  We have studied interacting and non-interacting quantum degenerate Fermi
gases in a three-dimensional optical lattice. We directly image the Fermi
surface of the atoms in the lattice by turning off the optical lattice
adiabatically. Due to the confining potential gradual filling of the lattice
transforms the system from a normal state into a band insulator which
constitutes a high fidelity quantum register. The dynamics of the transition
from a band insulator to a normal state is studied and the time scale is
measured to be an order of magnitude larger than the tunneling time in the
lattice. Using a Feshbach resonance we increase the interaction between atoms
in two different spin states and dynamically induce a coupling between the
lowest energy bands. We observe a shift of this coupling with respect to the
Feshbach resonance in free space which is anticipated for strongly confined
atoms.
