Ultracold Atoms in a Tunable Optical Kagome Lattice

We realize a two-dimensional kagome lattice for ultracold atoms by overlaying two commensurate triangular optical lattices generated by light at the wavelengths of 532 and 1064 nm. Stabilizing and tuning the relative position of the two lattices, we explore different lattice geometries including a kagome, a one-dimensional stripe, and a decorated triangular lattice. We characterize these geometries using Kapitza-Dirac diffraction and by analyzing the Bloch-state composition of a superfluid released suddenly from the lattice. The Bloch-state analysis also allows us to determine the ground-state distribution within the superlattice unit cell. The lattices implemented in this work offer a near-ideal realization of a paradigmatic model of many-body quantum physics, which can serve as a platform for future studies of geometric frustration.

Figure 1

Three bichromatic light beams intersecting at 120° form a kagome optical lattice for ultracold ${}^{87}\mathrm{Rb}$ atoms, with the two-dimensional potential $V(\mathbf{r})$ shown in (a). Profiles of the potential of the SW, LW, and combined lattices are shown in (b). Sites $D$ of the SW lattice are emptied as $\Delta V$ exceeds the chemical potential, so that the remaining sites $A$, $B$ and $C$ form the kagome geometry. (c) Different lattice geometries are created for intermediate LW-lattice depths ($V_{\mathrm{LW}}<9V_{\mathrm{SW}}$) by displacing the potential maxima of the SW lattice to the high-symmetry points $X$, $Y$ or $Z$ within the unit cell. For higher LW-lattice depths, a honeycomb geometry prevails.

Figure 2

Atom diffraction patterns, formed by a $\tau =8\mu \mathrm{s}$ pulse of the lattice potential (with $V_{\mathrm{SW}}/h\sim 80\mathrm{kHz}$ and $V_{\mathrm{LW}}/h\sim 50\mathrm{kHz}$) followed by 26 ms time of flight, exhibit left/right momentum asymmetry [defined as $(P_{+q}-P_{-q})/(P_{+q}+P_{-q})$] that varies with the displacement $\delta x$ between the LW- and SW-lattice intensity minima, in close agreement with the predicted behavior (solid line).

Figure 3

The real- and momentum-space composition of a superfluid for various lattices. (a) The kagome and decorated triangular lattices maintain threefold rotational symmetry in configuration and momentum space, while the symmetry of the 1D stripe lattice is reduced to a parity symmetry (left-right in the images). For each setting, a schematic distinguishes between sites of high (green) and low (red) atomic population. The expected momentum distribution for measured values of $V_{\mathrm{SW}}/h=40\mathrm{kHz}$ and $\Delta V/h=14\mathrm{kHz}$ is shown with the area of the black dot reflecting the fractional population. (b) Translating the LW-lattice potential maxima (marked as a star in the schematic) along the $A\mathrm{\text{−}}D$ axis tunes the lattice between kagome and 1D stripe geometries, as revealed by the population ratios ${\tilde{P}}_i$ identified according to the inset. The data (averages of 4–5 measurements) agree with calculations of the single-particle ground state (solid lines) with the lattice depth used in the experiment. Interaction effects are neglected since $\Delta V$ was higher than the chemical potential $\mu \sim h\times 3.5\mathrm{kHz}$ of the condensate in the SW-only lattice.

Figure 4

The superlattice was converted from a SW triangular to a kagome lattice by increasing $V_{\mathrm{LW}}$. As $\Delta V$ exceeds the condensate chemical potential ($\mu /h\simeq 3.5\mathrm{kHz}$), (a) the momentum population ratios reach the asymptotic value of $1/9$ expected for a kagome lattice, and (b) the $D$-site population is extinguished. Data points represent averages of 7–10 measurements. In (a), the dashed curve indicates the predicted ${\tilde{P}}_i$ while the shaded region indicates the expected variation in ${\tilde{P}}_i$ given a shot-to-shot instability of $\sim 20\mathrm{nm}$ in the relative position of the LW and SW lattices.