Abstract
The study of light–matter interaction has led to important advances in quantum optics and enabled numerous technologies. Over recent decades, progress has been made in increasing the strength of this interaction at the singlephoton level. More recently, a major achievement has been the demonstration of the socalled strong coupling regime^{1,2}, a key advancement enabling progress in quantum information science. Here, we demonstrate light–matter interaction over an order of magnitude stronger than previously reported, reaching the nonperturbative regime of ultrastrong coupling (USC). We achieve this using a superconducting artificial atom tunably coupled to the electromagnetic continuum of a onedimensional waveguide. For the largest coupling, the spontaneous emission rate of the atom exceeds its transition frequency. In this USC regime, the description of atom and light as distinct entities breaks down, and a new description in terms of hybrid states is required^{3,4}. Beyond light–matter interaction itself, the tunability of our system makes it a promising tool to study a number of important physical systems, such as the wellknown spinboson^{5} and Kondo models^{6}.
Main
Light propagating in a onedimensional (1D) waveguide is described by a 1D electromagnetic field with a continuous spectrum of frequencies. The strong coupling regime^{7} between an atom and such an electromagnetic continuum is defined as the regime in which the atom emits radiation predominantly into the waveguide with a rate Γ_{G} that significantly exceeds the decoherence rate of the atom as well as emission into any other channel. In this regime, the atomic transition frequency Δ far exceeds the emission rate Γ_{G} ≪ Δ. Achieving strong coupling to a continuum is a recent achievement in quantum optics^{8}. Strong atom–waveguide coupling has numerous applications, such as the development of quantum networks^{9} for quantum communication^{10} and quantum simulation^{11}. This technology, first demonstrated with superconducting qubits in open transmission lines^{8,10,12,13}, has also been implemented with both neutral atoms^{14} and quantum dots^{15} in photonic crystal waveguides. The distinctive signature of strong coupling is a decrease below 50% of the amplitude of transmitted signals due to coherent atomic scattering of photons.
A distinct regime of light–matter interaction is reached when Γ_{G} becomes comparable to the atomic transition frequency Γ_{G}/Δ ∼ 0.1, the ultrastrong coupling (USC) regime. Most studies involving atomfield interactions are in the regime Γ_{G} ≪ Δ, where the common rotatingwave approximation (RWA) applies. In the USC regime, the RWA breaks down but perturbative treatments still allow an effective atomfield description when Γ_{G}/Δ ∼ 0.1 (refs 16,17). A novel, unexplored regime of light–matter interaction is the nonperturbative USC regime, where Γ_{G} approaches or exceeds the atomic transition frequency Γ_{G}/Δ ∼ 1 and perturbation theory breaks down. This is a general definition also applicable to the case of discrete modes in cavity quantum electrodynamics systems^{18}. We note that the nonperturbative USC regime has also been referred to in the literature as the deep strong coupling regime^{19}. In the nonperturbative USC regime, the atom–photon system is described by photons dressing the atom even in the ground state^{3,4,18}. In this regime, the Markovian approximation also breaks down because the broad qubit linewidth Γ_{G} implies that the spectral density of the environment seen by the atom is not independent of frequency. The presence of a continuum of modes ultrastrongly coupled to an atom has the additional effect of renormalizing the atomic frequency from the bare value Δ_{0}, which is a generalization of the wellknown Lamb shift to arbitrary coupling strengths. These renormalization effects are also central to the wellknown spinboson model^{5}, which has been used to describe, for example, open quantum systems^{20}, quantum stochastic resonance^{21} and phase transitions in Josephson junctions^{22}. Reaching the nonperturbative USC regime allows the exploration of the ultimate limits in light–matter interaction strength and relativistic quantum information phenomena^{23}. In addition, ultrastrong couplings may have technological applications, such as singlephoton nonlinearities^{24} and broadband singlephoton sources^{3}.
Superconducting qubits are artificial atoms with transitions in the microwave range of frequencies. Recently, fluxtype superconducting qubits have been put forward as candidates to reach the nonperturbative USC regime^{3,25}, having demonstrated large galvanic couplings to resonators^{17} and a large anharmonicity that allows them to remain an effective twolevel system when Γ_{G} ∼ Δ. This is in contrast to other more weakly anharmonic qubits whose transitions would overlap for large enough Γ_{G}.
Here, we demonstrate nonperturbative ultrastrong coupling of a superconducting flux qubit^{26} coupled to an open 1D transmission line via a shared Josephson junction. As predicted^{3,25}, we observe that Γ_{G} increases with the inverse of the coupling junction size. For devices with a smallenough coupling junction we measure Γ_{G} ∼ Δ, indicating that we reach the nonperturbative USC regime. Our flux qubit has four Josephson junctions. Two reference junctions are designed with the same area, while the areas of the other two junctions are scaled by the factors α ∼ 0.6 and β > 1 with respect to the area of the reference junctions^{27}. The flux qubit is galvanically attached to the centre line of a 1D coplanar waveguide transmission line (Fig. 1a). To achieve ultrastrong couplings, we place the βjunction in parallel to the other three (Fig. 1b). The coupling to the line is then mainly determined^{3,25} by the matrix element between ground 0〉 and excited 1〉 qubit states of the superconducting phase operator across the βjunction , which is the dominant contribution to the coupling for β < 4. Further, we make the coupling tunable by turning the βjunction into a superconducting quantum interference device (SQUID) threaded by a flux ϕ_{β}, as shown in Fig. 1b (Methods).
The experiments are performed by applying a probe field with a variable frequency and recording the transmitted field amplitude and phase on a vector network analyzer. For emission rates Γ_{1}/Δ ≪ 1, where Γ_{1} is the total emission rate and, in the presence of thermal excitations, the transmitted coherent scattering amplitude at low driving power is given by^{8,28}:
Here Γ_{2} ≡ Γ_{ϕ} + (Γ_{1}/2)(1 + 2n_{th}) is the total decoherence rate, Γ_{ϕ} is the pure dephasing rate, δω = ω − Δ is the detuning of the probe field, and n_{th} is the thermal photon occupation number at the qubit frequency (Supplementary Information). The maximum reflection amplitude is r_{0} = Γ_{1}/[2Γ_{2}(1 + 2n_{th})]. As in other experiments on superconducting quantum circuits^{8,10}, relaxation into channels other than the waveguide is negligible. Therefore, we assume Γ_{1} = Γ_{G}. We note that equation (1) applies in the RWA. However, it has recently been shown^{4} that the scattering lineshapes are approximately Lorentzian in the USC regime up to Γ_{1}/Δ ∼ 1 if we consider Δ and Γ_{1} to be renormalized parameters. This can be shown using a polaron transformation, allowing us to interpret the scattering centre as an atom dressed by a cloud of photons.
We first show measurements on a device with a fixed coupling junction with β ≃ 3.5 (Fig. 1b). The transmission spectrum as a function of applied magnetic field (Fig. 2a) shows a maximum extinction at the symmetry point of 95%, indicating strong coupling. By fitting equation (1) (dashed line), we infer Γ_{1}/2π = 88 ± 11 MHz (see Methods), Δ/2π = 3.996 ± 0.001 GHz, giving Γ_{1}/Δ = 0.02, which is not in the USC regime. Flux qubit spectra in transmission lines similar to this one have previously been reported^{8,29}.
To enhance the coupling strength, we designed a second device where the size of the βjunction was decreased to β ≃ 1.8. The resulting qubit spectrum in Fig. 2b shows striking differences compared to the previous device with β ≃ 3.5. The qubit linewidth at the symmetry point is very large, comparable to the total measurement bandwidth of 3–11 GHz. The deviations from a Lorentzian lineshape are due to bandwidth limitations of our setup, still allowing us to infer a fullwidth at halfmaximum of 2Γ_{2}/2π ≃ 10.90 ± 0.44 GHz (see Methods). The extracted qubit emission rate Γ_{1}/2π ≃ 9.24 ± 0.52 GHz exceeds the qubit splitting Δ/2π = 7.68 ± 0.08 GHz, giving Γ_{1}/Δ = 1.20 ± 0.07, a clear indication that this device reaches the nonperturbative USC regime.
Having observed two devices with Γ_{1} ≪ Δ and Γ_{1} > Δ, we now explore the intermediate region using a device with tunable coupling (Fig. 1b) designed with a tunable range of β ∼ 1.6–3.6. In Fig. 3a–c, spectroscopy of the tunable coupling device is shown at three different values of ϕ_{β}. Using scanningelectron microscope (SEM) images of the measured device, we identify Fig. 3a–c as effectively having, respectively, β_{(a)} ≃ 3.6, β_{(b)} ≃ 2.3, β_{(c)} ≃ 1.6. Figure 3a corresponds to the highest effective βjunction size, therefore the lowest coupling strength. A flux qubit spectrum can be identified with Δ/2π = 5.20 ± 0.02 GHz and 2Γ_{2}/2π ≃ 2.40 ± 0.07 GHz. The maximum extinction at the symmetry point is over 95%. The quality of the signal below 4 GHz degrades due to the measurement taking place outside the optimal bandwidth of our amplifier and circulators (4–8 GHz, Supplementary Information). In Fig. 3b, the qubit gap decreases to Δ/2π ≃ 2.90 ± 0.05 GHz, as expected for a smaller βjunction. The width 2Γ_{2}/2π = 5.90 ± 0.22 GHz is clearly enhanced, with the extinction decreasing to 30%. In Fig. 3c, the qubit spectrum is barely discernible. The extinction is only 10%, with a response that appears featureless in our frequency range. Figure 3d, e shows the extracted values of r_{0} and Γ_{2} using equation (1). The value of 2Γ_{2}/2π ≃ 13 ± 3 GHz from Fig. 3c is an inferred bound due to the difficulty in fitting the transmission at this value of flux.
To understand the spectrum of the tunable coupling device and extract the corresponding emission rates Γ_{1}, we need to take into account finite temperature effects. We can set an upper bound on n_{th}, which is (Methods). Figure 3f shows that the values of n_{max} for β > 2 are consistent with a unique maximum effective temperature of T_{eff} = 90 mK, comparable to other superconducting qubit experiments. Using 0 < n_{th} < n_{max}, we then put bounds on Γ_{1}: . Using these bounds, we plot Γ_{1}/Δ in Fig. 4a. The plot clearly shows that we can tune the device from the regime of strong coupling all the way into the nonperturbative USC regime. The curve in Fig. 4a corresponds to the theoretical value of the normalized coupling strength (Supplementary Information)
with R_{Q} = h/(2e)^{2} = 6.5 kΩ the resistance quantum and Z_{0} the characteristic impedance of the line. The matrix element values of the phase operator across the coupling junction β, ϕ_{β}^{2}, are calculated using the methods of ref. 3. The observed values of Γ_{1}/Δ agree very well with the calculated values based on our circuit^{3} for an impedance close to the nominal 50 Ω. Above Γ_{1}/Δ ≃ π/2, equation (2) becomes a lower bound (Supplementary Information). This is consistent with data in the range β < 2 lying above equation (2). Including renormalization effects^{5} in equation (2) might further improve the agreement with the measurements for β < 2.
Our system allows us to explore the spinboson (SB) model in an ohmic bath. According to the SB model, the highfrequency modes of the transmission line renormalize the bare qubit splitting Δ_{0} to^{4,5}
α_{SB} is the SB normalized coupling strength that is related to the spectral density of the environment J(ω). For an ohmic system such as our transmission line, α_{SB} = J(ω)/πω. ω_{C} ≫ Δ_{0} is the cutoff frequency of the environment and p is a constant of order 1. Up to α_{SB} ∼ 0.5, we identify α_{SB} = Γ_{1}/πΔ. Above α_{SB} ≃ 0.5 (or Γ_{1}/Δ ≃ π/2) this relation becomes a lower bound for α_{SB} (Supplementary Information). In Fig. 4b we plot the experimental qubit splittings Δ (circles). Using qubit junction dimensions extracted from SEM images of the device, we diagonalize the qubit Hamiltonian at each flux ϕ_{β} (triangles) to give the bare qubit gaps Δ_{0}. We then renormalize the calculated Δ_{0} using equation (3) and a value of p = exp(1 + γ) ≃ 4.8, which is derived using an exponential cutoff model^{4,5}. γ is the Euler constant. We find the best fit to the measured Δ using a cutoff of ω_{C}/2π = 50 GHz, which is consistent with characteristic system frequencies such as the plasma frequency of the qubit junctions and the superconducting gap. The agreement between the observed qubit splittings Δ and our estimates of the renormalized gaps is clear^{3,4,5}.
As a prelude to future work, we can place our results in the context of the SB model. The SB model defines three dynamical regimes for the qubit: underdamped (α_{SB} < 0.5), overdamped (1 > α_{SB} > 0.5) and localized (α_{SB} > 1). The connection between Γ_{1}/Δ and α_{SB} allows us to draw the boundaries between these regimes in Fig. 4a. We see that our tunable device enters well into the overdamped regime, and very possibly into the localized regime for β < 2. More detailed measurements of the dynamics of the device in these regimes could further confirm the predictions of the SB model. Suggestively, the strong reduction of the qubit response seen in Fig. 3c (leftmost data points in Fig. 4a) with a flat response as a function of frequency is consistent with simulations of classical doublewell dynamics in the overdamped regime (P. FornDíaz, manuscript in preparation).
We have presented measurements of superconducting flux qubits in 1D open transmission lines in regimes of interaction starting at strong coupling and ranging deeply into the ultrastrong coupling regime. In particular, we observed qubits with emission rates exceeding their own frequency, a clear indication of nonperturbative ultrastrong coupling. These results are very relevant for the study of open systems in the USC regime, opening the door to the development of a new generation of quantum electronics with ultrahigh bandwidth for quantum and nonlinear optics applications. The tunability of our system also makes it wellsuited to the simulation of other quantum systems. In particular, we showed that the device can span the various transition regions of the SB model. With further development of our quantum circuit, the structure of the photon dressing cloud could also be directly detected, allowing the study of the physics of the Kondo model^{6} in a wellcontrolled setting. The ultrastrong coupling regime has other interesting intrinsic properties on its own, such as the entangled nature of the ground state.
Note added in proof: After acceptance of our paper, a related manuscript was published^{30} showing similar results to this work using a resonator instead of a transmission line.
Methods
Device details and fabrication.
We made the device with tunable coupling by replacing the βjunction with a SQUID threaded by a flux ϕ_{β}. The tunable coupling device then consists of two loops, the main loop that changes primarily the qubit magnetic energy through the flux ϕ_{ε} and the βloop that changes the effective coupling to the transmission line through ϕ_{β}. Changing β also modifies the minimum qubit splitting Δ. To minimize this effect, we make the SQUID junctions asymmetric, which lowers the sensitivity of Δ to ϕ_{β}. Similar tunable coupling architectures were already suggested in ref. 31. In the experiment, we sweep the global magnetic field, therefore simultaneously changing ϕ_{ε} and ϕ_{β}. The qubit spectrum shows minima near ϕ_{ε} ≍ ϕ_{0}(1/2 + n), with ϕ_{0} = h/2e the quantum of flux, n being an integer (Supplementary Information). Here, different n will correspond to different ϕ_{β}, leading to different coupling strengths. The loop areas A_{ε}/A_{β} are designed to have a large, incommensurate ratio, allowing the exploration of many different values of β.
The fabrication methods used are based on those of ref. 27. The fabrication of devices starts by patterning the transmission line using optical lithography followed by an evaporation of 200 nm of aluminium. A gap in the transmission line is left to place the qubit in a second lithography stage. We pattern the qubit using an electron beam writer. Prior to the second aluminium evaporation an Ar milling step is applied to remove the native oxide on the first aluminium layer, guaranteeing optimal conduction between the two aluminium layers. The qubit is evaporated using doubleangle shadow mask evaporation, resulting in a total thickness of 105 nm. After the first shadow evaporation step, we oxidize the film with dynamical flow at ∼0.01 mbar for 7 min, yielding critical current densities of ∼12 μA μm^{−2}. The chip is then diced and the transmission line is wirebonded to a printed circuit board connecting to the rest of the circuitry in our cryostat.
The transmission line consists of a 6.5 mm long onchip coplanar waveguide with a centre line and gaps 8 μm and 4 μm wide, respectively, resulting in a 50 Ω characteristic impedance. Numerical simulations are run to verify the impedance of the circuit. We use a squared webbed ground to reduce superconducting vortex motion on the ground plane.
Bounds on qubit emission rate.
The dependence of r_{0} and Γ_{2} on n_{th} shown below equation (1) does not allow the independent extraction of all parameters, Γ_{1}, Γ_{ϕ}, n_{th} at each value of β. However, we can set bounds on n_{th}. The lower bound case assumes no thermal excitations, therefore n_{th} = 0. If we instead set Γ_{ϕ} = Γ_{2} (1 − r_{0}(1 + 2n_{th})^{2}) ≥ 0, we identify an upper bound on the photon occupation number . In Fig. 3f, the values of n_{max} were extracted assuming Γ_{ϕ} = 0. If we were to assume Γ_{ϕ}/2π = 17 MHz as the nonthermal dephasing rate, extracted from the narrower linewidth of the device in Fig. 2a assuming n_{th} = 0, the resulting n_{th} would not differ significantly from n_{max}. Now, bounds on Γ_{1} = 2Γ_{2}r_{0}(1 + 2n_{th}) can be set as Γ_{1}(n_{th} = 0) and Γ_{1}(n_{th} = n_{max}), giving . The lower bound, n_{th} = 0, is close to the calculated value of n_{th} at the cryostat temperature of 10 mK for all qubit frequencies.
Spectroscopic analysis.
In all data shown, we use equation (1) to simultaneously fit the real and imaginary parts of the transmission. Supplementary Section 3 shows the full set of fitted resonances used in Figs 3 and 4 of the main text. Note that the baseline is fixed to a normalized value of 1 and is not adjusted. The baseline value is itself determined by measuring the transmitted background when the qubit is fluxtuned away from the frequency band of interest.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon request.
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Acknowledgements
We acknowledge financial support from NSERC of Canada, the Canadian Foundation for Innovation, the Ontario Ministry of Research and Innovation, Industry Canada, Canadian Microelectronics Corporation, EU FP7 FETOpen project PROMISCE, Spanish Mineco Project FIS201233022 and CAM Network QUITEMAD+. B.P. acknowledges the Air Force of Scientific Research for support under award FA95501210046. The University of Waterloo’s Quantum NanoFab was used for this work. We thank A. J. Leggett and A. Garg for fruitful discussions, and M. Otto, S. Chang, A. M. Vadiraj and C. Deng for help with device fabrication and with the measurement setups.
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P.F.D., C.M.W. and A.L. designed the experiment. P.F.D. designed the devices and fabricated them. P.F.D., C.M.W. and A.L. conducted the experiments. J.L.O. provided input to device design and fabrication. M.A.Y. and R.B. assisted in numerical modelling of devices. J.J.G.R. and B.P. provided theoretical support to interpret the measurements. P.F.D., C.M.W. and A.L. performed the data analysis and wrote the manuscript with feedback from all authors. C.M.W. and A.L. supervised the project.
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FornDíaz, P., GarcíaRipoll, J., Peropadre, B. et al. Ultrastrong coupling of a single artificial atom to an electromagnetic continuum in the nonperturbative regime. Nature Phys 13, 39–43 (2017). https://doi.org/10.1038/nphys3905
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DOI: https://doi.org/10.1038/nphys3905
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