## 2 Dyadic Green's Function for the Electric Field
### Polarizability of a Quantum Emitter in a Homogeneous Medium
We neglect the real part of \(\overset{\longleftrightarrow}{C_{0}}^{\mathrm{reg}}(\mathbf{r},\mathbf{r}^{ \prime},\omega)\) when compared to \(\overset{\longleftrightarrow}{C_{0}}^{\mathrm{SF}}(\mathbf{r}-\mathbf{r}^{ \prime},\omega)\), thereby approximating  
\[\overset{\longleftrightarrow}{C_{0}}\left(\mathbf{r},\mathbf{r}^{\prime}, \omega\right)\simeq-\frac{1}{3k_{1}^{2}}\overset{\longleftrightarrow}{I} \delta(\mathbf{r}-\mathbf{r}^{\prime})+i\frac{k_{1}}{6\pi}\overset{ \longleftrightarrow}{I}. \tag{41}\]  
Using the above approximation in Equation (39) and assuming that the electric field within the nanoparticle varies slowly, that is, \(\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}(\mathbf{r}_{0},\omega)\) throughout \(V\), we can solve the obtained algebraic equation for the total field \(\mathbf{E}(\mathbf{r}_{0},\omega)\) as a function of the external field \(\mathbf{E}_{\mathrm{ext}}(\mathbf{r}_{0},\omega)\), obtaining  
\[\mathbf{E}(\mathbf{r}_{0},\omega)=\frac{1}{1-\frac{1}{\epsilon_{1}}\left( \epsilon_{\mathrm{np}}(\omega)-\epsilon_{1}\right)\left(-\frac{1}{3}+i\frac{k _{1}^{3}}{6\pi}V\right)}\mathbf{E}_{\mathrm{ext}}(\mathbf{r}_{0},\omega). \tag{42}\]  
Therefore, the electric dipole moment follows from  
\[\mathbf{d} =\epsilon_{0}\left(\epsilon_{\mathrm{np}}(\omega)-\epsilon_{1} \right)\int_{V}d^{3}\mathbf{r}\mathbf{E}(\mathbf{r},\omega)\] \[\simeq\frac{\alpha_{\mathrm{CM}}}{1-i\frac{k_{1}^{3}}{6\pi \epsilon_{0}\epsilon_{1}}\alpha_{\mathrm{CM}}}\mathbf{E}_{\mathrm{ext}}( \mathbf{r}_{0},\omega), \tag{43}\]  
where we have introduced the Clausius-Mossotti polarizability [45]  
\[\alpha_{\mathrm{CM}}=4\pi\epsilon_{1}\epsilon_{0}R^{3}\frac{\epsilon_{\mathrm{ np}}(\omega)-\epsilon_{1}}{e_{\mathrm{np}}(\omega)+2\epsilon_{1}}. \tag{44}\]  
The polarizability of a nanoparticle embedded in a homogeneous medium with relative permittivity \(\epsilon_{1}\) can be read from Equation (43)  
\[\mathbf{j}_{f}(\mathbf{r},\omega)=-i\omega\left[\mathbf{P}_{\mathrm{np}}( \mathbf{r},\omega)-\mathbf{P}_{1}(\omega)\right]=-i\omega\epsilon_{0}\left( \epsilon_{\mathrm{np}}(\omega)-\epsilon_{1}\right)\mathbf{E}(\mathbf{r}, \omega), \tag{38}\]0  
It must be pointed out that the above equation is only approximate. As a matter of fact it is easy to see that if we had kept the term \(\propto 1/\left|\mathbf{R}\right|\) in \(\overset{\longleftrightarrow}{C_{0}}^{\mathrm{reg}}(\mathbf{r}-\mathbf{r}^{ \prime},\omega)\) we would have generated a real term, \(\propto k_{1}^{2}\), in the denominator of Equation (45), which can be interpreted as a dynamic depolarization effect [46]. The obtained term would still be incorrect, as additional terms of the same order in \(k_{1}\) would appear from taking into account that the electric field inside the nanoparticle is not constant. An exact treatment using [PERSON]'s scattering theory for a spherical particle would lead to [45; 47]\[\mathbf{j}_{f}(\mathbf{r},\omega)=-i\omega\left[\mathbf{P}_{\mathrm{np}}( \mathbf{r},\omega)-\mathbf{P}_{1}(\omega)\right]=-i\omega\epsilon_{0}\left( \epsilon_{\mathrm{np}}(\omega)-\epsilon_{1}\right)\mathbf{E}(\mathbf{r}, \omega), \tag{38}\]1  
There is indeed a term of order \(k_{1}^{2}\), but the term of order \(k_{1}^{3}\) is unchanged. The imaginary term of order \(k_{1}^{3}\) is usually denoted by radiation damping correction [1; 48] and is essentially to enforce the optical theorem for electromagnetic scattering in the lowest order [1; 49; 50; 51]. We note in passing that the radiation damping correction is also responsible for the decay rate of the dipole [1]. In the next sections, we will ignore the term of order \(k_{1}^{2}\) as it plays no significant role. However, it will become clear that it is essential to keep the radiation damping correction.