The Einstein rate equations govern the absorption and emission of photons from a two-level system of energy levels.

It states that the rate of absorption of photons is given by $$\textrm{N}_{1}\rho\left(\omega_{12}\right)\textrm{B}_{12}$$, the rate of stimulated emission is $$\textrm{N}_{2}\rho\left(\omega_{12}\right)\textrm{B}_{21}$$ and the rate of spontaneous emission is $$\textrm{N}_{2}\textrm{A}_{12}$$. Here, $$\textrm{N}_{1}, \textrm{N}_{2}$$ are the number of atoms in the ground and excited state of the two-level system respectively, $$\rho\left(\omega_{12}\right)$$ is the spectral energy density (the energy per unit volume integrated over time passing through the system) and $$\textrm{B}_{12}, \textrm{B}_{21}, \textrm{A}_{12}$$ are the Einstein coefficients.

Certain literature (1) (2) state that the light is broadband, but why is this required? What happens if the light is only at the resonant frequency of the transition ($$\omega_{12}$$)?

• Light cannot be a delta function at a single frequency. Apr 11 '19 at 10:08
• Oh noo then all the perfect plane waves that we studied do not exist :(( Apr 11 '19 at 12:32
• @RobJeffries In these cases, does lifetime-broadened light count as broadband? I've always assumed that broadband light is of the order of ~THz. Apr 12 '19 at 8:00