Controllable Production of Degenerate Fermi Gases of $^6$Li Atoms in the 2D-3D Crossover

The many-body physics in the dimensional crossover regime attracts much attention in cold atom experiments, but yet to explore systematically. One of the technical difficulties existed in the experiments is the lack of the experimental technique to quantitatively tune the atom occupation ratio of the different lattice bands. In this letter, we report such techniques in a process of transferring a 3D Fermi gas into a 1D optical lattice, where the capability of tuning the occupation of the energy band is realized by varying the trapping potentials of the optical dipole trap (ODT) and the lattice, respectively. We could tune a Fermi gas with the occupation in the lowest band from unity to 50$\%$ quantitatively. This provides a route to experimentally study the dependence of many-body interaction on the dimensionality in a Fermi gas.