Atoms at optical lattices Bose-Hubbard Hamiltonian H ∑ bb+∑6n+2 ∑(-1) D Jaksch et al(1998) 60 nm array of 1D tubes M. Greiner et al., Nature(2002) You can control almost all aspects of the periodic structure and the interactions between the atoms7 Atoms at optical lattices You can control almost all aspects of the periodic structure and the interactions between the atoms M. Greiner et al. , Nature (2002) D.Jaksch et al (1998)      + = − + + − i j i i i j i i ni ni U H J b b n ˆ (ˆ 1) 2 Bose  ˆ -Hubbard Hamiltonian