Figure 2: 〈J_z〉’s signature behaviours across the critical point in the ground-state quantum phase transition and in the experimental dynamics, calculated with Δ_r/2π=30 MHz and Ω/2π=4 MHz for example.

The quantum critical region, illustrated by the light-green background in all panels, happens around λ/λ_c=1 between the normal (n., the white region) and super-radiant (s., the green region) phases. (a) Numerical calculations of 〈J_z〉/(N/2) by solving equation (1) for the ground state at different number of qubits N as indicated. The cusp-like behaviour at the critical point λ/λ_c=1 occurs only in the thermodynamical limit, and the finite qubit cases (N=2, 4 and 8) display the drastic rise after traversing the critical point. Inset illustrates the maximal standard deviations (s.d.) of 〈J_x〉/(N/2) as calculated in a because of random noise (or inhomogeneity) for different number N of qubits involved. For illustrative purpose, here we only consider the frequency uncertainty in each qubit , relevant to our experimental set-up. It is seen that uncertainties do not give large errors and increasing the number of qubits yields better suppression of the random noise. (b) Numerical calculations of 〈J_z〉/(N/2) as function of λ/λ_c following the experimental pulse sequence in Fig. 1c at different durations as indicated. Sample decoherence is included in calculations. It is seen that 〈J_z〉/(N/2) curves rise around the same point as that in a. Inset illustrates λ/λ_c as a function of the ramping time t during the pulse sequence. (c) Numerically calculated energies of the lowest three energy eigenstates (top) and population distribution (in logarithmic scale) among these three states (bottom) of the four-qubit Hamiltonian system described in equation (1) as functions of λ/λ_c under decoherence, with τ=600 ns. Higher-energy states are omitted for clarity. Starting with all qubits and the resonator in their own ground states at λ/λ_c=0.5 (at this point the system’s ground state |0〉 is at E₀≈0 and takes the largest population as shown by the black line), E_n of the lowest few states significantly drop below 0 and the population distribution quickly evolves as λ/λ_c increases above 1 (in the light-green region), indicating a structural change of the eigenstates of the system crossing this critical point.