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Let $A=A_{21}$ and $r=g_2/g_1$. Before stimulated emission and stimulated absorption are added, the source and decay terms giveThe upper level is fed at a specified rate of addition to the number density, its spontaneous decay feeds the lower level, and lower-level removal leads to states outside this two-level system. This is therefore an open pumped system, not a closed two-level system with conserved population.

At steady state,The degeneracy-weighted population inversion compares population per magnetic sublevel. It is positive only if $rA{\tau }_1<1$; a positive pump rate alone cannot guarantee optical gain if lower-level removal is too slow. A short lower-state lifetime is consequently helpful even when the upper-state pumping is unchanged.

The equations assume finite positive $A$ and ${\tau }_1$, approximately constant pumping and no other specified sources, sinks or branching channels. A finite steady state would not exist for continuous pumping if the lower level had no escape route. The fixed source rate also presumes access to an external reservoir; it does not imply that the two populations can grow beyond all available active atoms in a real finite medium.

Resolve the weak broadband radiation into narrow frequency intervals whose relative phases are uncorrelated. The transition probabilities, rather than the coherent amplitudes, can then be added after ensemble or time averaging. For the x-polarized field, the energy density in one interval is $\rho (\omega )d\omega ={ϵ}_0{\mathcal{E}}_{\omega }^2/2$, so its squared coupling is $|d_{21}{|}^2{\mathcal{E}}_{\omega }^2/{\hslash }^2$. Inserting this into the weak rotating-wave result givesFor times long compared with the optical coherence time of the field, the narrow resonant kernel samples a nearly constant $\rho ({\omega }_0)$. Provided the resonance is well away from the zero-frequency boundary, extend the Rabi detuning integral to the full real line. Its value is $2\pi t$, yieldingThis is the stimulated absorption rate for the specified polarization and spectral radiation energy density per angular-frequency interval. The averaging window should also satisfy $Wt\ll 1$ while the lower state is treated as undepleted. A deterministic phase-locked broadband pulse would not in general permit adding these probabilities without its interference terms.

For isotropic unpolarized radiation with total spectral radiation energy density ${\rho }_{\mathrm{tot}}$, orientational averaging instead gives $W=\pi |{\mathbf{d}}_{21}{|}^2{\rho }_{\mathrm{tot}}({\omega }_0)/(3{ϵ}_0{\hslash }^2)$. The factor one third is not inserted into the original x-polarized formula. If ordinary frequency is used, ${\rho }_{\nu }=2\pi {\rho }_{\omega }$ and the isotropic coefficient becomes $|{\mathbf{d}}_{21}{|}^2/(6{ϵ}_0{\hslash }^2)$ multiplying ${\rho }_{\nu }({\nu }_0)$. The spectral convention and polarization must be stated together to avoid spurious factors of three or $2\pi$.

Weak incoherent frequency components contribute probabilities. The long-time sinc-squared kernel gives the displayed stimulated absorption rate for a fixed polarization and spectral energy density per angular frequency. Isotropic unpolarized radiation instead contributes $|{\mathbf{d}}_{21}{|}^2/3$. Converting to density per ordinary frequency changes the coefficient by $2\pi$.