Semiclassical spectrum of small Bose-Hubbard chains: A normal-form approach

We analyze the spectrum of the three-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.