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In ammonia, the nitrogen chemical atom can lie on either side of the plane of the three hydrogen chemical atoms. Quantum tunnelling couples these configurations; the resulting pair of energy levels permits an electric dipole transition. This geometrical inversion is distinct from population inversion, which concerns how many molecules occupy each level.^[ref]

For an idealized pair of energy levels with equal quantum degeneracy, the Boltzmann distribution gives $N_u/N_l=\exp [-h{\nu }_0/(k_{\mathrm{B}}T)]<1$ at positive temperature $T$. A warm source therefore supplies upper-state molecules together with an unwanted lower-state population. Selecting the former can produce population inversion in the transmitted beam without exciting each transmitted molecule during selection. The energy per net downward transition isHere $h$ is the Planck constant. Taking the reported ${\nu }_0=23.87\mathrm{GHz}$ as a rounded input gives $ϵ\approx 1.582\times 10^{-23}\mathrm{J}$. This calculation is a conversion of the reported frequency, not a new measurement of the ammonia inversion transition.

An ammonia maser uses an ammonia molecular beam, maser state selection and a maser cavity to obtain amplification or sustained microwave oscillation. In the 1954 experiment by James Power Gordon, Herbert Jack Zeiger and Charles Hard Townes, a quadrupolar static electric field focused upper-state molecules toward the cavity and deflected lower-state molecules away. The reported $J=K=3$ ammonia inversion transition lies near $23.87\mathrm{GHz}$.^[ref]

Multiplying the upper-level and photon equations by $h{\nu }_0$ makes the cancellation of internal transfer visible:Incoming excitation becomes unused excitation carried away, internal dissipation, useful output, or a change in stored energy. Translational kinetic energy, the source's other internal states and preparation losses lie outside this two-level budget. The output cannot exceed the incoming excitation power $h{\nu }_{0}F$.

Choose $G=10^{-4}{\mathrm{s}}^{-1}$ per unit inversion, $\gamma =10^4{\mathrm{s}}^{-1}$, ${\kappa }_{\mathrm{int}}={\kappa }_{\mathrm{out}}=10^6{\mathrm{s}}^{-1}$, and $F=4\times 10^{14}{\mathrm{s}}^{-1}$. These are stipulated teaching parameters. Then $N_0=4\times 10^{10}$, $F_{\mathrm{th}}=2\times 10^{14}{\mathrm{s}}^{-1}$, $N_{\mathrm{ss}}=2\times 10^{10}$ and $n_{\mathrm{ss}}=5\times 10^7$. Using the rounded ammonia inversion transition frequency, the loaded quality factor is approximately $7.50\times 10^4$. The incoming excitation power is $6.327\mathrm{nW}$: $4.745\mathrm{nW}$ leaves unused, $0.791\mathrm{nW}$ becomes internal dissipation and $0.791\mathrm{nW}$ is useful output, with rounding in these displayed values.

There is an instructive extraction optimum within this model. With the same $F$, $G$ and $\gamma$, put $\rho ={\kappa }_{\mathrm{out}}/{\kappa }_{\mathrm{int}}$ and $p=FG/(\gamma {\kappa }_{\mathrm{int}})=4$. For $0\le \rho <p-1$,Differentiation gives ${\rho }_{\mathrm{opt}}=\sqrt{p}-1=1$. At zero coupling no power is extracted; at $\rho =3$ the device reaches onset and the nonzero branch disappears. The optimum balances extraction against the inversion required to compensate that extraction. Equal residence rates and loss of molecular phase memory are simplifying assumptions; the calculated efficiency is not a universal bound for beam masers.