Probing Nearest-Neighbor Correlations of Ultracold Fermions in an Optical Lattice

We demonstrate a probe for nearest-neighbor correlations of fermionic quantum gases in optical lattices. It gives access to spin and density configurations of adjacent sites and relies on creating additional doubly occupied sites by perturbative lattice modulation. The measured correlations for different lattice temperatures are in good agreement with an ab initio calculation without any fitting parameters. This probe opens new prospects for studying the approach to magnetically ordered phases.

Figure 1

Probing nearest-neighbor correlations for different phases. A periodic lattice modulation causes tunneling of particles to neighboring sites. The number of created doublons strongly depends on the state of the many-body system (strongly interacting metal, paramagnetic Mott insulator, or antiferromagnet) and can be used to determine the nearest-neighbor correlator ${\mathcal{P}}_{i,i+1}$ [10].

Figure 2

Evolution of double occupancy as a function of the lattice modulation time $\tau$ for resonant excitation. The lattice depth is set to $7E_R$ ($U/6t=4.1$) and the modulation strength is $\delta V/V=0.1$. (a) The induced double occupancy saturates for large times, which is well captured by an exponential fit (solid line). (b) At low modulation times $D$ increases linearly, from which the doublon production rate $\Gamma$ is obtained by a linear fit. The lattice depth was set to $10E_R$ for this measurement ($U/6t=10.6$). (c) On the time scale of the modulation period $h/U$ the double occupancy shows an underlying low amplitude sinusoidal modulation. The solid line is a fit with fixed frequency $U/h$. For clarity, $\delta V/V$ was increased to 0.2. Error bars in double occupancy denote statistical errors from multiple measurements.

Figure 3

Doublon production rate $\Gamma$ as a function of the lattice modulation frequency $\nu$, measured for different modulation amplitudes $\tilde{\delta }t/t$. The experiments are performed at $U/6t=10.6$ and $V=10E_R$. The shaded areas are Gaussian fits to the spectra, which are used to extract the frequency integrated response $R$. The two vertical dashed lines denote twice the three-dimensional bandwidth $4zt$ (with $z=6$). The inset is a double-logarithmic plot of $R$ for various modulation amplitudes, where the dashed line is a power law fit. Error bars denote the fit errors.

Figure 4

Measurement of the nearest-neighbor correlator $\mathcal{P}$. (a) Dependence on entropy per particle in a trapped lattice system for $U/6t=4.1$, $V=7E_R$, and $\tilde{\delta }t/t=0.26$. Solid and open circles denote lower and upper bounds $s_{\mathrm{in}}$ and $s_{\mathrm{out}}$ for the entropy per particle in the lattice for each measured value of $\mathcal{P}$. The black solid line is the calculated nearest-neighbor correlator obtained from second order HTSE without any fitting parameters. The entropy $s$ inferred from comparing each measured value of $\mathcal{P}$ to theory lies in between the experimental bounds. For clarity the corresponding lattice temperatures $T$ and measured doublon production rates $\Gamma$ are also included. Error bars denote statistical errors from several measurements. Panels (b) and (c) show the calculated distribution of ${\mathcal{P}}_{i,i+1}$ in the trap for the lowest and highest entropies $s_{\mathrm{cold}}$ and $s_{\mathrm{hot}}$, while the area under these curves corresponds to the measured correlator $\mathcal{P}$.