 Solution

Let $A_{21}$ be the spontaneous emission coefficient, and let $B_{12}\rho$ and $B_{21}\rho$ be the absorption and stimulated emission rates per atom under the radiation spectral energy density $\rho$ at resonance. Because the two levels are nondegenerate, the Einstein coefficients satisfy $B_{12}=B_{21}=B$.

The radiative balance in a nondegenerate two-level atom isThe total spontaneous and stimulated emission rates are $R_{\mathrm{sp}}=N_2{A}_{21}$ and $R_{\mathrm{st}}=N_{2}B\rho$. Dividing the balance equation by $N_{2}B\rho$ givesThis presumes nonzero drive and upper-level population, and no other population-transfer mechanism. It does not require a thermal Boltzmann distribution. In this closed driven two-level system, spontaneous decay keeps $N_1>N_2$; arbitrarily intense driving approaches equal populations, rather than producing a population inversion.