Unified approach to the nonlinear Rabi models

TL;DR

This paper introduces a unified analytical framework for various nonlinear Rabi models using su(1,1) Lie algebra, revealing exact solutions, spectral properties, and the behavior of eigenstates near spectral collapse, advancing understanding in nonlinear quantum optics.

Contribution

It presents a unified Hamiltonian approach for multiple nonlinear Rabi models, identifying exact solutions and analyzing spectral and eigenstate properties with implications for quantum optics.

Findings

Existence of exact isolated solutions at level crossings.

Regular spectrum obtained via roots of the G-function.

Eigenstate decay rates decrease with increasing coupling, approaching zero near spectral collapse.

Abstract

An analytical approach is proposed to study the two-photon, two-mode and intensity-dependent Rabi models. By virtue of the su(1,1) Lie algebra, all of them can be unified to the same Hamiltonian with $\mathcal{Z}_2$ symmetry. There exist exact isolated solutions, which are located at the level crossings between different parities and correspond to eigenstates with finite dimension. Beyond the exact isolated solutions, the regular spectrum can be achieved by finding the roots of the G-function. The corresponding eigenstates are of infinite dimension. It is noteworthy that the expansion coefficients of the eigenstates present an exponential decay behavior. The decay rate decreases with increasing coupling strength. When the coupling strength tends to the spectral collapse point $g \rightarrow \omega / 2$, the decay rate tends to zero which prevents the convergence of the wave functions. This work paves a way for the analysis of novel physics in nonlinear quantum optics.