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

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.

13 pages, 13 figures

References in corpus (6)

• Extended Bose Hubbard model of interacting bosonic atoms in optical lattices: from superfluidity to density waves
• Exact number conserving phase-space dynamics of the M-site Bose-Hubbard model
• Semiclassical quantization of an N-particle Bose-Hubbard model
• Pair condensation of bosonic atoms induced by optical lattices
• Quantum phase transitions of atom-molecule Bose mixtures in a double-well potential
• Semiclassical quantisation for a bosonic atom-molecule conversion system

Cited by in corpus (4)

• Excited-state quantum phase transitions
• Complex density of continuum states in resonant quantum tunneling
• Effective and efficient resonant transitions in periodically modulated quantum systems
• Exact ground-state correlation functions of an atomic-molecular boson conversion model