# Is the Einstein A coefficient the same as the natural line width $\Gamma$?

Given a two-level system, the rate of decay from the excited state via spontaneous emission is given by the coefficient $A$ in Einstein's rate equations. However, there is also the natural line width $\Gamma$ (usually given in MHz$\times 2 \pi$) which can be experimentally obtained from spectroscopy experiments. The $\Gamma$ is related to the lifetime of the excited state by $\tau = 1/\Gamma$. Is $A = \Gamma$ or is it off by a factor of $2\pi$?

Take, e.g. the D2 line of $^{87}$Rb. Here, $\Gamma = 6$MHz $\times 2 \pi$, correspondingly $\tau = 26$ns. Is $A = 6$MHz or $A = 6$MHz $\times 2 \pi$?

We thus identify $$\Gamma = A_{21}$$ as the excited-state decay rate.
where $$A_{21}$$ was previously defined as the Einstein coefficient.