## 2 Dyadic Green's Function for the Electric Field
### Renormalized Polarizability of an Isotropic Quantum Emitter Near a Continuous Graphene Sheet  
In what follows we shall consider the case of an isotropic quantum emitter in close proximity to a graphene sheet. In the previous sections, we have seen how the effective polarizability of a nanoparticle depends on the reflected Green's function, \(\overleftrightarrow{G_{r}}(\mathbf{r}_{0},\mathbf{r}_{0},\omega)\), which can be reconstructed from its angular spectrum representation as  
\[\overleftrightarrow{G_{r}}(\mathbf{r}_{0},\mathbf{r}_{0},\omega)=\int\frac{d^ {2}\mathbf{p}_{\parallel}}{\left(2\pi\right)^{2}}\overleftrightarrow{G}_{r} \left(\mathbf{p}_{\parallel},z_{0},z_{0},\omega\right). \tag{54}\]  
As shown in Section 2.3, the reflected Green's function in the angular spectrum representation can be written in terms of the Fresnel reflection coefficients. For a planar interface covered by graphene, the reflection coefficients are given by [18; 52]  
\[r_{s} =\frac{\beta_{1}-\beta_{2}-\mu_{0}\omega\sigma_{T}(\omega)}{\beta _{1}+\beta_{2}+\mu_{0}\omega\sigma_{T}(\omega)}, \tag{55}\] \[r_{p} =\frac{\beta_{1}\epsilon_{2}-\beta_{2}\epsilon_{1}+\beta_{1}\beta _{2}\sigma_{L}/(\epsilon_{0}\omega)}{\beta_{1}\epsilon_{2}+\beta_{2}\epsilon_ {1}+\beta_{1}\beta_{2}\sigma_{L}/(\epsilon_{0}\omega)}, \tag{56}\]  
where \(\sigma_{T}(\omega)\) and \(\sigma_{L}(\omega)\) are the transverse and longitudinal optical conductivities of graphene and \(\beta_{1/2}\) given by Equation (21). Neglecting nonlocal effects in the conductivities we have \(\sigma_{T}(\omega)=\sigma_{L}(\omega)=\sigma(\omega)\), which we will model with a Drude-like term [18; 53; 54]  
\[\sigma(\omega)=\frac{e^{2}}{4\hbar}\frac{4}{\pi}\frac{\epsilon_{F}}{\hbar \gamma-i\hbar\omega}, \tag{57}\]  
where \(\epsilon_{F}\) is graphene's Fermi energy and \(\gamma\) is the broadening factor. The transmission coefficients \(t_{s}\) and \(t_{p}\) are related to the reflection coefficients via [18]  
\[t_{s} =1+r_{s}, \tag{58}\] \[t_{p} =\frac{\beta_{1}}{\beta_{2}}\sqrt{\frac{\epsilon_{2}}{\epsilon_{1 }}}(1-r_{p}). \tag{59}\]After performing the integration over the angular variable in Equation (54), we find that \(\overleftrightarrow{G_{r}}(\mathbf{r}_{0},\mathbf{r}_{0},\omega)\) is diagonal. Rotational invariance along the \(z\) direction imposes that \(G_{r}^{xx}(\mathbf{r}_{0},\mathbf{r}_{0},\omega)=G_{r}^{yy}(\mathbf{r}_{0}, \mathbf{r}_{0},\omega)\). The same will be true for the polarizability of the nanoparticle, which, using Equation (51), we can write as  
\[\alpha_{\text{eff}}^{xx}= \alpha_{\text{eff}}^{yy}=4\pi\epsilon_{0}\epsilon_{1}R^{3}\frac{ \bar{\alpha}_{0}}{1-\left(k_{1}R\right)^{3}Q_{r}^{\parallel}\left(\mathbf{r}_ {0},\mathbf{r}_{0},\omega\right)\bar{\alpha}_{0}}, \tag{60}\] \[\alpha_{\text{eff}}^{zz}=4\pi\epsilon_{0}\epsilon_{1}R^{3}\frac{ \bar{\alpha}_{0}}{1-\left(k_{1}R\right)^{3}Q_{r}^{zz}\left(\mathbf{r}_{0}, \mathbf{r}_{0},\omega\right)\bar{\alpha}_{0}}, \tag{61}\]  
where we have defined the dimensionless quantities \(\bar{\alpha}_{0}=\alpha_{0}/\left(4\pi\epsilon_{0}\epsilon_{1}R^{3}\right)\) with \(\alpha_{0}\) the diagonal element of the nanoparticle polarizability from Equation (45). \(\mathcal{G}_{r}^{\parallel}\left(\mathbf{r}_{0},\mathbf{r}_{0},\omega\right) =\left(4\pi/k_{1}\right)G_{r}^{yy}\left(\mathbf{r}_{0},\mathbf{r}_{0},\omega\right)\) and \(\mathcal{G}_{r}^{zz}\left(\mathbf{r}_{0},\mathbf{r}_{0},\omega\right)=\left( 4\pi/k_{1}\right)G_{r}^{zz}\left(\mathbf{r}_{0},\mathbf{r}_{0},\omega\right)\). More explicitly, these quantities can be evaluated from  
\[\mathcal{G}_{r}^{\parallel}\left(\mathbf{r}_{0},\mathbf{r}_{0}, \omega\right)=\frac{i}{2}\int_{0}^{\infty}dse^{2k_{1}z_{0}\sqrt{1-s^{2}}}s\left( \frac{1}{\sqrt{1-s^{2}}}r_{s}-\sqrt{1-s^{2}}r_{p}\right), \tag{62}\] \[\mathcal{G}_{r}^{zz}\left(\mathbf{r}_{0},\mathbf{r}_{0},\omega \right)=i\int_{0}^{\infty}dse^{2k_{1}d\sqrt{1-s^{2}}}\frac{s^{3}}{\sqrt{1-s^{ 2}}}r_{p}, \tag{63}\]  
where \(s=p_{\parallel}/k_{1}\).