Artificial Graphene with Tunable Interactions

We create an artificial graphene system with tunable interactions and study the crossover from metallic to Mott insulating regimes, both in isolated and coupled two-dimensional honeycomb layers. The artificial graphene consists of a two-component spin mixture of an ultracold atomic Fermi gas loaded into a hexagonal optical lattice. For strong repulsive interactions, we observe a suppression of double occupancy and measure a gapped excitation spectrum. We present a quantitative comparison between our measurements and theory, making use of a novel numerical method to obtain Wannier functions for complex lattice structures. Extending our studies to time-resolved measurements, we investigate the equilibration of the double occupancy as a function of lattice loading time.

Figure 1

Experimental setup used to create the artificial graphene system. Independent 2D layers of honeycomb geometry are realized using a tunable-geometry optical lattice. A sketch of the tunneling structure within the layers is shown on the right. The potential barriers between the sites are chosen such that the nearest-neighbor tunneling $t$ in the hexagonal planes is equal along all bonds, resulting in a band structure containing two Dirac points [23]. A repulsively interacting two-component spin mixture of fermionic ${}^{40}\mathrm{K}$ atoms (red and blue spheres) is loaded into the lattice. Gravity points along $y$.

Figure 2

Observing the metal to Mott insulator crossover in artificial graphene. (a) The measured double occupancy $D$ versus atom number $N$ for three different interaction strengths $U/3t$. For strong interactions, an incompressible Mott insulating core forms, leading to a strong suppression of $D$. Solid lines are theory predictions based on a high-temperature series expansion. (b) Excitation spectrum obtained by measuring $D$ after sinusoidal modulation of the lattice depth $V_Y$ for the same interaction strengths as above. The solid lines are Gaussian fits to the spectra. Arrows show the reference value without modulation. (c) Comparison of the extracted Hubbard parameters $U$ with those obtained from a calculation of the Wannier functions in the honeycomb lattice. Error bars in $D$ and $N$ show the standard deviation of 5 measurements. In (c), the uncertainty in $a$ and the fit error for the peak positions are smaller than the displayed data points. Data for additional interactions can be found in Ref. 23. Negative values of $D$ are caused by the subtraction of an independently measured offset.

Figure 3

The lattice loading process. The panels show $D$ after loading ramps with varying duration ${\tau }_L$ for two interactions and two initial atom numbers. The solid line is the expected $D$ from the high-temperature series expansion, taking atom loss and heating during lattice loading into account [23]. The insets show the calculated equilibrium density profiles for the atomic cloud in the optical dipole trap (dashed lines) and in the lattice (solid lines), illustrating the required density redistribution during the loading. Here, the initial atom number and entropies before loading into the lattice were used. Error bars in $D$ show the standard deviation of 3 measurements.

Figure 4

Coupled layers of artificial graphene. (a) Detail of the coupled-layer structure with $t=t_{\perp }$. The atoms populate about 80 layers. (b) The double occupancy $D$ versus atom number $N$ in the metallic and Mott insulating regime. Solid lines are theory predictions based on a high-temperature series expansion. (c) Excitation spectra for the interactions used in (b). The solid lines are Gaussian fits to the spectra. Arrows show the reference value without modulation. (d) The on-site interaction energy $U$ compared to the theoretical expectation. Error bars are as in Fig. 2. Data for additional interactions can be found in Ref. 23.