Observation of quantum many-body effects due to zero point fluctuations in superconducting circuits

Electromagnetic fields possess zero point fluctuations which lead to observable effects such as the Lamb shift and the Casimir effect. In the traditional quantum optics domain, these corrections remain perturbative due to the smallness of the fine structure constant. To provide a direct observation of non-perturbative effects driven by zero point fluctuations in an open quantum system we wire a highly non-linear Josephson junction to a high impedance transmission line, allowing large phase fluctuations across the junction. Consequently, the resonance of the former acquires a relative frequency shift that is orders of magnitude larger than for natural atoms. Detailed modeling confirms that this renormalization is non-linear and quantum. Remarkably, the junction transfers its non-linearity to about thirty environmental modes, a striking back-action effect that transcends the standard Caldeira-Leggett paradigm. This work opens many exciting prospects for longstanding quests such as the tailoring of many-body Hamiltonians in the strongly non-linear regime, the observation of Bloch oscillations, or the development of high-impedance qubits.


Supplementary Note 1: Experimental setup
The measurement setup is displayed in Supplementary Figure 1. The samples are put in a dilution refrigerator with a 25mK base temperature. |S 21 | is measured using a Vector Network Analyzer (VNA). An additional microwave source was used for two-tone measurements, while a global magnetic field was applied via an external superconducting coil. Both the coil and the sample were held inside a mu-metal magnetic shield coated on the inside with a light absorber made out of epoxy loaded with silicon and carbon powder. IR filters are 0.40mm thick stainless steel coaxial cables. The bandwidth of the measurement setup goes from 2.5 GHz to 12 GHz.

Supplementary Note 2: Odd and Even modes
Our device consists of two long Josephson chains of N + 1 sites tailored in the linear regime (with Josephson energy ( /2e) 2 /L much larger than the capacitive energy) interconnected via a smaller Josephson junction or weak-link (operating in the regime of small Josephson energy E J,bare ). Linearizing the tunneling term within each chain, but keeping the non-linear coupling between them, the Hamiltonian of the system reads: withn i,σ andφ i,σ the charge and phase operators on site i ∈ [1.
.N ] and in chain σ = L, R. These operators are canonically conjugate and obey at the quantum mechanical level the commutation rules φ i,σ ,n j,σ = iδ i,j δ σ,σ . The capacitance matrices can be read off the equivalent circuit in Supplementary Figure The total capacitance at the weak-link end of the chain amounts to C I = C J + C sh + C + C g , while the capacitance at the connecting output port is C O = C c + C c,I + C.
hhhh Supplementary Figure 2. Electrical circuit of the device. The capacitance network is indicated for the output ports (in black), the two chains (in blue) and the weak link (in red).
Due to the symmetry of our device, it is useful to define respectively even and odd modes: In this basis, the Hamiltonian decomposes in two uncoupled subsystems:Ĥ =Ĥ + +Ĥ − , where: H + reduces to the Hamiltonian of a linear chain, whileĤ − takes the form of a boundary Sine-Gordon-like model.

Supplementary Note 3: Fitting the transmission resonances
The transmission spectrum consists of pairs of peaks, that are fitted according to the model described in the Methods section of the main text. Close to a pair of even/odd resonances, the transmission is given by the formula: with ω o and ω e the even/odd resonance frequencies, κ e and κ o their respective intrinsic damping rate, and κ ext the broadening due to the 50 Ω output ports. A large selection of fitted spectra (for all three samples and various temperatures) is shown in Supplementary Figure 3.

Supplementary Note 4: The self consistent harmonic approximation
The hamiltonianĤ − describes a quantum many-body problem that cannot be solved analytically, and we therefore develop here an approximate yet microscopic approach to the problem. From now on, we will discard the -index in all fields, and replaceφ 0,− byφ J . The self consistent harmonic approximation (SCHA) is used to find the approximate ground state at thermal equilibrium [1,2]. This method consist of finding the best harmonic HamiltonianĤ t which satisfies the Gibbs-Bogoliubov inequality F ≤ F t + Ĥ −Ĥ t t , where: The trial HamiltonianĤ t is defined by replacing inĤ the non-linear tunneling term −E J cosφ J by a renormalized potential E * with the capacitance matrix: where C Σ = C I + C J + C sh = 2(C J + C sh ) + C + C g , and inductance matrix: Here L * = ( /2e) 2 /E * J is an effective inductance associated with the weak link. Let us define by E k = ω k the eigenvalues ofĤ t andâ † k the corresponding creation operators associated to its normal modes. AsĤ t is harmonic, one can write: The first term is evaluated as follows: where we used the fact that k|Ĥ t d dE * . (18)

Supplementary Note 5: Microscopic model
Let us now compute e iφJ t using Eq. (13) and the Baker-Campbell-Hausdorff formula : The terms where n = m are the only one different from 0 : Wick's theorem has been used between Eq. (21) and Eq. (22), and n k = 1/[exp( ω k /k B T ) − 1] is the Bose factor. One verifies easily that e −iφJ t = e iφJ t . We can finally simplify the term appearing in Eq. (18): We finally present the procedure to compute the normal mode expansion coefficients φ k , as obtained from the trial HamiltonianĤ t . The original charge and phase variables can be decomposed formally onto the normal modes: By imposing the canonical commutation relation for the bosonic operators and [φ p ,n m ] = iδ pm , we obtain the following normalization condition on the matrices [R] and [G] : Using Eq. (26) and Eq. (27) inĤ t , we obtain : In order to recover the usual harmonic form (12) ofĤ t , we firstly impose: with ω p = [Ω] pp . Once the [L −1 C −1 ] eigenvalue problem has been numerically solved, we can express the phase across the weak link in terms of the normal mode amplitudes so that the final self-consistent equation for E * J is : In practice, we determine E * J from the Hamiltonian formalism described here. Once the value has been determined (which in general depends also on temperature), it can be inserted in a full ABCD calculation [3], since the effect of the capacitive coupling to the output ports is very small in practice.

Supplementary Note 6: Phase shift induced by the small Josephson junction
Now that we have obtained the best harmonic approximation ofĤ by solving (36) self-consistently, we can investigate the effect of the small junction on the odd modes with respect to decoupled even modes. In frequency domain, the equations of motion for the classical phases φ j,− are given by : with [L −1 ] and [C] the inductance and capacitance matrices for the odd modes, and the columns of the matrix [φ] tabulate the phase configuration for different frequencies. The even modes form stationary cosine waves along the chain: with l = 0, 1, 2... the position in the chain and k the wavenumber. The dispersion relation reads with ω p = 1/ √ L(C + C g /4) the plasma frequency of the chain. In presence of the small junction (treated at the SCHA level), the odd modes have the same dispersion relation but experience an additional phase shift θ (we omit in our notation the fact that θ = θ k depends implicitely on k): The phase shift is determined from equation of motion that links sites 0 and 1 which we can rewrite using Eq. (40) as In the case where the junction is saturated (either at strong driving power, or for large thermal fluctuations), we have E * J = 0, and we use: Solving for θ, we find Supplementary Note 7: Splitting between odd and even modes Now that we have the analytic expression (46) for the phase shift induced by the small non-linear junction, we will see how it translates into the splitting between odd and even modes. For simplicity, we will assume here that C c and C c,I are big enough so that we can consider the last site N as grounded: with θ n the phase shift for the mode n, so that : with k • n the wave vector of the mode n in the bare chain (corresponding to the uncoupled even modes in the experiment). Using the dispersion relation, we find at order 1/N : We also have for the bare modes: Using Eq. (50) and Eq. (51), we obtain the connection between the relative odd-even splitting S induced by the small junction on the odd modes and the associated phase shift θ n on mode n: To make sure that approximating the site N as grounded is valid, we computed numerically the exact splitting obtained with and without these pads, using a full ABCD matrix calculation (shown in Supplementary Figure 4 with the parameters of sample B), and found very little effect of this approximation. In addition, we find that the theoretical phase shift Eq.  In the infinite system, the phase shift θ becomes a continuous function of frequency ω. It vanishes at the renormalized frequency of the weak link, as can be seen as follows. When θ = 0, Eq. (46) can be rewritten as From the definition of X follows that cot k = (X 2 − 1)/2X, and furthermore, that Using the definition (44) of λ and that of C Σ , we reduce Eq. (54) to implying that and hence ω = 1/ L * (C J + C sh ) ≡ ω * J .

Supplementary Note 8: Fitting the experimental splittings
We present in Supplementary Figure 5 the frequency-dependent splittings extracted from the analysis of the evenodd mode pairs (see Supplementary Figure 3), shown as dots for our three samples and various temperatures. Each of this data set is then fitted to the analytical formula (46), L * (T ), or equivalently E * J (T ) being the fitting parameter. The range of investigated temperature is restricted below 130 mK, since at too high temperatures, thermal fluctuations are so strong that the SCHA treatment breaks down. We find in Supplementary Figure 5 that the lineshape of the splitting is well reproduced by our calculations. The location of the zero of the splitting also allows to extract the value of the renormalized frequency ω * J of the small junction, a key quantity that is discussed in detail in the main text. Supplementary Note 9: Estimation of the shunting capacitance To determine a value of the unknown shunting capacitance C sh , we devised an original saturation technique. At high enough power, the fluctuations across the small junction can be so large that E * J renormalizes to zero, decoupling effectively the dynamics of the two chains, except for the remaining effect of C sh and C J . We can thus use formula (45), and since C J is known by design, one can directly infer C sh from an analysis of the even-odd splitting at high power. The evolution of the transmission as a function of power, and the resulting splittings are shown in Supplementary  Figure 6. From that measurement one can infer that C sh slightly increases (see Table I in the main text) when the size of the junction is increased, which is the expected behavior.
with Φ C the flux in the SQUID loops and d the asymmetry of the SQUID junctions [2]. As we can neglect the effect output port capacitances, the dispersion relation of the even modes is given by (39), which can be expressed as a function of ω: From Eq. (58), the free spectral range (namely the energy difference between two consecutive modes) is decreasing when Φ C /Φ 0 goes to π/2. This behavior is clearly seen in Supplementary Figure 7. One can also notice the absence of artifacts around Φ C = 0, which means that the chain is homogeneous and relatively exempt of disorder. By doing a two-tone spectroscopy at Φ C = 0, we can measure precisely the dispersion of the even modes up to 14GHz. From Eq. (59), we find C g and L for the three sample, C being known by design. This method allows an in-situ determination of the chain parameters.