Semiclassical analysis of Bose-Hubbard dynamics

TL;DR

This paper analytically investigates the two-site Bose-Hubbard model using semiclassical approximation and perturbation theory, providing explicit formulas for occupation differences and revealing collapse and revival phenomena with high accuracy.

Contribution

It offers a novel analytical approach to understanding Bose-Hubbard dynamics in the weak coupling and large particle number limit, extending previous numerical results.

Findings

Analytical expression for occupation difference between sites.

Excellent agreement with numerical solutions.

Description of collapse and revival phenomena.

Abstract

In this work the two site Bose-Hubbard model is studied analytically in the limit of weak coupling u and large number of particles N . The semiclassical approximation where \frac{1}{N} plays the role of Planck's constant was used and perturbation theory to order u^{2} was applied. In particular, the difference in the occupation between the two sites, where initially all particles are at one site was calculated analytically. Excellent agreement with the exact numerical solution was found. This quantity exhibits collapses and revivals that superimpose rapid oscillations. The occupation difference was calculated also for the case where initially both sites are occupied provided that the difference in occupation is sufficiently large. It provides an analytical description of results that were so far found only numerically. Similar behavior and analysis are expected for a large variety of physical situations in optics, atom optics and quantum dynamics of electrons in Rydberg atoms.