Abstract
The interaction between light and matter remains a central topic in modern physics despite decades of intensive research. Coupling an isolated emitter to a single mode of the electromagnetic field is now routinely achieved in the laboratory, and standard quantum optics provides a complete toolbox for describing such a setup. Current efforts aim to go further and explore the coherent dynamics of systems containing an emitter coupled to several electromagnetic degrees of freedom. Recently, ultrastrong coupling to a transmission line has been achieved where the emitter resonance broadens to a significant fraction of its frequency, and hybridizes with a continuum of electromagnetic (EM) modes. In this work we gain significantly improved control over this regime. We do so by combining the simplicity and robustness of a transmon qubit and a bespoke EM environment with a high density of discrete modes, hosted inside a superconducting metamaterial. This produces a unique device in which the hybridisation between the qubit and many modes (up to ten in the current device) of its environment can be monitored directly. Moreover the frequency and broadening of the qubit resonance can be tuned independently of each other in situ. We experimentally demonstrate that our device combines this tunability with ultrastrong coupling and a qubit nonlinearity comparable to the other relevant energy scales in the system. We also develop a quantitative theoretical description that does not contain any phenomenological parameters and that accurately takes into account vacuum fluctuations of our large scale quantum circuit in the regime of ultrastrong coupling and intermediate nonlinearity. The demonstration of this new platform combined with a quantitative modelling brings closer the prospect of experimentally studying manybody effects in quantum optics. A limitation of the current device is the intermediate nonlinearity of the qubit. Pushing it further will induce fully developed manybody effects, such as a giant Lamb shift or nonclassical states of multimode optical fields. Observing such effects would establish interesting links between quantum optics and the physics of quantum impurities
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Introduction
Due to strong interactions between elementary constituants, correlated solids^{1} and trapped cold atoms^{2} host fascinating manybody phenomena. Attempts to produce similar effects in purely optical systems are hampered by the obvious fact that photons do not naturally interact with each other. If this obstacle can be overcome, there is the tantalizing prospect of probing the manybody problem using the contents of the quantum optics toolbox, such as single photon sources and detectors, highorder correlations in timeresolved measurements, entanglement measures, and phase space tomographies to name a few^{3}.
One route to building a manybody quantum optical system is to rely on arrays of strongly nonlinear cavities or resonators^{4,5,6}, but minimising disorder in such architectures is a formidable challenge. Another route that circumvents these difficulties involves coupling a single wellcontrolled nonlinear element to a disorder free harmonic environment^{7}^{–}^{11}. If the difficult experimental challenge of engineering an ultrastrong coupling can be overcome^{12}^{–}^{17}, thus exceeding the boundaries of the standard single photon regime in quantum optics, this approach could pave the way to bosonic realizations of electronic impurity systems such as the famous Kondo and Anderson models^{4,18,19,20,21,22,23}. Our goal here is to achieve a large coupling between a sufficiently nonlinear qubit and a quantum coherent environment containing many harmonic degrees of freedom^{24–26}.
When coupling an impurity to a finite size electromagnetic environment, five important frequency scales have to be considered. The impurity is characterized by its qubit frequency ω_{qubit}, i.e., the excitation frequency between its two lowest internal states. Real impurities always possess more than two levels. The anharmonicity α, defined as the difference between ω_{qubit} and the frequency for excitation from the second to third internal impurity state, characterises the departure of an impurity from a trivial harmonic oscillator (α → 0) or a pure twolevel system (α → ∞). The coupling between the impurity and environment is characterized by the spontaneous emission rate Γ at which the impurity exchanges energy with its environment. The environment itself is characterized by its free spectral range δω, which measures the typical frequency spacing between environmental modes, and the spectral broadening κ of these modes due to their coupling to uncontrolled degrees of freedom. The soughtafter multimode regime is obtained when Γ is larger than δω so that the impurity is always coupled to several discrete environmental modes, producing a cluster of hybridized qubitenvironment resonances^{27}. There are several requirements for reaching the manybody regime. First, Γ must be a significant fraction of ω_{qubit} (ultrastrong coupling). This is a prerequisite for multiparticle decay^{19,22}. If the coupling is too weak, the system becomes trivial, since only numberconserving processes are relevant (Equivalently Markov and rotating wave approximations apply.) A second requirement is \(\alpha >rsim {\mathrm{\Gamma }}\). If this condition is not met, the nonlinearity of the impurity is swamped by the broadening of the impurity levels, and the same frequency will drive transitions between several impurity levels, so that the system as a whole behaves more like a set of coupled harmonic oscillators than like a twolevel system coupled to an environment^{28}. Within the manybody regime, two limits can be distinguished. In the case of a finitesize environment that we address here (namely, δω > κ), each mode of the system can be addressed and controlled individually, while in the limit of a thermodynamically large environment (δω/κ → 0) one recovers a smooth dissipationbroadened qubit resonance.
The manybody ultrastrong coupling regime (defined by the first two conditions above) is hard to reach in quantum optics experiments because the coupling to threedimensional vacuum fluctuations arises at order [α_{QED}]^{3}, with \(\alpha _{{\mathrm{QED}}} \simeq 1/137\) the fine structure constant^{3}. However, for superconducting qubits coupled to transmission lines, the scaling is much more favorable^{20,29–31} than in a vacuum. Indeed the ratio Γ/ω_{qubit} can essentially be made arbitrarily large, provided the impedance of the environment matches that of the qubit (see Sec. E of the Supplementary Information). Building on this ability of superconducting circuits to reach very large couplings, several experiments demonstrated the ultrastrong coupling regime in coupled qubit/cavity systems^{14,15,16,32}. The rich physics associated to this coupling regime has also been evidenced using quantum simulation^{33,34}. The condition Γ > δω has also been fulfilled by coupling superconducting qubits to open transmission lines^{35,36,37} or engineered resonators^{7,9}. However, it is only recently that the necessary conditions for the manybody regime were demonstrated concurrently^{12,13.} The device of^{12,13} consists of a flux qubit coupled to the continuum provided by a superconducting transmission line, which realises the thermodynamic limit (δω/κ → 0). Limitations of such setups include the lack of a microscopic model (since it is hard to characterize the waveguide properties of a transmission line outside the relatively narrow 4–8 GHz band where microwave transmission experiments can comfortably be performed), and importantly, that the transmission line is not an in situ tunable environment.
Results and discussion
In this work, we circumvent the above limitations, by designing circuits that provide independent tunability of both a qubit and a finite size but very large environment, while allowing highprecision spectroscopic measurements of the environment itself (δω > κ) and first principle modeling. Our device, shown in Fig. 1b, consists of a transmon qubit, which is relatively insensitive to both charge and flux noise, capacitively coupled to a long onedimensional Josephson metamaterial, comprising 4700 SQUIDs. Such chains have been studied since the early 90’s in the context of the superconductorinsulator transition^{38,39,40,41} or to explore dual of the Josephson effect^{42,43,44}. Our setup differs in two ways from these previous works. First, we took great care to produce a chain in the linear regime, far from the onset of nonlinear effects such as quantum phase slips^{45}, so that one of the basic benefits of quantum optics, i.e., the elimination of nonlinearities where they are not wanted, is realized. Second, we performed AC microwave spectroscopy of our device, instead of DC transport measurements. This allows us to characterize the electromagnetic degrees of freedom, also called MooijSchön plasma modes^{28,46,47,48,49,51}, microscopically. We managed to resolve as many as 50 individual low frequency electromagnetic modes of this nondissipative and fully tunable environment (see Fig. 2c). An essential property of the Josephson metamaterial is its high characteristic impedance \(Z_c = \sqrt {L_J/C_{\mathrm{g}}} \simeq 1590 \mathrm{\ \Omega}\), which being of the same order of magnitude as the effective impedance of our transmon qubit, \(Z_T = \hbar /(2e)^2\sqrt {2E_{C,{\mathrm{T}}}/E_{J,{\mathrm{T}}}} \simeq 760\,{\mathrm{\Omega }}\) (here at zero flux), allow us to reach multimode ultrastrong coupling. The simplicity of the transmon architecture enables us to either compute from first principles or to extract all the parameters necessary to construct a microscopic model of the full system, without dropping the socalled ‘A^{2}terms’, a routine approximation in optics^{52,53,54,55,56}, that however breaks down at ultrastrong coupling.
Our measurements are based on the frequencyresolved microwave transmission through the whole device. Figure 2b shows the amplitude of the transmitted field at a fixed value of the external magnetic field, and at low probe power. This spectrum reveals a set of resonances in our device, displaying a narrow spectral broadening κ/2π = 20 MHz (for the non hybridized modes of the chain) caused by the coupling to the 50 Ω contacts. As the external magnetic field is varied, two modulation periods are seen in the resonance spectrum (Fig. 2a). The short and long periods correspond respectively to a one quantum Φ_{0} = h/2e increase of the flux through the large transmon or through the small chain SQUID loops. This feature allows us to adjust independently the flux threading the transmon SQUID loop (Φ_{T}) from the one threading the chain SQUID loops (Φ_{C}). The former controls the qubit frequency, while the latter controls the impedance of the environment, and hence the qubitenvironment coupling strength. Before studying the hybridization between the transmon and chain, we characterize the chain on its own by setting Φ_{T} = Φ_{0}/2, so that the qubit decouples from it (See Fig. 2c and Sec. B of the Supplementary Information). We obtain a good fit between the dispersion relation predicted by a microscopic model and the one extracted from the measured resonances. This allows us to extract all of the parameters necessary to characterize the chain modes.
The transmon qubit becomes active when \(\Phi _{\mathrm{T}}\not = \Phi _0/2\), and is expected to hybridize with the chain. Figure 3a shows a lowpower spectroscopy of the system as a function of Φ_{T}, keeping Φ_{C} nearly constant. When a chain mode is not hybridized with the qubit, the corresponding spectral line runs nearly horizontally. When Φ_{T} is varied, the qubit frequency sweeps across the resonances of the chain modes and creates a clear pattern of several avoided level crossings (See Fig. 3a). We note that at fixed Φ_{T}, several chain modes in the vicinity of the transmon resonance are visibly displaced. Thus, the transmon simultaneously hybridizes with many modes. This is a signature of multimode ultrastrong coupling, a topic that will be further addressed below. Evidence that the transmon behaves as a qubit is provided by its saturation spectrum (Fig. 3b). Here the fluxes Φ_{T} and Φ_{C} are kept constant, while the transmission through the system is recorded at increasing probe power. For a harmonic system, the resonance positions are independent of driving power. In an anharmonically oscillating classical system, a gradual dependence on the driving power would appear. Experimentally, we observe that below a driving power ~−10 dBm, the resonance positions in the transmission spectrum are independent of the driving probe power. As the driving power increases beyond −10 dBm, the peak around 4.5 GHz disappears while the other peaks assume the positions they have when the qubit is inactive. (The horizontal lines correspond to the low power spectrum when Φ_{T} = Φ_{0}/2.) This is evidence of saturation, a clear qubit signature^{18,35,36}. The fact that several chain modes are shifted when the transmon is saturated constitutes an additional proof that the transmon hybridizes with many modes at once.
To quantify the hybridization of the transmon mode with the chain modes, we compare the normal mode spectrum of the full system at Φ_{T} ≠ Φ_{0}/2 to the spectrum at Φ_{T} = Φ_{0}/2. As mentioned above, in the latter case, the transmon decouples from the bare modes of the chain. The system with Φ_{T} ≠ Φ_{0}/2 therefore has one extra mode in the vicinity of the transmon frequency. We define the relative frequency shift δϕ_{n}(Φ_{T}, Φ_{C}) as the difference in frequency between the nth mode of the coupled and uncoupled chains, normalized to the free spectral range of the chain δω_{n}(Φ_{T}, Φ_{C}) = ω_{n}(Φ_{0}/2, Φ_{C}) − ω_{n−1}(Φ_{0}/2, Φ_{C}), and including a π factor for later convenience, i.e.,
where ω_{n}(Φ_{T}, Φ_{C}) is the frequency of the nth lowest nonzero mode for a given flux in the transmon and in the chain. This frequency shift is readily extracted from the peak positions in our global spectroscopic map (Fig. 2a). Remarkably (see Sec. H of the Supplementary Information for a derivation), δϕ_{n}(Φ_{T}, Φ_{C}) in Eq. (1) equals the phase shift experienced by mode n due to the presence of the nearby transmon mode:
where ϕ_{n}(Φ_{T}, Φ_{C}) is the phase shift of mode n of the full system at transmon flux Φ_{T} and chain flux Φ_{C}. From Eq. (1) it follows that the phase shift equals 0 (resp. π) for modes far below (resp. far above) the renormalized transmon frequency. For hybridized modes in the vicinity of the transmon line, δϕ_{n}(Φ_{T}, Φ_{C}) lies between 0 and π. This behavior is clearly observed in Fig. 4a where the measured relative frequency shifts are reported for a chain flux Φ_{C} = 0 and various transmon fluxes Φ_{T}. The wide frequency dispersion of intermediate δϕ_{n}(Φ_{T}, Φ_{C}) provides direct evidence for a hybridization with up to ten chain modes. In the thermodynamic limit of an infinite chain with perfect impedance matching to the measurement ports, the transmoninduced phase shift δϕ_{n}(Φ_{T}, Φ_{C}) becomes a continuous function δϕ(ω, Φ_{T}, Φ_{C}) of the mode frequency ω. Moreover, it can be shown that the frequency derivative of δϕ(ω, Φ_{T}, Φ_{C}) matches very precisely the theoretically expected lineshape of the dissipative response of the transmon coupled to an infinite environment. (See Sec. I of the Supplementary Information). This constitutes a central finding of our work: the renormalized transmon frequency ω_{T} and linewidth Γ_{T} can be directly inferred from a measurement of the phase shifts of the individual modes in the finite bath. In terms of measurement protocol however, there is a sharp difference between the chain mode phase shifts and the qubit response functions. Usually, the qubit response is obtained by observing the transmon, and its environment can be viewed as a black box that combines unmonitored decoherence channels as well as the physical ports used for measurement. This procedure constitutes the usual paradigm in the study of open quantum systems. Our protocol is unusual because information about an open quantum system is obtained by monitoring the discrete modes that constitute its dominant environment.
We finally turn to a quantitative analysis of our data, including a comparison to the predictions of a microscopic model, in order to determine if the requirements for reaching the manybody regime has been met. By extracting the maximum renormalized transmon frequency ω_{T,max} extracted from the phase shift data of Fig. 4a at chain flux Φ_{T} an integer multiples of Φ_{0}, we are able to infer the only remaining unknown system parameter, namely the maximum transmon Josehpson energy E_{J,T,max}. This allows us to estimate the anharmonicity of the transmon α, which ranges from 0.36 GHz at Φ_{T} = 0 to 0.44 GHz at Φ_{T} = 0.3Φ_{0}. We emphasize that the condition \(\alpha >rsim {\mathrm{\Gamma }}_{\mathrm{T}}\) for anharmonic manybody behavior is thus fulfilled (see Fig. 4b for the extracted transmon linewidth Γ_{T}, which lies in the range 0.2–0.4 GHz). Using the extracted parameters to calculate δϕ(ω, Φ_{T}, Φ_{C}) according to Eq. (11) we find the predicted theoretical lines in Fig. 4a. The excellent agreement between theory and experiment seen here for six different values of Φ_{T} persists for each of the hundreds of (Φ_{T}, Φ_{C}) combinations where we have made the comparison. (See Sec. C of the Supplementary Material for a further selection of results.) We stress that this agreement is obtained after all model parameters have been fixed, so that there is no fitting involved in comparing the predicted and measured phase shifts. The quantitative modeling of such a large quantum circuit clearly is an important landmark in the field of open quantum systems. In Fig. 4b we examine the transmon linewidth Γ_{T} that we extracted from the phase shift data, as a function of chain flux Φ_{C} for fixed transmon flux Φ_{T} = 0. Very good agreement (with no fitting parameters) is again obtained with the prediction of our model. These results demonstrate that we can tune the qubitenvironment coupling independently from ω_{T} using the flux in the chain, and that we achieved the ultrastrong coupling in our waveguide, i.e., coupling to a large number (here 10) of modes with a sizeable linewidth \({\mathrm{\Gamma }}_{\mathrm{T}}/\omega _{\mathrm{T}} \simeq 0.1\). A hallmark of ultrastrong coupling is the failure of the rotating wave approximation (RWA), as previously discussed in coupled qubit and cavity systems^{14}. We have examined the consequences of the RWA on our microscopic model (see Sec. J of the Supplementary Information), and found a discrepancy of 100 MHz in the transmon frequency ω_{T}, showing the quantitative importance of nonRWA terms. We would like to stress that demonstrating the relevance of these terms is much more than obtaining a good datatheory agreement. With counterrotating contributions in the few percent range, we expect a finite rate for parametric processes in which photonnumber is not conserved. In future we plan to use the current platform to observe these interesting manybody effects directly.
In conclusion, this work provides the first demonstration of manybody ultrastrong coupling between a transmon qubit and a large and tunable bath. To obtain full control over the environment, a superconducting metamaterial, comprising 4700 SQUIDs, was employed. Although this quantum circuit contains a huge number of degrees of freedom, we were able to characterise all its parameters in situ. This allows us to demonstrate unambiguously that our systems meets the three conditions required to reach the manybody regime, namely \({\mathrm{\Gamma }} >rsim \delta \omega\), \({\mathrm{\Gamma }} >rsim 0.1\omega _{{\mathrm{qubit}}}\), and \(\alpha >rsim {\mathrm{\Gamma }}\). A novel experimental methodology was implemented to analyze the qubit properties by means of the extraction of the phase shifts of the environmental modes. Despite the large size of our quantum circuit, we succeeded in providing a fully microscopic model which accounts for the transmon response without any fitting parameters. We also found that the qubit linewidth for the long chain agreed with results in the thermodynamic limit, showing that the finite environment has the same influence on the qubit as a truly macroscopic bath. The further possibility to tune the coupling to the environment in situ, demonstrated by a 50% fluxmodulation of the qubit linewidth, opens the way to controlled quantum optics experiments where manybody effects are fullydeveloped^{18,19,21,22,57,58}, as well as more advanced environmental engineering for superconducting qubits^{5}.
Methods
Sample fabrication and parameters
The sample was fabricated on a highly resistive silicon substrate, using a microstrip geometry. The ground is defined as the backside of the wafer which was goldplated, ensuring a good electrical conductivity. Interdigital capacitances were chosen to connect the transmon and the metamaterial. They do not provide the lowest surface participation factor^{59,60} but they allow us to maximize the coupling capacitances C_{c} of the transmon to the chain, while minimizing the capacitances to ground C_{g,T} and C_{g,T2}. (See panel a of Fig. 1 for the definitions of the these capacitances.) This system is probed via two 50 Ω transmission lines, one of which is capacitively coupled to the transmon, while the other is galvanically coupled to the chain. The whole device (Josephson junctions, capacitances and transmission lines) was fabricated in a single electronbeam lithography step, using a bridgefree fabrication technique^{61}. The Josephson elements of the chain are tailored to be deep in the linear regime (E_{J}/E_{C} = 8400), where E_{J} and E_{C} are the respectively the Josephson energy at Φ_{C} = 0 and the charging energy of a chain element), leaving the transmon as the main source of nonlinearity in the system. All parameters of the system are listed in Table 1. Here, L_{J,min} is the minimum inductance of a chain SQUID loop, (which occurs when Φ_{C} = 0). There is a slight asymmetry between the two Josephson junctions that constitute a single chain SQUID loop, which is quantified by the asymmetry parameter d. The fluxdependent inductance L_{J}(Φ_{C}) of a chain SQUID loop is given by
and the corresponding Josephson energy by \(E_J({\mathrm{\Phi }}_{\mathrm{C}}) = \varphi _0^2/L_J({\mathrm{\Phi }}_{\mathrm{C}})\) with φ_{0} = ℏ/2e the reduced flux quantum. The transmon Josephson junctions are symmetric, and hence the flux dependent transmon Josephson energy is given by
The transmon charging energy is not an independent parameter (see Eq (9)), but is listed in the table, due to the prominent role it plays in what follows. The meanings of the remaining parameters in Table 1 are explained in panel a of Fig. 1. The majority of the parameter values listed are obtained either using a finiteelement solver (Sonnet) or extracted from the measured dispersion relation of the chain at Φ_{T} = 0. (See Sec. B and D of the suplementary Material.) The only exceptions are C_{J}, that is obtained from knowledge of the junction areas via an empirical formula, and E_{J,T,max} that we extract via a procedure described in the last subsection below, which uses the data at Φ_{T} equal to multiples of Φ_{0} in panel a of Fig. 2.
Full model
The circuit diagram for the lumpedelement model is shown in Fig. 1. It consists of N + 2 nodes, where N is the number of SQUIDs in the chain. To describe the circuit, we use the Cooper pair number operator \(\hat n_j\), which gives the number of Cooper pairs in node j, and the superconducting phase operator \(\hat \varphi _j\), which gives the superconducting phase at node j. They satisfy the canonical commutation relations \([\hat n_j,\hat \varphi _l] = i\delta _{jl}\)^{62}. Here j, l ∈ [L, R, 1, 2, 3,…,N] with L and R referring to the left and right transmon nodes. As explained before, the SQUIDs of the chain are linear inductors, to a very good approximation. We define \(\hat n^T = (\hat n_L,\hat n_R,\hat n_1, \ldots ,\hat n_N)\) and \(\hat \varphi ^T = (\hat \varphi _L,\hat \varphi _R,\hat \varphi _1, \ldots ,\hat \varphi _N)\). In this notation, the Hamiltonian of the circuit is given by
\(\hat C\) is the capacitance matrix, such that elements \([\hat C]_{jl} = [\hat C]_{lj}\) equal the capacitive coupling between the charges on islands j and l. In the same way, elements \([\hat J]_{jl} = [\hat J]_{lj}\) of matrix \(\hat J\) contains the Josephson energy that couples the superconducting phase on island j and island l. Both matrices are (N + 2) × (N + 2). Their explicit forms are given below in Eqs. (6) and (7). In both matrices, the boundary conditions that determine entries (L, L) and (N, N) are obtained by assuming that the nodes to the left of node L and to the right of node N are grounded.
The elements in the capacitance matrix are given by
We define the operators \(\hat n_{\mathrm{T}} = (\hat n_R  \hat n_L)/2 +\) constant and \(\hat \varphi _{\mathrm{T}} = (\hat \varphi _R  \hat \varphi _L)/2\) associated with the transmon dynamics. Introducing these operators, and noting that the total transmon charge \(\hat n_R + \hat n_L\) is concerved, we can rewrite the Hamiltonian as
where we defined \(\hat \varphi _{N + 1} \equiv 0\). The transmon charging energy E_{C,T} is given by
The coupling of \(\hat n_{\mathrm{T}}\) to the charge on island j is given by
Chain modes phase shift in the thermodynamic limit
In Fig. 4a we compare the measured relative frequency shift δϕ_{n}(Φ_{T}, Φ_{C}) to the theoretically predicted transmon phase shift δϕ(ω, Φ_{T}, Φ_{C}) with which it is expected to agree in the thermodynamic limit. Here we provide the analytical formula for the phase shift ϕ(ω, Φ_{T}, Φ_{C}) of a mode with frequency ω. (See Sec. G of the Supplementary Information for the derivation.) It reads
In this expression \(\omega _{\mathrm{p}}({\mathrm{\Phi }}_C) = 1/\sqrt {L_J({\mathrm{\Phi }}_{\mathrm{C}})\left( {C_J + C_{\mathrm{g}}/4} \right)}\) is the plasma frequency of the chain, and
has dimensions of capacitance. Finally, E_{S}(Φ_{T}) is an effective linear inductor energy associated with the Josephson junctions in the transmon, which nonetheless incorporates the transmon nonlinearity, and is given by
(See Sec. F of the Supplementary Material for further detail.) The theoretical δϕ(ω, Φ_{T}, Φ_{C}) curves plotted in panel a of Fig. 4 were obtained from ϕ(ω, Φ_{T}, Φ_{C}) similarly to Eq. (2) as the difference
Analysis of the experimental data
To extract the relative frequency shift δϕ_{n}(Φ_{T}, Φ_{C}) from the data presented in Fig. 2 of the main text, we go about as follows. At a fixed value of the magnetic field that determines Φ_{T} and Φ_{C}, we fit each of the peaks in the transmission spectrum individually with a Lorentzian. This gives the center frequency of the peaks. From these peak positions, we obtain δϕ_{n}(Φ_{T}, Φ_{C}) experimentally using Eq. (1) at a particular Φ_{T} and Φ_{C}. Next we extract the transmon frequency ω_{T} for the transmon coupled to the chain from the experimentally determined δϕ_{n}(Φ_{T}, Φ_{C}). The details are as follows. Empirically, we find that the experimentally determined δϕ_{n}(Φ_{T}, Φ_{C}) vs. ω_{n} datapoints fit an arctangent lineshape
very well, for suitable choices of the parameters A, ω_{T}, and Γ_{T}. (In the parameter regime where our device operates, the theoretically predicted phase shift δϕ(ω, Φ_{T}, Φ_{C}) also closely approximates this line shape.) We therefore fit the measured δϕ_{n} vs. ω_{n} at fixed Φ_{T} and Φ_{C} to Eq. (15), interpreting ω_{T} as the frequency and Γ_{T} as the resonance width of the transmon when it is coupled to the chain. Before we can quantitatively compare the experimental results for δϕ_{n}(Φ_{T}, Φ_{C}) to the theoretically predicted δϕ(ω, Φ_{T}, Φ_{C}) (Eq. (11)), one final model parameter, namely the maximum transmon Josephson energy E_{J,T,max} must be determined from the experimental data. The general procedure is as follows. Our theoretical model predicts that in the regime where the actual device operates, this transmon frequency is very nearly equal to the isolated (L_{J} → ∞) transmon frequency, i.e., the chain only slightly renormalizes the transmon frequency, and indeed, we see little Φ_{C} dependence in the extracted ω_{T}. At Φ_{T} = nΦ_{0}, n = 0, ±1, ±2,…, where the transmon Josephson energy is maximal, we therefore use the isolated transmon result (see Sec. F of the Supplementary Information):
Taking the average over n of the experimentally determined ω_{T}(Φ_{T} = nΦ_{0}), and using our first principle estimate for E_{C,T} in Table 1, we obtain
The theoretical curves in Fig. 4a were then obtained using the system parameters in Table 1 in Eqs. (11), (14), (13), and (4). The full data set covers many transmon periods. Within a given transmon period, we generally analyze data at several values of Φ_{T} in the interval from −0.3 ϕ_{0} to 0.3 ϕ_{0}. Each transmon period is measured at different Φ_{C}. We also take into account the small variation in Φ_{C} as the transmon flux sweeps through one flux quantum. The experimental points in Fig. 4b are obtained as the transmon width Γ_{T} closest to Φ_{T} = 0. The error bars come from the least square fit using Eq. (15). The theoretical width is obtained from a fit of the phase shift δϕ(ω, Φ_{T}, Φ_{C}) with the arctangent of Eq. (15).
Acknowledgements
The authors would like to thank F. Balestro, D. Dufeu, E. Eyraud, J. Jarreau, J. P. Girard, T. Meunier, and W. Wernsdorfer, for early support with the experimental setup. Very fruitful discussions with C. K. Andersen, H. U. Baranger, D. Basko, S. Bera, J. J. Garcia Ripoll, S. M. Girvin, F. Hekking, K. Le Hur, C. Mueller, A. Parra, and E. Solano are strongly acknowledged. The sample was fabricated in the clean rooms Nanofab and PTA (Upstream Technological Platform).This research was supported by the ANR under contracts CLOUD (project number ANR16CE240005), GEARED (project number ANR14CE260018), by the UGA AGIR program, by the National Research Foundation of South Africa (Grant No. 90657), and by the PICS contract FERMICATS. J.P.M. acknowledges support from the Laboratoire d’ excellence LANEF in Grenoble (ANR10LABX5101). R.D. and S.L. acknowledge support from the CFM foundation. N.G. acknowledges support from the Fondation Nanosciences de Grenoble.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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J.P.M., S.F., and N.R designed the experiment. J.P.M. and N.R. fabricated the device. J.P.M. and S.L. performed the experiment and analysed the data with help from S.F, N.R., and I.S., while S.F. and I.S. provided the theoretical support. All coauthors participated in setting up the experimental platform, and took part in writing the paper.
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Puertas Martínez, J., Léger, S., Gheeraert, N. et al. A tunable Josephson platform to explore manybody quantum optics in circuitQED. npj Quantum Inf 5, 19 (2019). https://doi.org/10.1038/s4153401801040
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DOI: https://doi.org/10.1038/s4153401801040
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