Bose-Hubbard dimers, Viviani's windows and pendulum dynamics

TL;DR

This paper explores the geometric interpretation of the two-mode Bose-Hubbard model, linking classical curves to quantum states, and demonstrates its relation to pendulum dynamics and Viviani's curve.

Contribution

It introduces a geometric perspective of the Bose-Hubbard dimer using intersection curves, generalizes Viviani's curve, and connects classical dynamics with quantum quantization.

Findings

Orbits as intersection curves of sphere and cylinder

Dynamics match those of a simple mathematical pendulum

Action integrals used for semiclassical quantization

Abstract

The two-mode Bose-Hubbard model in the mean-field approximation is revisited emphasizing a geometric interpretation where the system orbits appear as intersection curves of a (Bloch) sphere and a cylinder oriented parallel to the mode axis, which provide a generalization of Viviani's curve studied already in 1692. In addition, the dynamics is shown to agree with the simple mathematical pendulum. The areas enclosed by the generalised Viviani curves, the action integrals, which can be used to semiclassically quantize the N-particle eigenstates, are evaluated. Furthermore the significance of the original Viviani curve for the quantum system is demonstrated.