Bloch Band And Bloch Waves Of Superfluid Fermi Gases In Optical Lattices

Currently, experiments on ultracold Fermi gases are rapidly advancing toward therealization of super?uid states in optical lattices. The Fermi gases can be changed contin-uously from a Bose-Einstein condensate(BEC) to a BCS super?uid by using the Feshbachresonance technique. Now, the super?uidity of ultracold fermions in optical lattices hasbeen established. In such a super?uid system, people have given many theoretical e?ortson the crossover from the BEC state to the BCS state in optical lattices, such as the Blochoscillations, the super?uid-insulator transition, the collective excitations, the thermody-namic properties, and the stability. The intrinsically localized excitations [i.e., gap solitonin one- (1D) and two-dimensional (2D) optical lattices, self-trapping, and stable-movingsoliton and breather in three-dimensional (3D) optical lattices] are also presented.In the first part, Bloch band and stabilities of Bloch waves of super?uid Fermi gasesin one-dimensional periodic optical lattices are discussed. Within the hydrodynamicaltheory and the two-mode approximation, the Bloch band structure, the energetic anddynamical instabilities of Bloch waves at the first Brilliouin zone are presented. Theresults show that, when the atom density is beyond a critical value, a loop structurein the Bloch band at the zone edge is developed along the BCS-BEC crossover. TheBloch band structure and the stabilities of Bloch waves are modified dramatically whenthe system crosses from the BCS side to the BEC side, and they can be adjusted to therequired characteristics by changing the atom s interaction (with the Feshbach resonancetechnique), the atom density, and the lattice parameters. The analytical expressions ofthe critical atom density for exciting the loop structure and maintaining the stabilities ofBloch waves are obtained.Secondly, the existence and stability of gap soliton(GS) and gap wave of super?uidFermi gases in 1D periodic optical lattices are discussed. Within the framework of mean-field- hydrodynamic model and with a periodic potential. We present several GS speciesin first two band gaps of the optical lattices induced spectrum. These include fundamentalgap solitons(FGSs) and their bound states, as well as a subfundamental gap soliton in thesecond gap, featuring two peaks of opposite signs in a single well of the periodic potential.We found this kind of gap soliton exist both in the BEC limit and at unitary regime.Linear-stability analysis demonstrate that the stability property are di?erent between inthe BEC limit and at the unitary regime. However, fundamental solitons in first bandgap and second band gap are always stable not only in BEC limit but also at unitaryregime. Lastly, our results show that there also exists a composition relation betweenBloch waves and GSs or gap waves in super?uid Fermi gases like that in Bose system, and GSs is viewed as a segment of the Bloch waves. Furthermore, we also reveal thatself-trapping e?ect in one-dimensional optical lattices can be associated with the existenceof a localized gap state(gap wave) in super?uid Fermi gas.Finally, we summarize the main result of this paper and give an outlook of the futurein this field.