Symmetry of asymmetric quantum Rabi models

The (symmetric) quantum Rabi model appears ubiquitously in various quantum systems and its applications include quantum information technology. In this talk, using the representation theory of the Lie algebra sl2, we present a picture of the asymmetric quantum Rabi model equivalent to the one drawn by confluent Heun ordinary differential equations. We show the existence of spectral degeneracies (level crossings in the spectral graph) of the asymmetric quantum Rabi model when the symmetry-breaking parameter equals 1/2 by studying the constraint polynomials, and give a conjectural formula that ensures the presence of level crossings for general half-integers. This result on level crossings was demonstrated numerically by physicists Li and Batchelor in 2015, investigating an earlier empirical observation by Braak (2011). In the picture, we find also a certain reciprocity described by sl2.