Floquet topological states in shaking optical lattices

In this Rapid Communication we propose realistic schemes to realize topologically nontrivial Floquet states by shaking optical lattices, using both the one-dimensional lattice and two-dimensional honeycomb lattice as examples. The topological phase in the two-dimensional model exhibits quantum anomalous Hall effect. The transition between topological trivial and nontrivial states can be easily controlled by both shaking frequency and shaking amplitude. Our schemes have two major advantages. First, both the static Hamiltonian and the shaking scheme are sufficiently simple to implement. Secondly, it requires relatively small shaking amplitude and therefore heating can be minimized. These two advantages make our schemes much more practical.

Figure 1

(a) The typical energy structure under consideration. (b) The laser setup of the one-dimensional shaking optical lattice. Solid and dashed lines represent lattice potential at two different times. (c) The laser setup of the two-dimensional honeycomb optical lattice. The dashed circle with arrow indicates how each lattice potential rotates in time.

Figure 2

(a) Phase diagram in terms of shaking frequency ${\Delta }_0/E_r$ and shaking amplitude $k_{r}b$. $V=3E_r$ is fixed. (b), (d) Quasienergy spectrum of a finite size one-dimensional shaking optical lattice. ${\Delta }_0/E_r=0$ and $k_{r}b=0.5$ for (b); and ${\Delta }_0/E_r=0.8$ and $k_{r}b=0.5$ for (d), as marked in (a). Inset: Winding of $\mathbf{B}\left(k_x\right)$ in the $yz$ plane as $k_x$ changes from $-\pi$ to $\pi$. See text for a definition of $\mathbf{B}\left(k_x\right)$. (c) The wave functions for the in-gap states of (b).

Figure 3

(a) Phase diagram in terms of the on-site energy difference of the $AB$ sublattice of a honeycomb lattice $M/E_r$ and the shaking amplitude $k_{r}b$. $\omega /E_r$ is fixed at $0.2$. (b), (d) Quasienergy spectrum of a finite size two-dimensional shaking honeycomb lattice with armchair edge. $M/E_r=0$ and $k_{r}b=0.1$ for (b) and $M/E_r=0.003$ and $k_{r}b=0.1$ for (d), as marked in (a). (c) The wave functions for the in-gap states of (b). The lattice potential $V_{\overline{X}}/E_r=5$, $V_X/E_r=0.65$, $V_Y/E_r=2$, and $\alpha =0.8$.