Spectral Degeneracies in the Asymmetric Quantum Rabi Model

Abstract

The aim of this article is to investigate certain family of (so-called constraint) polynomials which determine the quasi-exact spectrum of the asymmetric quantum Rabi model. The quantum Rabi model appears ubiquitously in various quantum systems and its potential applications include quantum computing and quantum cryptography. In (Wakayama, Symmetry of Asymmetric Quantum Rabi Models) [30], using the representation theory of the Lie algebra $\mathfrak {sl}_2$, we presented a picture of the asymmetric quantum Rabi model equivalent to the one drawn by confluent Heun ordinary differential equations. Using this description, we proved the existence of spectral degeneracies (level crossings in the spectral graph) of the asymmetric quantum Rabi model when the symmetry-breaking parameter $\varepsilon$ equals $\frac{1}{2}$ by studying the constraint polynomials, and conjectured a formula that ensures the presence of level crossings for general $\varepsilon \in \frac{1}{2}\mathbb {Z}$. These results on level crossings generalize a result on the degenerate spectrum, given first by Kuś in 1985 for the (symmetric) quantum Rabi model. It was demonstrated numerically by Li and Batchelor in 2015, investigating an earlier empirical observation by (Braak, Phys. Rev. Lett. 107, 100401–100404, 2011) [3]. In this paper, although the proof of the conjecture has not been obtained, we deepen this conjecture and give insights together with new formulas for the target constraint polynomials.