Ultracold atomic gases can be prepared in laboratory environments with unrivaled control and purity. In one and a half decades the field has evolved from the first Bose condensates in dilute alkali gases to multiple species mixtures and degenerate Fermi systems. Combining quantum degenerate fermions and optical lattices enables the simulation of relevant condensed matter systems. By observing the final state of an atomic sample in a tailored optical potential the system can be used as an analog quantum computer to evaluate Hamiltonians that are computationally impossible to tackle on classical computers. A versatile optical dipole trap has been constructed and characterized. The trap can be converted to an optical lattice, allowing for the investigation of the phase diagram of the two-component, one-dimensional, imbalanced Fermi gas.