Microscopic Derivation of Hubbard Parameters for Cold Atomic Gases

We study the exact solution for two atomic particles in an optical lattice interacting via a Feshbach resonance. The analysis includes the influence of all higher bands, as well as the proper renormalization of molecular energy in the closed channel. This exact solution allows for the precise determination of the parameters in the Hubbard model and the two-particle bound state energy. We identify the regime, where a single band Hubbard model fails to describe the scattering properties of the atoms as well as the bound states energies.

Figure 1

Exact bound state energies (red line) for two atoms in an optical lattice of strength $V=12E_r$ and $\mathbf{K}=0$. The additional bound states (dashed gray lines) are weakly coupled to atoms in the lowest Bloch band, i.e., $|w_{\alpha }{|}^2\lesssim {10}^{-4}$. The green dotted line denotes the bound state energies predicted from the Hubbard model with the on-site interaction determined by Eq. (8). The blue dotted line denotes the bound states neglecting the correction $\gamma =3.5/E_r$.

Figure 2

Bound state energies (red line) for different lattice depths: (a) For weak optical lattices with $V=4E_r$, the Bloch bands in three dimensions nearly overlap. (b) For deep optical lattices with $V=50E_r$, the bound state energies are compared with the energies obtained by replacing the optical lattice by a harmonic well with trapping frequency ${\omega }_p=\sqrt{4V{E}_r}/\hslash$ (dashed blue line).