Fig. 4. Nonclassical corrections probed by AGPs.

a AGP’s wavevector as a function of the graphene–metal separation t, contrasting the metal’s response based on classical (d_⊥ = 0) and quantum (d_⊥ = ±4 Å) treatments. Inset: corresponding group velocity $v_{\mathrm{p}}=\partial \omega /\partial q{\mid }_{q=q\left({\omega }_0\right)}$. b Dependence of the renormalized graphene–metal separation $\tilde{t}\equiv t-s$ versus t. Setup parameters: r_s = 3, $ħ$γ_m = 0.1 eV, ϵ_d = 4, E_F = 0.3 eV, and $ħ$γ = 8 meV; we assume an excitation at λ₀ = 11.28 μm ($ħ$ω₀ ≈ 110 meV or f₀ ≈ 26.6 THz)³². c Relative quantum shift of the AGP wavevector, $\mathrm{Re}\mathrm{\Delta }q/\mathrm{Re}{q}_0$, with Δq ≡ q₀ − q where q₀ and q denote the AGP wavevector associated with d_⊥ = 0 and d_⊥ = −4 Å, respectively. The results presented in both a and c are based on the exact, numerical solution of Eq. (1).