Generalized Rabi models: diagonalization in the spin subspace and differential operators of Dunkl type

Abstract

A discrete parity ${\mathbb{Z}}_2$ symmetry of a two parameter extension of the quantum Rabi model which smoothly interpolates between the latter and the Jaynes-Cummings model, and of the two-photon and the two-mode quantum Rabi models enables their diagonalization in the spin subspace. A more general statement is that the respective sets of $2\times 2$ hermitian operators of the Fulton-Gouterman type and those diagonal in the spin subspace are unitary equivalent. The diagonalized representation makes it transparent that any question about integrability and solvability can be addressed only at the level of ordinary differential operators of Dunkl type. Braak's definition of integrability is shown (i) to contradict earlier numerical studies and (ii) to imply that any physically reasonable differential operator of Fulton-Gouterman type is integrable.

6 pages, minor fine-tuning

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Cited by in corpus (12)

• The quantum Rabi model: solution and dynamics
• Nonlinear Quantum Rabi Model in Trapped Ions
• Spectral Collapse in the two-photon quantum Rabi model
• Dissipative quantum phase transition of light in a generalized Jaynes-Cummings-Rabi model
• Quantum simulation of driven para-Bose oscillators
• Simulating para-Fermi oscillators
• Cross-cavity quantum Rabi model
• Squeezed displaced entangled states in the quantum Rabi model
• The asymmetric quantum Rabi model in the polaron picture
• A unified treatment of polynomial sectors and constraint polynomials of the Rabi models
• A Generalization of the Parametric Amplifier with Dunkl Derivative: Spectral and Statistical Properties
• Unconventional supersymmetric quantum mechanics in spin systems