Semiclassical Spectrum of Small Bose-Hubbard Chains: A Normal Form Approach

TL;DR

This paper uses a semiclassical normal form approach to accurately analyze the low-energy spectrum of a small three-site Bose-Hubbard model, providing insights relevant to current experiments with few particles.

Contribution

It introduces a resonance normal form method to derive an effective classical Hamiltonian that reproduces the quantum spectrum of a small Bose-Hubbard chain.

Findings

Accurately reproduces the low-energy spectrum for small particle numbers

Demonstrates the effectiveness of semiclassical methods in quantum regimes

Provides a classical perspective on quantum many-body systems

Abstract

We analyze the spectrum of the 3-site Bose-Hubbard model with periodic boundary conditions using a semiclassical method. The Bohr-Sommerfeld quantization is applied to an effective classical Hamiltonian which we derive using resonance normal form theory. The derivation takes into account the 1:1 resonance between frequencies of a linearized classical system, and brings nonlinear terms into a corresponding normal form. The obtained expressions reproduce the exact low-energy spectrum of the system remarkably well even for a small number of particles N corresponding to fillings of just two particles per site. Such small fillings are often used in current experiments, and it is inspiring to get insight into this quantum regime using essentially classical calculations.