Fig. 2: Single Coupler Behavior.

a The ratio of the Josephson inductance to the geometrical inductance, β_c, dictates the shape of the potential energy of the tunable rf-SQUID coupler circuit. When the geometrical inductance dominates, β_c ≪ 1, the potential energy landscape is approximately harmonic. When the Josephson inductance dominates, β_c ≫ 1, the energy landscape becomes double-welled with each minimum representing oppositely circulating current states. Due to the large energy barrier between the states, moderate changes in f_z do not change the current state of the circuit. The coupling is optimized when the geometrical and Josephson inductances are approximately equal. This results in a wide, shallow energy minimum where even slight changes in f_z can induce strong fluctuations between the oppositely circulating current states of the coupler circuit. b Both the single coupler transverse field, Δ_c/2, and the intercoupler interaction energy, \({J}_{{{{{\rm{c}}}}}_{i}{{{{\rm{c}}}}}_{i+1}}={M}_{{{{{\rm{c}}}}}_{i}{{{{\rm{c}}}}}_{i+1}}{I}_{{{{{\rm{c}}}}}_{i}}^{z}{I}_{{{{{\rm{c}}}}}_{i+1}}^{z}\), are displayed as a function of the coupler f_x when f_z = Φ₀/2. These parameters are calculated in single coupler simulations and then transcribed into spin model parameters. The equality of these two terms appearing in the transverse field Ising model for f_x ≃ 0.14 Φ₀ signals the location of the quantum critical point, in the vicinity of which we expect long-range correlations to emerge. In addition, the dependence of β_c is displayed as a function of the coupler's f_x for f_z = Φ₀/2. By design, the optimum coupling point, β_c ≈ 1, coincides with the coupler f_x value where we expect critical behavior in the coupler chain.