Supersymmetry journey from the Jaynes–Cummings to the anisotropic Rabi model

Abstract

We revisit the Jaynes–Cummings and anti-Jaynes–Cummings model through the lens of the Lie theory, aiming to highlight the efficacy of an operator-based approach for diagonalization. We focus on explicitly delineating the steps from an underlying abstract supersymmetry, provided by the u(1|1) superalgebra, into concrete proper states and energies in the laboratory frame. Additionally, we explore the anisotropic Rabi model possessing an underlying supersymmetry, provided by the osp(2|2) superalgebra, in a squeezed reference frame, where it is possible to approximate its spectral characteristics by an effective Jaynes–Cummings model. Finally, we identify a regime for a factorizable anisotropic Rabi model, exhibiting an equally spaced, double degenerate energy spectrum with a unique ground state energy. Our work aims to merge mathematical physics with practical quantum optics, underscoring the critical role of the Lie theory.