Where H ^ ^ is our semi-classical hamiltonian. It is written as H ^ ^ = ℏ ℏ ω ω 0 | e ⟩ ⟩ ⟨ ⟨ e | + V e . d . . And Γ Γ = d 0 2 ω ω 0 3 3 π π ϵ ϵ 0 ℏ ℏ c 3 . Γ Γ represents the linewidth of the transition, and thus you can see Γ Γ − − 1 as the half-life of the given transition.