The relationship between classical and quantum three one-mode systems interacting in a non-linear way is described. We investigate the integrability of these systems by using the reduction procedure. The reduced coherent states for the quantum system are constructed. We find the explicit formulas for the reproducing measure for these states. Examples of some applications of the obtained results in non-linear quantum optics are presented.
A four-wave mixing Hamiltonian system on the classical as well as on the quantum level is investigated. In the classical case, if one assumes the frequency resonance condition of the form ω0 − ω1 + ω2 − ω3 = 0, this Hamiltonian system is integrated in quadratures, and the explicit formulas of solutions are presented. Under the same condition, the spectral decomposition of quantum Hamiltonian is found, and thus, the Heisenberg equation for this system is solved. Some applications of the obtained results in non-linear optics are discussed.
In the framework of the Poisson geometry of twistor space we consider a family of perturbed 3-dimensional Kepler systems. We show that Hamilton equations of this systems are integrated by quadratures. Their solutions for some subcases are given explicitly in terms of Jacobi elliptic functions.
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