Purcell effect

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_{\rm P} = \frac{3}{4\pi^2}\left(\frac{\lambda_{\rm free}}{n}\right)^3 \frac{Q}{V}\,, where \lambda_{\rm free} is the vacuum wavelength, n is the refractive index of the cavity material (so \lambda_{\rm free}/n is the wavelength inside the cavity), and Q and V are the cavity quality factor and mode volume, respectively.

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_{\rm P} = \frac{3}{4\pi^2}\left(\frac{\lambda_{\rm free}}{n}\right)^3 \frac{Q}{V}\,, where \lambda_{\rm free} is the vacuum wavelength, n is the refractive index of the cavity material (so \lambda_{\rm free}/n is the wavelength inside the cavity), and Q and V are the cavity quality factor and mode volume, respectively. \rho_{\rm c} = \frac{1}{V \Delta \nu} to that of the free space density of states \rho_{\rm f} = \frac{8\pi n^3 \nu^2}{c^3}\,. Here, \nu and \Delta \nu are the resonance frequency and bandwidth, respectively. Using Q = \frac{\nu}{\Delta \nu}\,, one gets \frac{\rho_{\rm c}}{\rho_{\rm f}}

References

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