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We study the ground state phase diagram of population balanced and imbalancedultracold atomic Fermi gases with a short range attractive interactionthroughout the crossover from BCS to Bose-Einstein condensation (BEC), in atwo-dimensional optical lattice (2DOL) comprised of two lattice and onecontinuum dimensions. We find that the mixing of lattice and continuumdimensions, together with population imbalance, has an extraordinary effect onpairing and the superfluidity of atomic Fermi gases. In the balanced case, thesuperfluid ground state prevails the majority of the phase space. However, forrelatively small lattice hopping integral $t$ and large lattice constant $d$, apair density wave (PDW) emerges unexpectedly at intermediate coupling strength,and the nature of the in-plane and overall pairing changes from particle-liketo hole-like in the BCS and unitary regimes, associated with an abnormalincrease in the Fermi volume with the pairing strength. In the imbalanced case,the stable polarized superfluid phase shrinks to only a small portion of theentire phase space spanned by $t$, $d$, imbalance $p$ and interaction strength$U$, mainly in the bosonic regime of low $p$, moderately strong pairing, andrelatively large $t$ and small $d$. Due to the Pauli exclusion between pairedand excessive fermions within the confined momentum space, a PDW phase emergesand the overall pairing evolves from particle-like into hole-like, as thepairing strength grows stronger in the BEC regime. In both cases, the groundstate property is largely governed by the Fermi surface topology. Thesefindings are very different from the cases of pure 3D continuum, 3D lattice or1DOL.

Ground states of atomic Fermi gases in a two-dimensional optical lattice with and without population imbalance