Mean-Field Scaling of the Superfluid to Mott Insulator Transition in a 2D Optical Superlattice

The mean-field treatment of the Bose-Hubbard model predicts properties of lattice-trapped gases to be insensitive to the specific lattice geometry once system energies are scaled by the lattice coordination number $z$. We test this scaling directly by comparing coherence properties of ${}^{87}\mathrm{Rb}$ gases that are driven across the superfluid to Mott insulator transition within optical lattices of either the kagome ($z=4$) or the triangular ($z=6$) geometries. The coherent fraction measured for atoms in the kagome lattice is lower than for those in a triangular lattice with the same interaction and tunneling energies. A comparison of measurements from both lattices agrees quantitatively with the scaling prediction. We also study the response of the gas to a change in lattice geometry, and observe the dynamics as a strongly interacting kagome-lattice gas is suddenly “hole doped” by introducing the additional sites of the triangular lattice.

Figure 1

Triangular and kagome optical lattices. (a) Triangular lattices created by light at wavelengths 532 nm (left) and 1064 nm (right). Intensity (indicated by color saturation) minima are separated by lattice spacings $a=355\mathrm{nm}$ and $2a$, respectively. (b) Overlapping the 1064-nm intensity minima with those of the 532-nm lattice (sites $D$) yields a kagome lattice. (c) Potential energy along a path between sites of the unit cell: 532-nm lattice only (green dashed line) and bichromatic lattice (blue line). Atoms are confined to the kagome geometry when $\mathrm{\Delta }V$ exceeds the chemical potential $\mu$ and the tunneling energy $J$.

Figure 2

Comparing the superfluid to Mott insulator transition in lattices with different coordination numbers. (a) Momentum-space images of atoms released from the triangular (upper row) and kagome (lower row) lattices for $V_{532}/h=38,54$ and 78 kHz. Diffraction of the superfluid produces sharp peaks at reciprocal lattice vectors; those from kagome lattice show additional peaks at wave number $k_{\mathrm{kag}}=2\pi \sqrt{3}/(1064\mathrm{nm})$. On-site interactions in deeper lattices drive a phase transition to the Mott insulating state, as indicated by loss of coherence. (b) We measure coherent fraction by summing the populations at all coherent diffraction peaks and dividing by the total atom number. Data are shown as a function of $V_{532}$ (lower axis) and of $U/J$ (upper axis). Each point represents 3–5 experimental iterations and the estimated standard error in the mean for each point is shown. See Ref. [26] for measurements in triangular lattices at higher filling.

Figure 3

A test of the scaling hypothesis in which the coherent fractions measured in the triangular and kagome lattices [Fig. 2] are plotted against the scaled interaction energy $\tilde{U}=U/zJ$. The overlap of the two data sets at all $\tilde{U}$ indicates agreement with the scaling prediction.

Figure 4

Response of a degenerate Bose gas to a structural lattice change. (a) The experimental sequence. (b) Coherent fraction as a function of ramp time after the gas equilibrates in the final lattice. Lines show the coherent fraction observed for samples loaded directly into and held within a constant-strength triangular or kagome lattice; line thickness shows the estimated standard error. (c) Time-resolved response following fastest ramp. Three one-dimensional line cuts through momentum-space images are summed. The coherent diffraction at wave number $\pm k_{\mathrm{kag}}$ (shaded red to guide the eye) is transiently enhanced as superfluid flows into once-vacant $D$ sites in the unit cell. First-order diffraction of equilibrated superfluid from triangular lattice occurs at $\pm 2k_{\mathrm{kag}}$ (shaded green).