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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$?

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I refer you to p170 on Daniel Stecke Quantum Optics note

We thus identify $\Gamma = A_{21}$ as the excited-state decay rate.

where $A_{21}$ is previsouly defined as the Einstein coefficient.

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