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According to https://en.wikipedia.org/wiki/Spontaneous_emission#cite_note-9, the spontaneous emission rate in homogenous non-dispersive medium with refractive index $n$ is given by

$\Gamma = n\left( \frac{\omega^3 |\mu_{12}|^2}{3\pi\epsilon_0\hbar c^3} \right)$

Please could anyone explain why $\Gamma \propto n$ instead of $\Gamma \propto n^3$ since the speed of light in the medium is $c/n$? I found several textbooks which derives this rate in vacuum but not in a homogenous medium. Thank you.

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