Polynomial Lie Algebras \(\hat E_{R_1 }^\mathcal{P} (u(n);(m))\) in Nonlinear Models of Quantum Optics: Basic Ideas and Cluster Dynamics in the Heisenberg Picture

Abstract

We consider applications of polynomial Lie algebras \(\hat E_{R_1 }^\mathcal{P} (u(n);(m))\), which are extensions of the unitary Lie algebras u(n) by their symmetric tensors vm of mth rank, to examine a class of nonlinear models of quantum optics and laser physics. Particular attention is paid to examining integrability of evolution equations arising from the Heisenberg equations for collective (cluster) dynamic variables related to the \(\hat E_{R_1 }^\mathcal{P} (u(2);(2))\) generators. This allows one to examine some peculiarities of cluster dynamics of models incorporating second-harmonic generation together with interference phenomena. Full integrability of such equations at the quantum and quasiclassical levels is shown to be possible for particular values of the model parameters. In general, these equations are integrable only in the quasiclassical (cluster mean-field) approximation. They do not admit a full separation of variables and consequently correspond to very irregular dynamic regimes.