In the $\{|L⟩,|R⟩\}$ basis:

Predict before reading on. Before any numbers: at $\epsilon =0$ the diagonal is degenerate. What must the eigenstates look like then, and what is the splitting in terms of $A$?

Zero field. $E_{\pm }=E_0\pm A$: splitting $2A=h\times 23.87$ GHz, i.e. $A=49.4\mu$eV. Eigenstates are the fully mixed combinations $(|L⟩\pm |R⟩)/\sqrt{2}$ — mixing angle $\theta =45^{\circ }$. Neither eigenstate has a dipole moment: the nitrogen is on both sides at once.

Crossover. The character changes when the diagonal term ties the coupling, $d\epsilon =A$:

Predict before reading on. Predict before computing 50 kV/cm: that is $\sim 3\times$ the crossover field. Will the splitting be closer to $2A$ (tunneling-dominated) or to $2d\epsilon$ (field-dominated)? And will the eigenstates be mixed or localized?

At 50 kV/cm: $d\epsilon =153\mu$eV $=3.1A$. The splitting opens to

and the mixing angle $\frac{1}{2}\arctan (A/d\epsilon )=8.9^{\circ }$: the eigenstates have re-localized to nearly pure “nitrogen up” and “nitrogen down,” now WITH dipole moments — which is exactly how the maser’s electrostatic state selector separates upper-state molecules from lower-state ones.

Check the limits. $\epsilon \to 0$: splitting $\to 2A$ ✓. $d\epsilon \gg A$: splitting $\to 2d\epsilon$, linear in field ✓. The quadratic-at-small-field, linear-at-large-field Stark curve is the square root interpolating between its two limits.

Predict before reading on. Articulate the recipe in one sentence: given any two nearby levels and a knob, what three quantities do you extract and what formula combines them?