Quasi-exactly solvable models in nonlinear optics

TL;DR

This paper explores a broad class of nonlinear optical models with polynomial Hamiltonians, demonstrating their quasi-exact solvability by constructing commuting integrals of motion and explicit reduced Hamiltonians.

Contribution

It introduces a method to identify quasi-exactly solvable structures in nonlinear optics models with multiple degrees of freedom.

Findings

Constructed complete sets of commuting integrals of motion.

Characterized common eigenspaces of the integrals.

Explicitly expressed reduced Hamiltonians in sl(2) form.

Abstract

We study a large class of models with an arbitrary (finite) number of degrees of freedom, described by Hamiltonians which are polynomial in bosonic creation and annihilation operators, and including as particular cases n-th harmonic generation and photon cascades. For each model, we construct a complete set of commuting integrals of motion of the Hamiltonian, fully characterize the common eigenspaces of the integrals of motion, and show that the action of the Hamiltonian in these common eigenspaces can be represented by a quasi-exactly solvable reduced Hamiltonian, whose expression in terms of the usual generators of sl(2) is computed explicitly.