Generic two-mode quantum Rabi model: analytical energies and degeneracies

Abstract

We present an exact analytical solution of a generic two-mode quantum Rabi model with non-identical bosonic modes. By means of a Bogoliubov transformation, the Hamiltonian is reduced to invariant subspaces labeled by a conserved quantity, and a transcendental G-function is derived, whose zeros determine the regular energy spectrum. We show that the spectra in different subspaces are related by a simple energy shift, revealing the crucial role of mode asymmetry in shaping the spectral structure. In the symmetric limit, the model exhibits an enhanced degeneracy, including fourfold degenerate energy levels arising from the merging of invariant subspaces. The structure of exceptional solutions is analyzed in detail. We identify finite-dimensional Juddian solutions associated with level crossings, as well as special nondegenerate solutions that give rise to isolated states. The degeneracy of these solutions is shown to be controlled by the underlying symmetry of the model. Furthermore, we investigate spectral collapse in the generic two-mode quantum Rabi model. Unlike in the two-photon quantum Rabi model, the collapse energy depends explicitly on the subspace, leading to a richer collapse scenario. We demonstrate that spectral collapse originates from the vanishing spacing between adjacent poles of the G-function, and that the collapse point supports an infinite sequence of bound states. These states exhibit exponential accumulation toward the collapse energy, revealing a hierarchical structure of collapse-induced bound states.