The Purcell effect is the enhancement of a quantum system's spontaneous emission rate by its environment. In the 1940s Edward Mills Purcell discovered the enhancement of spontaneous emission rates of atoms when they are incorporated into a resonant cavity. In terms of quantum electrodynamics the Purcell effect is a consequence of enhancement (or decreasing) of local density of photonic states at the emitter position. It can also be considered as an interference effect. The oscillator radiates the wave which is reflected from the environment. In turn the reflection excites the oscillator either out of phase resulting in higher damping rate accompanied with the radiation enhancement or in phase with the oscillator mode leading to the radiation suppression. For an emitter tuned to the fundamental mode of a cavity and placed at its center the magnitude of the enhancement is given by the Purcell factor
F P = 3 4 π 2 ( λ f r e e n ) 3 Q V , {\displaystyle F_{\rm {P}}={\frac {3}{4\pi ^{2}}}\left({\frac {\lambda _{\rm {free}}}{n}}\right)^{3}{\frac {Q}{V}}\,,}
where λ f r e e {\displaystyle \lambda _{\rm {free}}} is the vacuum wavelength, n {\displaystyle n} is the refractive index of the cavity material (so λ f r e e / n {\displaystyle \lambda _{\rm {free}}/n} is the wavelength inside the cavity), and Q {\displaystyle Q} and V {\displaystyle V} are the cavity quality factor and mode volume, respectively.
Heuristic derivation One way of seeing why the Purcell effect arises is by using cavity quantum electrodynamics. Fermi's golden rule dictates that the transition rate for the atom–vacuum (or atom–cavity) system is proportional to the density of final states. In a cavity at resonance, the density of final states is enhanced (though the number of final states may not be). The Purcell factor is then just the ratio of the cavity density of states
ρ c = 1 V Δ ν {\displaystyle \rho _{\rm {c}}={\frac {1}{V\Delta \nu }}}
to that of the free space density of states
ρ f = 8 π n 3 ν 2 c 3 . {\displaystyle \rho _{\rm {f}}={\frac {8\pi n^{3}\nu ^{2}}{c^{3}}}\,.}
Here, ν {\displaystyle \nu } and Δ ν {\displaystyle \Delta \nu } are the resonance frequency and bandwidth, respectively. Using
Q = ν Δ ν , {\displaystyle Q={\frac {\nu }{\Delta \nu }}\,,}
one gets
ρ c ρ f = c 3 8 π n 3 ν 2 Q ν V = 1 8 π ( λ f r e e n ) 3 Q V , {\displaystyle {\frac {\rho _{\rm {c}}}{\rho _{\rm {f}}}}={\frac {c^{3}}{8\pi n^{3}\nu ^{2}}}{\frac {Q}{\nu V}}={\frac {1}{8\pi }}\left({\frac {\lambda _{\rm {free}}}{n}}\right)^{3}{\frac {Q}{V}}\,,}
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