# Purcell enhancement and transitional dipole moment of a quantum dot?

Let us consider a quantum emitter with a transitional dipole moment $\mathbf{p}$ and transitional frequency $\omega_0$ which is placed in an inhomogeneous environment. Its spontaneous decay rate is enhanced by the Purcell effect and can be described mathematically as [Principle of Nano-Optics, L. Novotny and B. Hecht, page 247]: $$\gamma = \frac{\pi \omega_0}{3\hbar\epsilon_0}|\mathbf{p}|^2\rho_{\text{p}}(\mathbf{r}_0,\omega_0)$$ $$\rho_{\text{p}}(\mathbf{r}_0,\omega_0)=\frac{6\omega_0}{\pi c^2 |\mathbf{p}|^2}\left[ \mathbf{p} \cdot \text{Im} \left[\overleftrightarrow{\mathbf{G}}(\mathbf{r}_0,\mathbf{r}_0;\omega_0) \right] \cdot\mathbf{p} \right]$$ How does this apply to a semi-conductor quantum dot (QD)? If I understand correctly, a (spherical) QD does not have a well-defined transitional dipole moment vector $\mathbf{p}$ and can be excited by an electric field in any direction, so the above equations are no longer valid. Thank you for the answers.

• Please define your notation. For example, what the heck is $\overleftrightarrow{\mathbf{G}}$? – DanielSank Sep 7 '17 at 20:49
• Thank you. It is the dyadic green's function of the electromagnetic field of the system. Please see link – ErwinNK Sep 11 '17 at 17:09