Pair condensation in the BCS-BEC crossover of ultracold atoms loaded onto a two-dimensional square lattice

We investigate the crossover from the Bardeen-Cooper-Schrieffer (BCS) state of weakly bound Cooper pairs to the Bose-Einstein condensate (BEC) of strongly bound molecular dimers in a gas of ultracold atoms loaded on a two-dimensional optical lattice. By using the mean-field BCS equations of the emerging Hubbard model and the concept of off-diagonal long-range order for fermions we calculate analytically and numerically the pair binding energy, the energy gap, and the condensate fraction of Cooper pairs as a function of interaction strength and filling factor of atoms in the lattice at zero temperature.