Floquet edge states with ultracold atoms

We describe an experimental setup for imaging topologically protected Floquet edge states using ultracold bosons in an optical lattice. Our setup involves a deep two-dimensional optical lattice with a time-dependent superlattice that modulates the hopping between neighboring sites. The finite waist of the superlattice beam yields regions with different topological numbers. One can observe chiral edge states by imaging the real-space density of a bosonic packet launched from the boundary between two topologically distinct regions.

Figure 1

Snapshots of the potentials used to produce the tight-binding model for trapped atoms studied in this paper [Eq. (1)]. Each potential is applied sequentially for a fixed period of time. As shown in the key, blue and red respectively correspond to low and high potential. As depicted by the white arrows, during time steps 1, 2, 3, and 4, hopping only occurs between closely spaced “dimers.” No hopping occurs during time step 5. The labeling of sites $A_{ij}$ and $B_{ij}$ are illustrated in panel 5. For the protocol described here, step 5 plays no role, but is convenient for generalizations that include a potential bias between A sites and B sites [17].

Figure 2

Floquet band structure of the model in Eq. (1) with open boundary conditions in the $x$ direction and periodic boundary conditions in the $y$ direction. For weak hopping [$JT<J_{c}T=\frac{5}{4}\pi$: (a) and (b)] there are two overlapping bulk bands and the winding number $W=0$ [see Eq. (6)]. For larger $J$, two edge modes are apparent (shown in red), and $W=1$ (cf. Ref. [17]).

Figure 3

(a) Spatially dependent hopping $J\left(x\right)$. (b), (c), (d) The corresponding local density of states $\rho (\epsilon ,x)$ at $\epsilon T=\pi ,\pi /2,0$ [see Eq. (8)]. The red dashed lines in the top graph separate spatial regions that are in different topological phases. Midgap edge states at $\epsilon T=\pi$ are spatially localized at the boundary $\pm x_o$ between the different phases. Dashed horizontal lines are drawn to help visualize the strongest features.

Figure 4

Dispersion relation $\epsilon \left(k_y\right)$ for a system with spatially dependent hopping $J\left(x\right)$ (see Fig. 3). The red points specify the energies of “massless” edge states with dispersion $[\epsilon \left(k_y\right)-\frac{\pi }{T}]\propto k_y$. These lie at the boundary between the $W=1$ and $W=0$ regions in Fig. 3. The edge-state group velocity $V_g$ is given by slope of these lines $V_g=\frac{\partial \epsilon }{\partial k_y}\approx 1.34a/T$. The blue points specify the energies of “massive” states with dispersion $[{\epsilon }_n\left(k_y\right)-\frac{\pi }{T}]\propto \sqrt{{\widetilde{m_n}}^2+V_g^2{k}_y^2}$ [see Eq. (10)].

Figure 5

Real-space density showing chiral edge modes. Darker colors represent higher density. Column (a) shows the dynamics resulting from the experimental protocol described in Sec. 3 B with spatially dependent hopping $J(x,y)T=2\pi exp[-12\frac{{(x-L/2)}^2+{(y-L/2)}^2}{L^2}]$, where $L=40a$. At time $t=0$ a condensate is placed in the lattice. A wave packet moves in the clockwise direction along the circular boundary separating two topologically distinct regions (dashed red line). Column (b) shows the dynamics resulting from running the steps shown in Fig. 1 in reverse. The edge-state wave packet now moves in the counterclockwise direction. Column (c) shows the dynamics resulting after initializing a wave packet away from the circular boundary. In this case there is no observable edge-state propagation.