Generalized n-level Jaynes–Cummings and Dicke models, classical rational r-matrices and algebraic Bethe ansatz

Abstract

We construct integrable n-level generalizations of Jaynes–Cummings and Dicke models corresponding to the simple (reductive) Lie algebras of rank n. We show that for each such Lie algebra there exist many integrable Jaynes–Cummings and Dicke-type models each of which is associated with the reductive subalgebras containing Cartan subalgebra and is obtained via reduction from Jaynes–Cummings or Dicke-type models with a maximal number of bosons. We diagonalize the constructed Jaynes–Cummings and Dicke-type Hamiltonians in the physically most interesting case of using a nested algebraic Bethe ansatz.