Seeing Topological Order in Time-of-Flight Measurements

In this Letter, we provide a general methodology to directly measure topological order in cold atom systems. As an application, we propose the realization of a characteristic topological model, introduced by Haldane, using optical lattices loaded with fermionic atoms in two internal states. We demonstrate that time-of-flight measurements directly reveal the topological order of the system in the form of momentum-space Skyrmions.

Figure 1

Haldane-type model. (a) Two triangular optical lattices ($A$ and $B$) are Raman-coupled by a laser that makes an atom switch sublattice ($A\leftrightarrow B$). This allows for both next-nearest-neighbor hoppings, $t_{a,b}$, and a complex nearest-neighbor hopping, $t_{jk}$, whose phase depends on the momentum imparted by the Raman laser, $\Delta \mathbf{p}$. (b) Phase diagram of the zero-energy ground states, as a function of the energy imbalance between lattices, $\epsilon /t_a$, and the momentum imparted by the laser, $\Delta \mathbf{p}=(0,\Delta p_y)$. We plot the exact phase boundary in the thermodynamic limit (solid line), together with a color-graded simulation of the Chern number for a finite lattice with $2\times 17\times 17$ sites and $20\times 20$ pixels. The diagrams on the right-hand side show how the Dirac points are displaced on the distribution of Bloch vectors $S_z(\mathbf{k})$ induced by the Hamiltonian [left (blue) positive, right (red) negative].

Figure 2

(a) Spin texture of the Haldane model, interpreted as a mapping from momentum space, $(k_x,k_y)$, onto the Bloch sphere, $\mathbf{S}=⟨\mathit{\sigma }⟩\propto (\cos \varphi \sin \theta ,\sin \varphi \sin \theta ,\cos \theta )$. The colors and arrows show the polar angle, $\theta (\mathbf{k})$, and the azimuthal component of the spin, $(S_x,S_y)$, for a topological phase $\nu =+1$. (b) Interference pictures for this phase which result in momentum density distributions for the $a$ (up) and $b$ (down) particles. A square lattice of $20\times 20$ pixels partitioned into triangles is used to compute the estimate ${\nu }_D=0.9$ [Eq. (6)]. The enclosed area corresponds to (a).

Figure 3

(a) Density of states and (b) Chern number simulation at $\epsilon /t_a=-0.5$ for a lattice with $r=0$ (solid line), ${10}^{-3}$ (dashed line), and 0.02 (dotted line). (c) Chern number for $\epsilon /t_a=-0.5$ and $p_y=3\pi /4a$ [uppermost horizontal solid (blue) line] or $2\pi /a$ [lowermost horizontal solid (black) line] vs confining potential strength, $r$, starting with $1/2$ (solid) or $1/4$ particles per site (dashed) on a lattice with $50\times 50$ sites. The values obtained using $50\times 50$ pixels are compared with those from a $20\times 20$ matrix [middle horizontal (gray) line]. (d) Density plots for the points marked in (c), showing the wedding cake structure where regions with $n=1$ do not contribute to the Chern number.