Editing Spontaneous emission (section)

==Rate of spontaneous emission==
The rate of spontaneous emission (i.e., the radiative rate) can be described by [[Fermi's golden rule]].<ref>B. Henderson and G. Imbusch, Optical Spectroscopy of Inorganic Solids (Clarendon Press, Oxford, UK, 1989).</ref> The rate of emission depends on two factors: an 'atomic part', which describes
the internal structure of the light source and a 'field part', which describes the density of electromagnetic modes of the environment. The atomic part describes the strength of a transition between two states in terms of transition moments. In a homogeneous medium, such as [[free space]], the rate of spontaneous emission in the dipole approximation is given by:

:<math> \Gamma_{\text{rad}}(\omega)= \frac{\omega^3n|\mu_{12}|^2}    {3\pi\varepsilon_{0}\hbar c^3}      
= \frac{4 \alpha \omega^3n| \langle 1|\mathbf{r}|2\rangle |^2}    {3 c^2}    </math>

:<math> \frac{|\mu_{12}|^2}    {\pi\varepsilon_{0}\hbar c}      
= 4 \alpha | \langle 1|\mathbf{r}|2\rangle |^2 </math>

where <math>\omega</math> is the emission frequency, <math>n</math> is the [[index of refraction]], <math>\mu_{12}</math> is the [[transition dipole moment]], <math>\varepsilon_0</math> is the [[vacuum permittivity]], <math>\hbar</math> is the [[reduced Planck constant]], <math>c</math> is the vacuum [[speed of light]], and <math>\alpha</math> is the [[fine-structure constant]]. The expression <math>|\langle 1|\mathbf{r}|2\rangle|</math> stands for the definition of the transition dipole moment <math>|\mu_{12}|=|\langle 1|\mathbf{d}|2\rangle|</math> for dipole moment operator <math>\mathbf{d}=q\mathbf{r}</math>, where <math>q</math> is the elementary charge and <math>\mathbf{r}</math> stands for position operator. (This approximation breaks down in the case of inner shell electrons in high-Z atoms.) The above equation clearly shows that the rate of spontaneous emission in free space increases proportionally to <math>\omega^3</math>.

In contrast with atoms, which have a discrete emission spectrum, [[quantum dots]] can be tuned continuously by changing their size. This property has been used to check the <math>\omega^3</math>-frequency dependence of the spontaneous emission rate as described by Fermi's golden rule.<ref>A. F. van Driel, G. Allan, C. Delerue, P. Lodahl, W. L. Vos and D. Vanmaekelbergh,
Frequency-dependent spontaneous emission rate from CdSe
and CdTe nanocrystals: Influence of dark states, Physical Review Letters,
95, 236804 (2005).[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.95.236804 Phys. Rev. Lett. 95, 236804 (2005) - Frequency-Dependent Spontaneous Emission Rate from CdSe and CdTe Nanocrystals: Influence of Dark States (aps.org)]</ref>