Spontaneous Emission

Atoms in an excited state n$_u$ changes its state from n$_u$ to n spontaneously. The number of transitions is proportional to the number of atoms in the n$_u$ state $N_{n_u}$ as

$$A_{n_u\rightarrow n}N_{n_u},$$ (2.137)

where $A_{n_u\rightarrow n}$ represents the probability of transition from n$_u$ to n in unit time. This coefficient is called Einstein's $A$ coefficient. The lifetime of n$_u$ level is related to this coefficient as

$$\tau_{\rm life}(n_u)=\frac{1}{A_{n_u\rightarrow n}+A_{n_u\rightarrow n_d}},$$ (2.138)

for this three-level atom.