## 2 Dyadic Green's Function for the Electric Field
### Renormalized Polarizability of an Isotropic Quantum Emitter Near a Continuous Graphene Sheet
We have found that nonlocal effects in the graphene conductivity (its dependence on the wavevector) play no visible role in the position or the intensity of the resonance in the effective polarizability of the nanoparticle (when \(z_{0}=151\) nm). We also expect nonlocal effects akin to the nanoparticle to be negligible provided that \(z_{0}\gtrsim 10\,\text{nm}\), below which nonlocal contributions in metallic nanoparticles usually arise [57; 58] (the situation is different for semiconductor nanoparticles [58]). In Figure 4 we present the spectrum of the polarizability of a CdSe nanoparticle in the presence of graphene on a substrate. As in Figure 3, the observed broad band resonance in the imaginary part of the polarizability tensor components is due to the excitation of surface plasmons in graphene. As discussed previously, the order of magnitude of the plasmonic resonance frequency can be estimated from the relation \(k_{\mathrm{spp}}(\omega)z_{0}\simeq 1\), which is independent of the nanoparticle's material. Indeed, we observe the resonance here approximately at the same frequency as for the metallic nanoparticle (see Figure 3). In order to further access the plasmonic nature of the broad band resonance, we have studied its position as a function of the Fermi energy and found a complete agreement with the above relation, that is, in the peak of the resonance scales with the Fermi energy \(\sqrt{E_{F}}\). Interestingly, the intensity of the resonance in Figure 4 is smaller by a factor of 2.5 when compared to the case of the metallic nanoparticle. This can be understood from the following simple consideration. We notice that in the imaginary part of the bare polarizability, \(\Im\tilde{\pi}_{0}\) is negligible for both metallic and CdSe nanoparticles in the considered frequency range. Therefore, we can write for the imaginary part of the renormalized polarizability (e.g., the \(xx\) component):  
\[r_{s} =\frac{\beta_{1}-\beta_{2}-\mu_{0}\omega\sigma_{T}(\omega)}{\beta _{1}+\beta_{2}+\mu_{0}\omega\sigma_{T}(\omega)}, \tag{55}\]4 Top panels: Real (**a**) and imaginary (**b**) parts of the renormalized polarizability of a gold nanoparticle with radius \(R=50\,\mathrm{nm}\) located at a distance of \(z_{0}=151\,\mathrm{nm}\) from a graphene sheet with a Fermi energy of \(1\,\mathrm{eV}\) and damping parameter of \(h\gamma=4.1\,\mathrm{meV}\) supported by a dielectric of permittivity \(\epsilon_{2}=2\). The solid red line represents the \(xx\) component, and the black dotted line represents the \(zz\) component of the polarizability in the presence of graphene. For comparison, the \(xx\) component of the polarizability of the nanoparticle is also represented in the absence of graphene (but in the presence of the dielectric interface), as \(\alpha_{xx}^{NG}\). One can appreciate the increase in the imaginary part of the polarizability by about two orders of magnitude when the particle is near doped graphene. (**c**) The imaginary part of the nanoparticle polarizability in a vacuum. In all the panels, the parameters used in the Drude model for dielectric function of gold are: \(h\omega_{p}=7.9\,\mathrm{eV}\) and \(\Gamma_{0}=0.053\,\mathrm{eV}\). \[r_{s} =\frac{\beta_{1}-\beta_{2}-\mu_{0}\omega\sigma_{T}(\omega)}{\beta _{1}+\beta_{2}+\mu_{0}\omega\sigma_{T}(\omega)}, \tag{55}\]5 \[r_{s} =\frac{\beta_{1}-\beta_{2}-\mu_{0}\omega\sigma_{T}(\omega)}{\beta _{1}+\beta_{2}+\mu_{0}\omega\sigma_{T}(\omega)}, \tag{55}\]6  
Since \(\epsilon_{CdSe}\approx\epsilon_{\infty}\) for \(\omega>15\) THz, we have \(\tilde{\alpha}_{0}\approx 0.684\) for CdSe, while for the metal particles \(\tilde{\alpha}_{0}\approx 1\), because the metal permittivity is negative and large in modulus in the far-infrared region. The denominator is close to unity in both cases, so in the case of a metal nanoparticle, we have a stronger renormalization of the polarizability (by a factor of \(\approx\)2.5) in the presence of a graphene sheet. This is the maximum renormalization that one can obtain if the bare polarizability is dispersionless.