Topological phase transitions in tilted optical lattices

We analyze the energy spectrum and eigenstates of cold atoms in a tilted brick-wall optical lattice. When the tilt is applied, the system exhibits a sequence of topological phase transitions reflected in an abrupt change of the eigenstates. It is demonstrated that these topological phase transitions can be easily detected in a laboratory experiment by observing Bloch oscillations of cold atoms.

Figure 1

The brick-wall lattice. The bonds marked by the solid and dashed lines corresponds to the different hoping matrix elements $J_1$ and $J_2$. Notice that for $J_2=0$ the brick-wall lattice transforms into the deformed honeycomb lattice with $A$ and $B$ sites marked by asterisks and open circles.

Figure 2

Energy spectrum of the tilted brick-wall lattice. Parameters are $J_1=0.5$, $J_2=0$, and $dF=1/1.9$ (left panel), $dF=1/2$ (middle panel), and $dF=1/2.1$ (right panel). Through the paper, we use arbitrary energy units since only the ratio between the hopping and Stark energies matters.

Figure 3

Upper panel: Winding number of the phase $\chi$ vs the inverse field magnitude. Lower panel: The energy band $E_{n=0}^{\left(1\right)}$ as a color map. The limits of the energy axis are $E_{min}=0$ (dark blue) and $E_{max}=dF/2$ (bright yellow). The values of the hopping matrix elements are the same as in Fig. 2.

Figure 4

Bloch oscillations of a localized wave packet for $F=1/1.9$ (upper panel) and $F=1/2.1$ (lower panel). Time is measured in units of the tunneling period $T=h/J_1$. Shown are the mean value $x\left(t\right)=\langle \psi \left(t\right)\left|\widehat{x}\right|\psi \left(t\right)\rangle$, blue solid lines, and $y\left(t\right)=\langle \psi \left(t\right)\left|\widehat{y}\right|\psi \left(t\right)\rangle$, red dotted lines. The ratio of the field components $F_x/F_y=0.005$.