Polynomial deformations of Lie algebras in quantum optics

Abstract

A new approach is proposed for solving nonlinear problems of quantum optics with internal symmetries. Generalized Weyl-Howe dual pairs of algebras are introduced which simultaneously describe both invariance of Hamiltonians H and dynamic symmetries (DS) of models under study. In the general case the approach leads to polynomial deformations gd of Lie algebras as DS algebras. We give two schemes of the application of algebras gd to solving concrete physical tasks. One of them is based on the use of the defining relations of algebras gd and in another one finds exactly solvable analogs of original problems via mappings of algebras gd into some familiar Lie algebras.