far realized only in the context of classical waveguides.²⁴ These devices are passive without the possibility to in situ control the speed of light and to ultimately realize a quantum memory protocol.

Here, we experimentally demonstrate a first realization of slow light in a superconducting wQED system consisting of eight locally tunable transmon qubits coupled to a one-dimensional waveguide. We engineer the required flatband structure by using qubits directly as controllable dispersive elements. First, we consider the standard case of dressed state based slow light and demonstrate that even in the case of imperfect EIT, where the physics is merely governed by the ATS, a moderate retardation of the group velocities by a factor down to 1500 compared to vacuum can be achieved. Second, we engineer a similar band structure based on detuned collective resonances of the participating qubits. This allows for three times larger efficiencies of the slow-light medium. The demonstrated slow light effect can be used for a fixed-delay quantum memory in superconducting wQED and paves way to more general applications such as on demand storage-and-retrieval memory for quantum information processing.

The qubit metamaterial presented in this work consists of eight superconducting transmon qubits,²⁵ capacitively coupled to the mode continuum of a coplanar waveguide, see Fig. 1(a). Each qubit has an integrated SQUID loop and is thereby individually tunable between 3 and 8 GHz by applying local magnetic flux via a flux bias line. We fulfill the metamaterial limit of sub-wavelength dimensions by choosing a dense qubit spacing of $d=400 μ$ m, which corresponds to a fraction of the light wavelength λ with the phase delay of $φ=2πλd=0.05−0.16$⁠. As shown in former works, in such a setting the qubits obtain an infinite-range photon-mediated effective interaction, which is almost exclusively of collective dissipative nature.^12,13 For a detailed study of the mode structure of this metamaterial, its super- and subradiant polaritonic excitations, we guide the reader to Ref. 14. Figure 1(b) shows the first three ladder-type energy levels of the transmon qubits. If the $2→1$ transition is driven with a microwave control tone with the amplitude (Rabi-strength) $Ωc$⁠, these levels hybridize and split into two dressed states separated by $Ωc$⁠, forming the ATS.²⁶ For our experiments, we use the ATS of the individual qubits to calibrate the absolute value of the control power $Pc=Ωc24Γ10ℏωc$²⁷ (see the supplementary material). With reference to a weak probe tone and assuming a $∝ exp (−iωt)$ time dependence, the reflection coefficient of a single qubit is given by¹⁰ 

The corresponding transmission coefficient is given by $t=1+r$⁠, referenced in this article as element of the scattering matrix $S21(ω)$⁠. Here, the qubits are strongly coupled to the waveguide, ensuring a multi-mode Purcell limited average relaxation rate of $Γ10/2π=12$ MHz. The average decoherence rate of the $1→0$ transition is $γ10/2π=(Γ10/2+Γnr)/2π=6.9$ MHz with $Γnr$ accounting for pure dephasing and radiative losses to unguided modes. The corresponding average decoherence rate of the $2→0$ transition is $γ20/2π=6.9$ MHz, which is here by coincidence the same value. We note that this rate is in agreement with other experiments with superconducting qubits,^10,28 significantly higher than that of cold atoms.³ The increased rate is due to the ladder-type level structure of the transmon, where it can be shown that $γ20>Γ21/2$ always holds.²⁹ This necessarily leads to imperfect EIT since the associated dark state obtains a finite lifetime.³ The corresponding implications for the creation of slow light are discussed in the following paragraph. The expected band structure for the metamaterial in the lossless case (⁠$Γnr=0, γ20=0$⁠) based on Eq. (1) for different values of $Ωc$ is shown in Fig. 1(c). The expected group velocity of light $vg=dωdk$ in the center band strongly depends on the control tone strength $Ωc$⁠. In the regime of small $Ωc$⁠, the slope of the band is almost flat, giving rise to slowing down the electromagnetic waves in the waveguide.

In order to create a slow-light medium, the qubits are consecutively tuned to a common resonance frequency $ω10/2π=7.812$ GHz in the vicinity of the qubits' upper sweet spot. A continuous microwave control tone is applied, driving resonantly the $2→1$ transition at the frequency $ωc/2π=7.533$ GHz and Rabi-strength $Ωc$⁠. Figure 2 shows the collective ATS, for N = 7 resonant qubits. For low control powers $Pc→0$⁠, only a single bandgap of strongly suppressed transmission above ω₁₀ is present, which gradually splits for higher $Ωc$ into two separate bandgaps, with a transparency window of finite transmission opening up in between [compare band structure calculation in Fig. 1(c)]. It is noted that due to the finite system size, the observed bandgap $Δω$ is smaller than for the infinite case $Δω/2π=Γ10cω10d/2π≈76 MHz$⁠. A more detailed discussion on finite size effects is provided in Ref. 14. In this region, the phase of the complex transmission coefficient Arg $(S21(ω))$ features a steep roll-off [Fig. 2(b)], indicating low group velocities. The observed transmission coefficient is in agreement with a numerical transfer-matrix calculation,^14,30 which takes into account a cable resonance caused by impedance mismatches in the cryostat wiring. Due to the aforementioned large $2→0$ decoherence rate $γ20≈γ10$⁠, the expected sharp EIT feature at ω₁₀ with near unity transmission for small $Ωc$ is smeared out, and transmission is reduced. The expected effective group velocity $vg$ of a signal sent through the transparency window can be calculated from the phase gradient^20,30

The traversal time $τ=(N−1)d/vg$ of a pulse through the medium with N = 7 qubits inferred from the spectroscopic data of Fig. 2(a) at $ω=ω10$ is shown in Fig. 3(b). In good agreement with the numerical transfer-matrix model, the expected traversal time, or conversely, the effective group index $ng$⁠, can be tuned over a large range with the applied control power $Pc$⁠. In contrast to the textbook case of EIT in the limit of $γ20≈0$⁠, where $τ∝1/Ωc2$ diverges for small $Ωc$⁠,³ we observe τ approaching a maximum of 15 ns (⁠$ng≈1900$⁠) at $Pc≈−124$ dBm before it decreases again for even lower control tone strengths. A similar behavior was observed in room temperature vapor of ⁴He.³¹ An analytical expression for the expected delay $τ(ω10)$ in the case of a resonant control tone can be found from the effective dispersion relation $(kd)2≈φ2−2χφ$ with $χ=ir1+r$ and $φ=ωcd$⁠:³² 

Figure 3(b) shows both asymptotes of Eq. (3), corresponding to the strong and weak Rabi-strength $Ωc$ regime (⁠$φ≫χ, φ≪χ$⁠). In the limit of $γ20→0$⁠, the condition $φ≫χ$ is always fulfilled, and the corresponding $τ(ω10,Ωc)$ is reduced to the formulas of textbook EIT.³ In this case. we find good agreement between the measured delays and Eq. (3). In the opposite limit $φ≪χ$⁠, only qualitative agreement is observed. The systematic offset to the measured data is rooted in the band structure calculation of Eq. (3), which assumes an infinite metamaterial deviates, in particular, for small $Ωc$⁠. More details on the derivation of $τ(ω10,Ωc)$ are provided in the supplementary material. Formally, EIT and ATS should be treated as two distinct, but closely related phenomena: EIT is a destructive Fano-type interference effect between two excitation pathways, which can only occur if $Ωc<ξγ10$⁠, smoothly transitioning to the ATS-regime for $Ωc>ξγ10$⁠, with two independent resonances of the dressed state doublet.¹¹ The coefficient $ξ=(N2−1)φ/3$ is the approximate width of the bandgap for a finite system with $N<π/φ≈20$ qubits in units of γ₁₀. In this work, due to large γ₂₀, the accessible control strengths are in the transient regime $Ωc≳ξγ10$⁠. Here, a quantitative method to distinguish between ATS and EIT based on Akaike's information criterion of Ref. 11 cannot directly be applied to the measured data, since it features an asymmetric line shape due to the microwave background. Using the method for the calculated single-qubit splitting based on Eq. (1) in conjunction with the experimentally extracted qubit decoherence rates yields that the observed line shape is by almost 100% certainty described by the ATS (not shown).

We have verified the spectroscopically inferred time delays τ in pulsed time-domain measurements by using an FPGA-based heterodyne microwave setup (see the supplementary material). Here, we generated Gaussian pulses of width $σ=50$ ns at resonant center-frequency ω₁₀ with a continuously applied control tone at frequency ω₂₁ and strength $Ωc$⁠. The pulse amplitude is kept at the single photon level (⁠$Pp<ℏω10Γ10$⁠) in order to prevent saturation of the qubits. The digitized transmitted pulses for different control tone strengths are shown in Fig. 3(a). By fitting Gaussians to the measured data, we extracted the temporal position of each pulse center. The extracted pulse arrival times were compared to a reference pulse, measured in the case of far detuned qubits, thus giving access to the pulse delay τ. The delays measured in this way are presented in Fig. 3(b) showing good agreement with the spectroscopically inferred data and numeric. Due to the finite spectral width $1/σ=20$ MHz of the Gaussian pulses [indicated by dashed purple lines in Fig. 2(a)] and that of the transparency window, the maximal accessible delay here is limited to $τ≈12$ ns (⁠$ng≈1500$⁠). In qualitative agreement with the transmission measurement data of Fig. 2(a), the amplitude of the measured pulses becomes smaller when reducing the control tone amplitude. This effect is most prominent in the sector of large delays; however, exactly here a high transmission is desired to realize an efficient quantum memory protocol.³³ At maximal delay, the efficiency, i.e., the energy ratio of the transmitted and reference pulse, is about 16%. Due to small average delay-bandwidth products of 0.2, the device under investigation appears suitable for a fixed-delay quantum memory,³⁴ rather than a general purpose on-demand storage procedure, where the pulse has to fit spatially completely into the metamaterial for high storage efficiency.³³ Reducing the decoherence rate γ₂₀ would allow for significantly larger group indices and improve the bandwidth-delay product toward values larger than unity. Experimentally, this can be achieved by employing Λ-type three-level systems such as flux- or fluxonium qubits.²⁰ 

As pointed out by Shen et al.,³⁵ ATS-like band structures can be realized not only by dressed states but also equivalently by two detuned distinct resonances coupled to the same mode continuum. For this approach, which we call dispersion engineering, neither a third qubit level nor an additional microwave control tone is required. Here, we use the individual qubit frequency control to tune every second qubit to a frequency f₁ and every other qubit to f₂, creating two collective four-qubit resonances. The frequency $f2=7.882$ GHz is kept constant throughout the experiment, while the frequency f₁ of the other four qubits is varied. The transmission S₂₁ [Figs. 4(a) and 4(b)] resembles that of the dressed state ATS in Fig. 2; the main ATS feature, being a transparency window with a steep phase roll-off between two bandgaps, is intact. Since the large γ₂₀ has no influence on the first two transmon levels, the brightest of the collective four qubit subradiant states are visible as pronounced peaks several MHz below f₁ and f₂¹⁴ [compare Fig. 4(b)]. The observed line shape is in good agreement with a numerical transfer matrix calculation. The pulse delays τ and effective group indices $ng$⁠, calculated from Arg $(S21(ω))$ with Eq. (2) with respect to the frequency f₁, are shown in Fig. 4(c). Since the phase in the transparency window shows a varying slope, an average of the slope in a bandwidth of 10 MHz around the center is used to estimate τ. Analogously to the previous paragraph, we use Gaussian pulses with $σ=50$ ns to directly measure and validate the expected pulse delay τ. The center frequency is adjusted close to $(f1+f2)/2$ for each trace. Figure 4(c) compares the delay τ of pulsed and spectroscopic measurements with numerical results from a transfer-matrix model. Here, we find delays up to $τ=17$ ns, corresponding to group indices of $ng=1850$⁠, which is comparable to the group indices found in the dressed-state ATS case for a similar splitting between the dressed states. In contrast to the ATS measurement, we achieve an efficiency of $≈45–50$%, as the transmittance of the transparency window is solely limited by the non-radiative decoherence $Γnr$ and not influenced by γ₂₀. This also implies potentially possible much higher delays for narrower transparency windows. Even though the dispersion-engineered approach lacks the possibility of fast and simple control via an additional microwave tone in contrast to the ATS, it can be used as a fixed delay quantum memory and showcases the ability to tailor the band structure on demand in superconducting wQED. For further studies, one can think of more complex band structures going beyond the capabilities of the ATS, e.g., engineering several spectral regions of different group indices, referred to as multi-color slow light.³⁶ 

In this work, we demonstrated slow light in a superconducting waveguide QED system. A metamaterial consisting of eight densely spaced transmon qubits was used to engineer a band structure with a flat dispersion profile to obtain reduced group velocities of electromagnetic waves. Using the well-known dressed-state ATS, we observed group velocities reduced by a factor of down to 1500, both in spectroscopic and direct time-resolved pulsed measurements. The maximum achievable delay and efficiency is limited by the strong dephasing of the $2→0$ transition, which is inherent to ladder-type three-level systems. Moreover, we demonstrated slow light with a tailored band structure based on distinct detuned collective resonances of the qubits. At comparable retardation, we observed three times larger efficiencies, not limited by large γ₂₀. The demonstrated slow-light metamaterial can be employed as a fixed-delay quantum memory or a tunable pulse retarder to synchronize pulses in quantum information processors. Based on our proof-of principle experiment, further improvements which significantly ameliorate the device performance can be made. A similar metamaterial based on flux or fluxonium qubits, which feature a Λ-type level structure, is expected to reach the EIT regime and may ultimately realize a general purpose and on-demand superconducting quantum memory.

See the supplementary material for additional information on qubit characterization, the experimental setup, power calibration, and band structure calculations.

This work has received funding from the Deutsche Forschungsgemeinschaft (DFG) by Grant No. U.S. 18/15-1, the European Union's Horizon 2020 Research and Innovation Programme under Grant Agreement No. 863313 (SUPERGALAX), BMBF Grant “PtQUBE” via No. 13N15016, and by the Initiative and Networking Fund of the Helmholtz Association, within the Helmholtz Future Project “Scalable solid state quantum computing.” J.D.B. acknowledges financial support from Studienstiftung des Deutschen Volkes, A.S. was supported by Landesgraduiertenförderung-Karlsruhe, and A.N.P. gratefully acknowledges support from Russian Science Foundation via Grant No. 20-12-00194.