Formation and Dynamics of Antiferromagnetic Correlations in Tunable Optical Lattices

We report on the observation of antiferromagnetic correlations of ultracold fermions in a variety of optical lattice geometries that are well described by the Hubbard model, including dimers, 1D chains, ladders, isolated and coupled honeycomb planes, as well as square and cubic lattices. The dependence of the strength of spin correlations on the specific geometry is experimentally studied by measuring the correlations along different lattice tunneling links, where a redistribution of correlations between the different lattice links is observed. By measuring the correlations in a crossover between distinct geometries, we demonstrate an effective reduction of the dimensionality for our atom numbers and temperatures. We also investigate the formation and redistribution time of spin correlations by dynamically changing the lattice geometry and studying the time evolution of the system. Time scales ranging from a sudden quench of the lattice geometry to an adiabatic evolution are probed.

Figure 1

Experimental observation of the dependence of antiferromagnetic correlations on lattice geometry. The trap averaged correlator $-⟨{\widehat{S}}_i^x{\widehat{S}}_{i+1}^x⟩-⟨{\widehat{S}}_i^y{\widehat{S}}_{i+1}^y⟩$ is measured along the strong links for various lattice geometries, which differ in the number of strong nearest neighbor links $Z$. Additionally, a schematic view of the lattice geometries is shown, where the strong tunneling links are indicated by a bar. For all data points, the bandwidth $W/h=2.6(1)\mathrm{kHz}$ and the on-site interaction $U/h=0.87(2)\mathrm{kHz}$ is constant. Error bars denote the standard error of 50 measurements.

Figure 2

Spin correlations for a crossover between two different lattice geometries. (a) Scan between a square ($Z=4$) and a 1D chain ($Z=2$) geometry and (b) scan between a 1D chain ($Z=2$) and a dimerized ($Z=1$) geometry. In both cases, the strong to weak tunneling ratio $t_s/t$ varies between 1 and 5. Blue and red symbols denote the measured spin correlations along the strong and weak links, respectively. Gray symbols show their sum. The error bar on the tunneling ratio denotes the uncertainty on the lattice parameters, while the data are the mean $\pm$ the standard error of at least 25 measurements.

Figure 3

Dynamics of spin correlations. The lattice is ramped within a time $\tau$ (a) from a 1D chain geometry to the same 1D chain geometry rotated by 90° and (b) from a dimerized to 1D chain geometry [24]. Blue and red points denote the measured spin correlations along the previously and new strong links. For all closed symbols, the correlations are measured immediately after the ramp, whereas open symbols include an additional wait time of 50 ms. The gray solid lines indicate the reference value for the correlations in the initial geometry without the lattice ramp, and the shading denotes the error on this value. Error bars are the standard error of at least 25 measurements.