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Experiments with ultracold atoms in optical lattice have become a versatile

testing ground to study diverse quantum many-body Hamiltonians. A single-band

Bose-Hubbard (BH) Hamiltonian was first proposed to describe these systems in

1998 and its associated quantum phase-transition was subsequently observed in

  1. Over the years, there has been a rapid progress in experimental realizations

of more complex lattice geometries, leading to more exotic BH Hamiltonians with

contributions from excited bands, and modified tunneling and interaction energies.

There has also been interesting theoretical insights and experimental studies on “un-

conventional” Bose-Einstein condensates in optical lattices and predictions of rich

orbital physics in higher bands. In this thesis, I present our results on several multi-

band BH models and emergent quantum phenomena. In particular, I study optical

lattices with two local minima per unit cell and show that the low energy states of a

multi-band BH Hamiltonian with only pairwise interactions is equivalent to an effec-

tive single-band Hamiltonian with strong three-body interactions. I also propose a

second method to create three-body interactions in ultracold gases of bosonic atoms

in a optical lattice. In this case, this is achieved by a careful cancellation of two

contributions in the pair-wise interaction between the atoms, one proportional to

the zero-energy scattering length and a second proportional to the effective range.

I subsequently study the physics of Bose-Einstein condensation in the second band

of a double-well 2D lattice and show that the collision aided decay rate of the con-

densate to the ground band is smaller than the tunneling rate between neighboring

unit cells. Finally, I propose a numerical method using the discrete variable repre-

sentation for constructing real-valued Wannier functions localized in a unit cell for

optical lattices. The developed numerical method is general and can be applied to

a wide array of optical lattice geometries in one, two or three dimensions.