$Z_N$ symmetric chiral Rabi model: a new $N$-level system

Abstract: We present a new tractable quantum Rabi model for $N$-level atoms by extending the $\mathbb Z_2$ symmetry of the two-state Rabi model. The Hamiltonian is $\mathbb Z_N$ symmetric and allows the parameters in the level separation terms to be complex while remaining hermitian. This latter property means that the new model is {\em chiral}, which makes it differ from any existing $N$-state Rabi models in the literature. The $\mathbb Z_N$ symmetry provides partial diagonalization of the general Hamiltonian. The exact isolated (i.e. exceptional) energies of the model have the Rabi-like form but are $N$-fold degenerate. For the three-state case ($N=3$), we obtain three transcendental functions whose zeros give the regular (i.e. non-exceptional) energies of the model.