Crossover resonance
= Crossover resonance
{title2=$\nu_{\rm co}=(\nu_a+\nu_b)/2$}
Two allowed optical transitions sharing a lower <quantum state> can produce an extra <saturation spectroscopy> feature even where there is no additional <energy level> difference. Approximate their <wavenumbers> by the same $k$, and let their <angular frequencies> be $\omega_a,\omega_b$. A pump resonant with one transition and an opposed probe resonant with the other satisfy
$$
\omega_L-kv=\omega_a,\qquad \omega_L+kv=\omega_b,
$$
so
$$
\omega_L=\frac{\omega_a+\omega_b}{2},\qquad
v=\frac{\omega_b-\omega_a}{2k}.
$$
Exchanging the two transitions selects the opposite <velocity>. Both beams can then modify the same shared population, producing a crossover near the midpoint. This requires the relevant <velocities> to be occupied and the transitions to be connected by the population dynamics; not every pair must produce a visible feature. Strength and sign depend on <optical pumping> and the multilevel structure. Assigning every narrow feature to a separate atomic transition would therefore give false <energy levels>.