Stokes graphs of the Rabi problem with real parameters

TL;DR

This work recasts the quantum Rabi problem as a real-coefficient quadratic differential $Q_0(z)dz^2$ with double poles at $z=\pm1$ and a quartic numerator, linking the physical parameters $Δ$, $E$, and $g$ to the coefficients $c_3,c_2,c_1,c_0$ through explicit formulas. It delivers a complete taxonomy of Stokes graphs and domain configurations based on the number and placement of real zeros of $P_0(z)$, using discriminant criteria to map parameter regions to topological types and noting mirror symmetry constraints. An extensive asymptotic analysis as $|g|\to\infty$ yields a limiting differential $Q_a(z)dz^2$ with a structured classification into four primary regimes, illuminating connections to isomonodromic deformations and the Painlevé V tau-function. Together, these results provide a rigorous framework for understanding the qualitative behavior of solutions to the Rabi problem across coupling regimes and establish a foundation for linking spectral data to Stokes geometry.

Abstract

The goal of this paper is to study the geometry of the Stokes graphs associated with the problem, which was introduced by Isidor Rabi in 1937 to model reactions of atoms to the harmonic electric field with frequency close to the natural frequency of the atoms. In the standard Garnier form, the Rabi model is a matrix linear differential equation with three physical parameters, which are: the level of separation of the fermion mode $Δ$, the boson-fermion coupling $g$, and the eigenvalue $E$ of the Hamiltonian relevant to this model. The qualitative behavior of solutions of this type of problems is often described in terms of the Stokes graphs of associated quadratic differential, which in the case of Rabi problem can be represented in the form $Q_0(z)\, dz^2 = -\frac{z^4+c_3z^3+c_2z^2+c_1z+c_0}{(z-1)^2(z+1)^2}\, dz^2$ with the coefficients $c_k$, $k=0,1,2,3$, depending on the parameters $Δ$, $g$, and $E$. In this paper, we first give a complete classification of possible generic topological types of domain configurations and Stokes graphs of this quadratic differential assuming that its coefficients $c_k$ are real and the zeros of its numerator are distinct from its poles. Then we identify the set of coefficients $(c_3,c_2,c_1,c_0)\in \mathbb{R}^4$, which correspond to particular choices of the physical parameters $Δ$, $g$, and $E$. The structure of Stokes graphs and domain configurations of quadratic differentials, which appear as asymptotic cases when the parameters of the Rabi problem tend to infinity, also will be discussed.

Figures (53)

• Figure 1: Portion of the cylinder $S(a)$ associated to the Rabi problem with parameters $g^{2}=a^{-1}=1$, $\Delta\ge 0$, and $E\in \mathbb{R}$.
• Figure 2: Sets $\mathcal{I}_4$, $\mathcal{C}_4$, $\mathcal{IR}$ and $\mathcal{R}_4$ with different types of Stokes graphs of $Q_a(z)\,dz^2$.
• Figure 3: Case I-1 and case I-2.
• Figure 4: Case I-3 and case I-3-m
• Figure 5: Case I-3-deg.

Theorems & Definitions (14)

• Lemma 3.1
• Proposition 5.1: Generic cases
• Proposition 5.2: Degenerate cases with full set of critical points and multiple zeros of $P_0(z)$
• Theorem 5.3
• Theorem 5.4
• Lemma 5.5
• Lemma 5.6
• Lemma 5.7
• Corollary 5.8
• Lemma 5.9