We give a complete classification of generic topological types of domain configurations and Stokes graphs of the quadratic differential
\({Q}_{0}\left(z\right)d{z}^{2}=-\frac{{z}^{4}+{c}_{3}{z}^{3}+{c}_{2}{z}^{2}+{c}_{1}z+{c}_{0}}{{\left(z-1\right)}^{2}{\left(z+1\right)}^{2}}d{z}^{2}\)
with ck ∈ \({\mathbb{R}}\) , assuming that the zeros are distinct from ±1. We identify the set of coefficients (c3, c2, c1, c0) ∈ \({\mathbb{R}}^{4}\) , with the particular choices of physical parameters Δ, g, and E describing the Rabi model of a reaction of atoms to the harmonic electric field with a frequency close to the natural one of the atoms. The asymptotic structure of Stokes graphs and domain configurations of quadratic differentials, when the Rabi parameters tend to infinity, is also discussed.
