Topological Fractional Pumping with Alkaline-Earth-Like Atoms in Synthetic Lattices

Alkaline-earth(-like) atoms, trapped in optical lattices and in the presence of an external gauge field, can form insulating states at given fractional fillings. We will show that, by exploiting these properties, it is possible to realize a topological fractional pump. Our analysis is based on a many-body adiabatic expansion, on simulations with time-dependent matrix product states, and, for a specific form of atom-atom interaction, on an exactly solvable model of fractional pump. The numerical simulations allow us to consider a realistic setup amenable of an experimental realization. As a further consequence, the measure of the center-of-mass shift of the atomic cloud would constitute the first measurement of a many-body Chern number in a cold-atom experiment.

Figure 1

Left: Three contiguous nuclear spin states of ${}^{173}\mathrm{Yb}$ ($F=5/2$) are coupled through independent Raman couplings which adiabatically eliminate long-lived excited states (${}^3{P}_1$, $F^{\prime }=7/2$). Energies are not in scale. By coupling only a subset of the nuclear spin states, an effective spin $\mathcal{I}=1$ is obtained. Right: The system geometry with periodic boundaries is a thin torus with a long real dimension and a length-three synthetic one. Two fluxes, ${\mathrm{\Phi }}_R$ and ${\mathrm{\Phi }}_S$, pierce the noncontractible circles. A flux $\gamma$ trades every synthetic plaquette.

Figure 2

Pumped charge after one period for a system with hard-wall boundaries, $L=48$, $\gamma =\pi /3$, $U/t=\infty$, $V/t=10$; $\mathrm{\Omega }/t=1$ for $\nu =1/2$ and $\mathrm{\Omega }/t=0.25$ for $\nu =1/3$. Left: Space-resolved charge $Q_j^{(h)}(\tau )$ for $\nu =1/2$ and $T=24$. Right: Pumped charge $Q_{\text{av}}(T)$ as a function of period $T$ for $\nu =1/2$ (top) and $\nu =1/3$ (bottom). Dashed lines show quantized values. The pumped charge is independent of the initial eigenstate.

Figure 3

Left: Time evolution of the density profile at $\nu =1/3$. The parameters are specified in the caption of Fig. 2; $T=100t^{-1}$. For such a value of $T$, the system is essentially in the instantaneous ground state at all times. Right: Flux dependence of ${\epsilon }_{j,\lambda =0}({\mathrm{\Phi }}_S)$.

Figure 4

Center-of-mass displacement of the cloud ${\mathrm{\Delta }}_{\mathrm{c}.\mathrm{m}.}(T)/\ell$ (top) and pumped charge $Q_{\text{av}}(T)$ (bottom) versus $T$ for a magnetic crystal with $\nu =1/2$ in a harmonic potential. Dashed lines show quantized values. In order to observe the oscillations due to nonadiabatic corrections, a time resolution $\sim t^{-1}$ is needed. The figure indicates that the small metallic wings at the crystal edges, due to the harmonic potential, do not significantly affect the quantization. Parameters: $\gamma =2\pi /3$, $U/t=\infty$, $\mathrm{\Omega }/t=1$, $w_0/t=0.01$, $L=48$, $N=10$.