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.

Figure 1

Energy levels of the three-site BHM (red solid lines) and semiclassical levels given by Eq. (17) (black dashed lines) as a function of the interaction strength $g$ for $N=24$ atoms. The energy levels are rescaled to the Bogoliubov frequencies [i.e., we show $(E-E_0)/\Omega$, where $E_0$ is the energy of the ground state, and $\Omega =\sqrt{A}=\sqrt{9+2g}$]. The Bogoliubov levels are denoted by the (blue) dotted lines.

Figure 2

Energy levels of the three-site BHM (solid red lines) and semiclassical levels given by Eq. (17) (dashed black line) as a function of the interaction strength $g$ for $N=6$. The energy levels are rescaled to the Bogoliubov frequencies. The corresponding Bogoliubov levels are denoted by dotted (blue) lines.

Figure 3

Energy levels of the three-site BHM (black squares) and semiclassical energy levels given by Eq. (24) (red circles) as a function of the interaction strength $g$, for $N=12$. The levels are almost indistinguishable on the scale of (a). Panel (b) represents an enlarged view of a part of (a).

Figure 4

Energy levels of the two-site BHM (red solid lines) and the first five semiclassical levels (A15) (black dashed lines) for $N=9$ atoms.

Figure 5

Phase portraits of the Hamiltonian (A10). Left: $g=1.$ Right: $g=100$; the separatrix divides the phase space into domains of oscillatory and rotational motion. The size of the separatrix loop diminishes with increasing $g$, so at large enough values of $g$ the area of the oscillatory domain divided by $2\pi$ becomes smaller than the minimal action of the semiclassical system. Even before that, the phase trajectory with the minimal classical action becomes so elongated along the $\phi$ axis that an expansion around the origin becomes inaccurate.