Rigorous interpretation of U(1)-symmetry remnants in S2/S2′ phases of the Extended Rabi Model

Develop a rigorous interpretation of the observed remnants of U(1) symmetry in the S2 and S2′ excited superradiant phases of the Extended Rabi Model (Hamiltonian (6)) for interaction-asymmetry parameter |δ| ≠ 1, potentially using the framework of quasi-dynamical and partial-dynamical symmetries, to explain the semiclassical features near the origin associated with the Mexican-hat structure of Hamiltonian (7).

Background

In the excited superradiant phases S2 (Jaynes–Cummings-like) and S2′ (anti–Jaynes–Cummings-like) of the Extended Rabi Model, the semiclassical Hamiltonian exhibits a local Mexican-hat-like structure near the origin, leading to emergent states whose Wigner functions concentrate around the local maximum and separatrix. The authors note that these phases display remnants of the U(1) symmetry characteristic of the pure (anti) Jaynes–Cummings limits.

A formal, symmetry-based explanation for these remnants is not yet established. The paper points to the possible applicability of quasi-dynamical and partial-dynamical symmetries to account for this structure, and explicitly identifies the need for a more rigorous interpretation connecting the observed spectral and phase-space features to such symmetry frameworks.