Effective three-body interaction in an asymmetric double-well optical lattice

We study ultracold atoms in a double-well optical lattice, with a view to creating an effective Hamiltonian that has large three-body interaction energy. The lattice has an asymmetric double-well geometry along the $x$ axis and single wells along the perpendicular axes. We obtain tunneling and two-body interaction energies using numerically constructed Wannier functions from an exact band structure calculation. This gives a Bose-Hubbard (BH) Hamiltonian spanning the lowest two bands along the $x$ axis, and the ground band along the perpendicular axes. We then obtain the many-particle (MP) states, $|\nu,N\rangle$, where $N(\le 3)$ is the particle number per site, and $\nu\in \{1,N+1\}$, by diagonalizing the on-site BH Hamiltonian in the particle number basis. Starting with the ground MP states, we show that tunneling is predominantly confined to the ground state in each $N$ sector. We thus create an effective Hamiltonian ($H_{\rm eff}$) in the ground MP states, and show that $H_{\rm eff}$ has large three-body interaction energy ($\Gamma_3$), comparable to or larger than the two-body term ($\Gamma_2$). The ratio $\Gamma_3/\Gamma_2$ can be tuned by changing the lattice parameters. We are now investigating the possibility of having unique many-body ground states for such systems.