  We study ultracold atoms in an optical lattice with two local minima per unit
cell and show that the low energy states of a multi-band Bose-Hubbard (BH)
Hamiltonian with only pair-wise interactions is equivalent to an effective
single-band Hamiltonian with strong three-body interactions. We focus on a
double-well optical lattice with a symmetric double well along the $x$ axis and
single well structure along the perpendicular directions. Tunneling and
two-body interaction energies are obtained from an exact band-structure
calculation and numerically-constructed Wannier functions in order to construct
a BH Hamiltonian spanning the lowest two bands. Our effective Hamiltonian is
constructed from the ground state of the $N$-atom Hamiltonian for each unit
cell obtained within the subspace spanned by the Wannier functions of two
lowest bands. The model includes hopping between ground states of neighboring
unit cells. We show that such an effective Hamiltonian has strong three-body
interactions that can be easily tuned by changing the lattice parameters.
Finally, relying on numerical mean-field simulations, we show that the
effective Hamiltonian is an excellent approximation of the two-band BH
Hamiltonian over a wide range of lattice parameters, both in the superfluid and
Mott insulator regions.
