Sharma and Fulde-Farrell-Larkin-Ovchinnikov states in an optical lattice

We study an imbalance mixture of atomic Fermi gas of two hyperfine states loaded into an optical lattice. We solve the self- consistent equations for the filling factors and the gap equation to investigate the existence of Sharma and Fulde-Farrell-Larkin-Ovchinnikov (FFLO) superfluid states assuming a contact interaction between the atoms (Hubard model). The order parameter in the case of Fulde-Farrell-Larkin-Ovchinnikov (FFLO) superfluid is chosen to be $Delta_{\textbf{q}}=\Delta_0 \exp\left(\imath \textbf{q.r}\right)$, where $2\textbf{q}$ is the pair momentum in a single plane wave FFLO state.