## 4 Extension of the Formalism When the Quantum Emitter Has Both an Electric and a Magnetic Dipole
### Free-Space Electric, Magnetic and Mixed Green's Functions
The factor of \(2/3\) is well-known as being the demagnetization factor of a spherical particle [63], however, to the best of our knowledge, this term has not been discussed in the literature before in the context of the application of [PERSON]'s functions to electromagnetic problems. Correctly taking this term into account is essentially to describe self-field effects in the magnetization of a particle (analogous to the self-field effects in the polarization in the (electric-only) case considered before). For the case of nanoparticle characterized by a permittivity \(\epsilon_{\mathrm{np}}\) and permeability \(\mu_{\mathrm{np}}\), the free polarization and magnetization densities inside the nanoparticle volume read  
\[\mathbf{P}_{f}(\mathbf{r},\omega) =\mathbf{P}_{\mathrm{np}}(\mathbf{r},\omega)-\mathbf{P}_{n}( \omega)=\epsilon_{0}\left(\epsilon_{\mathrm{np}}-\epsilon_{n}\right)\mathbf{E}( \mathbf{r},\omega), \tag{90}\] \[\mathbf{M}_{f}(\mathbf{r},\omega) =\mathbf{M}_{\mathrm{np}}(\mathbf{r},\omega)-\mathbf{M}_{n}( \omega)=\mu_{0}^{-1}\left(\mu_{n}^{-1}-\mu_{\mathrm{np}}^{-1}\right)\mathbf{B}( \mathbf{r},\omega), \tag{91}\]  
where \(\mathbf{P}_{\mathrm{np}}(\mathbf{r},\omega)\) and \(\mathbf{P}_{n}(\omega)\) are the polarization densities of the nanoparticle and host medium, and \(\mathbf{M}_{\mathrm{np}}(\mathbf{r},\omega)\) and \(\mathbf{M}_{n}(\omega)\) are their densities, respectively. We used the linear constitutive relations\(\mathbf{P}_{n}(\mathbf{r},\omega)=e_{0}\left(\epsilon_{n}-1\right)\mathbf{E}( \mathbf{r},\omega)\) and \(\mathbf{M}_{n}(\mathbf{r},\omega)=\mu_{0}^{-1}\left(1-\mu_{n}^{-1}\right) \mathbf{B}(\mathbf{r},\omega)\). The previous relation between the magnetic field induction and the magnetization follows from the equations \(\mathbf{H}=\mathbf{B}/\mu_{0}-\mathbf{M}\) and \(\mathbf{M}=\chi\mathbf{H}=\chi(\mathbf{B}/\mu_{0}-\mathbf{M})\), where \(\chi\) is the magnetic susceptibility, then \(\mathbf{M}(1+\chi)=\chi\mathbf{B}/\mu_{0}\Leftrightarrow\mathbf{M}=\frac{ \chi}{1+\chi}\mathbf{B}/\mu_{0}\Leftrightarrow\mathbf{M}=(1-\mu^{-1})\mathbf{ B}/\mu_{0}\). The same reasoning provides the relation between the polarization and the electric field. Inserting the two previous equations in Equations (85) and (86), we obtain  
\[\mathbf{E}(\mathbf{r},\omega) =\mathbf{E}_{0}(\mathbf{r},\omega)+\omega^{2}\mu_{n}\mu_{0}e_{0} \left(\epsilon_{\text{np}}-\epsilon_{n}\right)\int_{V}d^{3}\mathbf{r}^{ \prime}\overleftrightarrow{G}_{0}^{EE}\left(\mathbf{r},\mathbf{r}^{\prime}, \omega\right)\cdot\mathbf{E}(\mathbf{r}^{\prime},\omega)\] \[+\omega\mu_{n}k_{n}\left(\mu_{n}^{-1}-\mu_{\text{np}}^{-1}\right) \int_{V}d^{3}\mathbf{r}^{\prime}\overleftrightarrow{G}_{0}^{EM}\left( \mathbf{r},\mathbf{r}^{\prime},\omega\right)\cdot\mathbf{B}(\mathbf{r}^{\prime },\omega), \tag{92}\] \[\mathbf{B}(\mathbf{r},\omega) =\mathbf{B}_{0}(\mathbf{r},\omega)+\mu_{n}k_{n}^{2}\left(\mu_{n}^ {-1}-\mu_{\text{np}}^{-1}\right)\int_{V}d^{3}\mathbf{r}^{\prime}\overleftrightarrow {G}_{0}^{MM}\left(\mathbf{r},\mathbf{r}^{\prime},\omega\right)\cdot\mathbf{B}( \mathbf{r}^{\prime},\omega)\] \[-\omega\mu_{n}\mu_{0}k_{n}\epsilon_{0}\left(\epsilon_{\text{np}}- \epsilon_{n}\right)\int_{V}d^{3}\mathbf{r}^{\prime}\overleftrightarrow{G}_{ 0}^{ME}\left(\mathbf{r},\mathbf{r}^{\prime},\omega\right)\cdot\mathbf{E}( \mathbf{r}^{\prime},\omega). \tag{93}\]  
The set of coupled Equations (92) and (93) are the Lippmann-Schwinger equations for electromagnetic scattering. Solving them, we can obtain the electric and magnetic fields scattered by the nanoparticle.