Ultracold atoms, especially when deposited in optical lattices, have vast applications in research fields ranging from atomic and molecular physics over solidstate physics to quantuminformation processing. The flexibility and precise controllability of the trapping potential and the interaction strength between the atoms enables the quantum simulation of a large class of Hamiltonians which allows for studying, e.g., (i) molecular dynamics in confined geometries, (ii) quantum phases and quantumphase transitions, (iii) quantum transport and scattering in 1D, 2D and 3D confinement or in discretized space. The understanding of these processes could eventually render the possibility to implement quantum computers with ultracold atoms in optical lattices.
A magnetic Feshbach resonance (MFR) occurs if a resonant molecular bound state (RBS) has an energy close to the scattering energy of the unbound atoms. MFRs are the major tool to tune the interaction strength between atoms.
Although many channels take part in the scattering process, twochannel and even singlechannel models can be used to reproduce the scattering behaviour and molecular processes at an MFR [1].
Confinement of atoms leads to MFR phenomena which are not present in free space and which might offer additional control over the system [2]:

The interaction strength can differ drastically for different trap levels.

The resonance position is shifted from the one observed in free space.

Confinementinduced molecules can be generated in exited trap states.
[1] PRA 80 061401 (2009), PRA 81 022719 (2010)
[2] arXiv:1005.5306 (2010) 

The BoseHubbard model (BHM, formally more correct would be Hubbard model for Bosons) is regularly used to describe ultracold Bosonic atoms in the first Bloch band of optical lattices. It parameterizes the Hamiltonian in terms of the hopping amplitude J, the onsite interaction strength U and onsite energies E_{i} [1].
The comparison of the predictions of the BHM with the eigenenergies of an exact diagonalization of the Hamiltonian of two interacting atoms confined to three sites of an optical lattice reveals the range of applicability of the BHM. The analysis shows that the traditional BHM reproduced the exact energies only for weak interaction. For strong interaction the coupling to many Bloch bands leads to a breakdown of the BHM. Surprisingly, the applicability can be extended to arbitrarily strong interactions (i.e. infinite scattering length) by replacing U with a corrected parameter U^{corr}.
[1] Phys. Rev. A 80, 013404 (2009) 

The exact description of dynamical processes of atoms in optical lattices is numerically demanding, since several Bloch bands can be involved. It is therefore in general hard to answer even simple questions like which of the two processes depicted on the left is dominant, e.g., for large repulsive interaction between the atoms.
By implementing a time propagation based on the exact eigen solutions of a double or triplewell potential we aim to analyze quantum transport and entangling of atoms in an optical lattice on a fundamental level.


We solved analytically the problem of two identical particles on a 1D superlattice with a period twice the natural lattice spacing and found that the scattering process of the particles exhibits a very interesting behavior: In the case that one sub Bloch band is doubly occupied the particles can form a bound state in the continuum due to the collision [1]. These exotic states do not appear in homogeneous lattices and imply a singleparticle interband transition in spite of absence of external forces or dissipation.
A common but spectacular effect in lattice physics is the existence of bound states also for repulsive interactions. On a superlattice one can find additionally bound states in the band gaps of the twoparticle continuumenergy spectrum (see red lines in the energy spectrum) which could be important for analytical descriptions of twoqubit gates.
[1] EPL 92, 10001 (2010)

Quantum systems in reduced dimensionalty show a broad range of interesting phenomena and offer a long list of open questions. In optical lattices reduced dimensions are realized via tight confinement and thus freezing out motion in some spatial directions. The resulting systems are quasi one or two dimensional. The presently most famous system of reduced dimension is surely graphene. Other interesting phenomena in systems with one or two dimensions are:

Quasi 1d:

Fermionization of Bosons, TonksGirardeau gas.

Strongly correlated quantum gas phase, SuperTonksGirardeau gas.

Confinementinduced resonances, i.e. 1d effective interaction diverges.

Quasi 2d:

Confinementinduced resonances, i.e. 2d effective interaction diverges.

Inheritance of exclusive repulsive interactions.

