Fig. 2: Role of electron-phonon coupling in the equilibrium system at atomic limit t₀ = 0.

a Electron number N_e in equilibrium as a function of local chemical potential ε/U₁ and electron-phonon coupling strength \({g}_{0}/\sqrt{\hslash {\omega }_{0}{U}_{1}}\). b Energy of different electron configurations with fixed N_e = 2. Here we have used the single-mode approximation, which is verified by comparing the results obtained from single mode (solid lines) and multi (e.g., 100 considered here) modes (dots). The single- and multi-mode scenarios are found to agree with each other qualitatively well. The transition points between patterns ●○○● (○●●○) and ●○●○/○●○● are also presented in the two-electron region of (a) by dashed lines, respectively. Here we set U₁/2πℏ = 200 GHz, U₂/2πℏ = 20 GHz, U₃/2πℏ = 2 GHz, and ω₀/2π = 3 GHz, which are within the typical tuning range of parameters shown in Fig. 1b.