Shaping topological properties of the band structures in a shaken optical lattice

The realization of band structures with nontrivial topological properties in an optical lattice is an exciting topic in current studies of ultracold atoms. Here we point out that this lofty goal can be achieved by using a simple scheme of shaking an optical lattice, which is directly applicable in current experiments. The photon-assisted band hybridization leads to the production of an effective spin-orbit coupling, in which the band index represents the pseudospin. When this spin-orbit coupling has finite strengths in multiple directions, nontrivial topological structures emerge in the Brillouin zone, such as topological defects with a winding number of 1 or 2 in a shaken square lattice. The shaken lattice also allows one to study the transition between two band structures with distinct topological properties.