The two-site Bose--Hubbard model

TL;DR

This paper provides a mathematical overview of the two-site Bose-Hubbard model, analyzing classical and quantum dynamics, and discusses its exact solvability, highlighting a threshold coupling between different phases.

Contribution

It offers a comprehensive analysis of the mathematical properties and phase transitions of the two-site Bose-Hubbard model, including classical, quantum, and algebraic Bethe ansatz approaches.

Findings

Existence of a threshold coupling between delocalized and self-trapped phases

Qualitative agreement with experimental observations

Discussion of the model's exact solvability via algebraic Bethe ansatz

Abstract

The two-site Bose--Hubbard model is a simple model used to study Josephson tunneling between two Bose--Einstein condensates. In this work we give an overview of some mathematical aspects of this model. Using a classical analysis, we study the equations of motion and the level curves of the Hamiltonian. Then, the quantum dynamics of the model is investigated using direct diagonalisation of the Hamiltonian. In both of these analyses, the existence of a threshold coupling between a delocalised and a self-trapped phase is evident, in qualitative agreement with experiments. We end with a discussion of the exact solvability of the model via the algebraic Bethe ansatz.