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

TL;DR

This paper explores the classical-quantum correspondence in bosonic two-mode conversion systems, using polynomial algebras and Kummer shapes to analyze the dynamics, eigenvalues, and state densities in the large particle number limit.

Contribution

It introduces a framework connecting many-particle quantum systems with their classical mean-field limits via polynomial algebra deformations and geometric Kummer shapes.

Findings

Eigenvalues can be derived from mean-field dynamics using WKB quantization.

State densities are approximated by periods of classical orbits.

Bifurcation analysis reveals characteristic steps and singularities in the dynamics.

Abstract

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.