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Stimulated Emission

The process by which an excited atom emits a photon when stimulated by the presence of another photon of the same energy, key to laser operation.
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The statement of the theorem

Let the atomic system be described by the Hamiltonian H0H_0, with eigenstates ∣e⟩|e\rangle and ∣g⟩|g\rangle corresponding to energy levels EeE_e and EgE_g. The interaction with the quantized electromagnetic field is given by Hint=−12∇×∇×d∙E(r,t)H_{int} = -\frac{1}{2} \boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \boldsymbol{d} \bullet \boldsymbol{E}(\boldsymbol{r}, t), where d\boldsymbol{d} is the dipole moment operator and E(r,t)\boldsymbol{E}(\boldsymbol{r}, t) is the quantized electric field operator. The transition rate WegW_{eg} for stimulated emission is derived from Fermi's Golden Rule and is given by:\n\nΓe→g=Weg=ω33ϵ0ℏc3∣dge∣2ρ(ω)\Gamma_{e\to g} = W_{eg} = \frac{\omega^3}{3\epsilon_0 \hbar c^3} |\boldsymbol{d}_{ge}|^2 \rho(\omega) \n\nwhere ω=(Ee−Eg)/ℏ\omega = (E_e - E_g)/\hbar is the transition angular frequency, dge=⟨g∣d∣e⟩\boldsymbol{d}_{ge} = \langle g | \boldsymbol{d} | e \rangle is the transition dipole matrix element, and ρ(ω)\rho(\omega) is the spectral energy density of the incident radiation field evaluated at ω\omega. The evolution of the excited state population NeN_e is governed by the rate equation:\n\ndNedt=−WegNe+WgeNg\frac{d N_e}{d t} = -W_{eg} N_e + W_{ge} N_g
Source: Wikipedia