Classical-quantum correspondence in bosonic two-mode conversion systems: polynomial algebras and Kummer shapes

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Bosonic quantum conversion systems can be modeled by many-particle single-mode Hamiltonians describing a conversion of n molecules of type A into m molecules of type B and vice versa. These Hamiltonians are analyzed in terms of generators of a polynomially deformed su(2) algebra. In the mean-field limit of large particle numbers, these systems become classical and their Hamiltonian dynamics can again be described by polynomial deformations of a Lie algebra, where quantum commutators are replaced by Poisson brackets. The Casimir operator restricts the motion to Kummer shapes, deformed Bloch spheres with cusp singularities depending on m and n. It is demonstrated that the many-particle eigenvalues can be recovered from the mean-field dynamics using a WKB type quantization condition. The many-particle state densities can be semiclassically approximated by the time-periods of periodic orbits, which show characteristic steps and singularities related to the fixed points, whose bifurcation properties are analyzed.