A unified treatment of polynomial sectors and constraint polynomials of the Rabi models

TL;DR

This paper introduces a unified framework using gradation slicing to analyze polynomial solutions of differential equations, applying it to Rabi models to derive known and new constraint polynomials for eigenvalue determination.

Contribution

It develops a general theory for polynomial solutions of ODEs with polynomial coefficients, applicable to Rabi models, and derives new constraint polynomials for complex models.

Findings

Reproduces known constraint polynomials for standard Rabi models.

Generates new constraint polynomials for multi-photon and generalized Rabi models.

Shows a common set of grading parameters characterizes various Rabi models.

Abstract

General concept of a gradation slicing is used to analyze polynomial solutions of ordinary differential equations (ODE) with polynomial coefficients, ${\cal L}\psi=0$, where ${\cal L}=\sum_l p_l(z) d_z^l$, $p_l(z)$ are polynomials, $z$ is a one-dimensional coordinate, and $d_z=d/dz$. It is not required that ODE is either (i) Fuchsian or (ii) leads to a usual Sturm-Liouville eigenvalue problem. General necessary and sufficient conditions for the existence of a polynomial solution are formulated involving constraint relations. The necessary condition for a polynomial solution of $n$th degree to exist forces energy to a $n$th baseline. Once the constraint relations on the $n$th baseline can be solved, a polynomial solution is in principle possible even in the absence of any underlying algebraic structure. The usefulness of theory is demonstrated on the examples of various Rabi models. For those models, a baseline is known as a Juddian baseline (e.g. in the case of the Rabi model the curve described by the $n$th energy level of a displaced harmonic oscillator with varying coupling $g$). The corresponding constraint relations are shown to (i) reproduce known constraint polynomials for the usual and driven Rabi models and (ii) generate hitherto unknown constraint polynomials for the two-mode, two-photon, and generalized Rabi models, implying that the eigenvalues of corresponding polynomial eigenfunctions can be determined algebraically. Interestingly, the ODE of the above Rabi models are shown to be characterized, at least for some parameter range, by the same unique set of grading parameters.