Integrability and superintegrability of the generalized n-level many-mode Jaynes–Cummings and Dicke models

Abstract

We analyze symmetries of the integrable generalizations of Jaynes–Cummings and Dicke models associated with simple Lie algebras g and their reductive subalgebras gK [T. Skrypnyk, “Generalized n-level Jaynes-Cummings and Dicke models, classical rational r-matrices and nested Bethe ansatz,” J. Phys. A: Math. Theor. 41, 475202 (2008)]. We show that their symmetry algebras contain commutative subalgebras isomorphic to the Cartan subalgebras of g, which can be added to the commutative algebras of quantum integrals generated with the help of the quantum Lax operators. We diagonalize additional commuting integrals and constructed with their help the most general integrable quantum Hamiltonian of the generalized n-level many-mode Jaynes–Cummings and Dicke-type models using nested algebraic Bethe ansatz.