Chern numbers hiding in time-of-flight images

We present a technique for detecting topological invariants—Chern numbers—from time-of-flight images of ultracold atoms. We show that the Chern numbers of integer quantum Hall states of lattice fermions leave their fingerprints in the atoms’ momentum distribution. We analytically demonstrate that the number of local maxima in the momentum distribution is equal to the Chern number in two limiting cases, for large hopping anisotropy and in the continuum limit. In addition, our numerical simulations beyond these two limits show that these local maxima persist for a range of parameters. Thus, an everyday observable in cold atom experiments can serve as a useful tool to characterize and visualize quantum states with nontrivial topology.

Figure 1

Relation between the Chern number $c_r$ and structure in TOF data as a function of magnetic flux per unit cell $\alpha$ in units of the flux quantum $h/e$ and hopping anisotropy $\lambda$. (a) Schematic map of the $\alpha$-$\lambda$ plane showing the Chern dimer and the continuum regimes. Numerical results for the remainder of the parameter space will be shown in Figs. 2 and 3. (b and c) Normalized 1D crystal momentum distributions $n\left(k_x\right)$ [see Eq. (4)], computed at three different chemical potentials corresponding to $c_r=2,3,4$; in each case, the number of peaks or oscillations is equal to $c_r$. The data in (b) is for an isotropic $\lambda =1$ lattice at $\alpha =0.05$, and the data in (c) is for an anisotropic $\lambda =1000$ lattice at $\alpha =0.45$. $k_x$ is measured in units of $1/d$, where $d$ is the lattice spacing.

Figure 2

1D crystal momentum distributions $n\left(k_x\right)$. (a) Energy spectrum of Eqs. (1) and (2) for $\lambda =2$. (b) Evolution of $n\left(k_x\right)$ with increasing chemical potential $\mu$ at $\lambda =1$ and $\alpha =0.05$ showing $c_r$ increasing in tandem with the number of oscillations in $n\left(k_x\right)$. (c and d) Evolution of $n\left(k_x\right)$ with increasing anisotropy $\lambda$ at $\alpha =0.45$ for (c) $c_r=2$ and (d) $c_r=3$. These curves depict the distributions approaching the cosine function with increasing $\lambda$. The unit for $k_x$ is $1/d$.

Figure 3

Energies in the anisotropic limit for $p=1$, $q=5$ showing the perfect sinusoidal energies when $\lambda \to \infty$ (solid black curves), and the avoided crossings when $\lambda =10$ (solid blue curves). The hybridized states at these crossings correspond to Chern dimers, and the red-circled example has the spatial extent $\delta s_x=1$. The unit for $k_y$ is $1/d$.

Figure 4

Momentum distribution function $n\left(k_x\right)$ for the isotropic lattice at temperature $T=0.2t$ and a trap potential with $m{\omega }^2{d}^2={10}^{-3}t$. ${\mu }_0$ at the trap center is set to $-0.75t,-0.5t$ for $\alpha =0.2,0.3$, respectively. The twin peaks, indicated by the arrows, allow us to determine the Chern number near the trap center to be 2. The unit for $k_x$ is $1/d$.