Interaction-Enhanced Group Velocity of Bosons in the Flat Band of an Optical Kagome Lattice

Geometric frustration of particle motion in a kagome lattice causes the single-particle band structure to have a flat $s$-orbital band. We probe this band structure by placing a Bose-Einstein condensate into excited Bloch states of an optical kagome lattice, and then measuring the group velocity through the atomic momentum distribution. We find that interactions renormalize the band structure, greatly increasing the dispersion of the third band, which is nearly non-dispersing the single-particle treatment. Calculations based on the lattice Gross-Pitaevskii equation indicate that band structure renormalization is caused by the distortion of the overall lattice potential away from the kagome geometry by interactions.

Figure 1

Experimental scheme. (a) Illustration of the scheme with an example of one-dimensional system. The band structure of a 1D lattice, with lattice wave number $2\times q_0$, at zero (left) and nonzero (right) depth in the extended zone scheme. A particle accelerated to the $n$th Brillouin zone ($n-1<|k|/q_0<n$) is loaded adiabatically into the $n$th band as the lattice. (b) The optical kagome lattice is constructed by overlaying triangular lattices using short- (532 nm, green) and long-wavelength (1064 nm, shown in red) light. Primitive lattice vectors ${\mathbf{a}}_{1,2}$ are shown. The four sites in a unit cell are labeled A–D. (c) The first four Brillouin zones of the kagome lattice. Acceleration along $\mathbf{y}$ maps atoms sequentially into the $n=1$ (light blue, $\mathrm{\Gamma }$ to $K$), $n=3$ (green, $K$ to $M$), and then $n=4$ (purple, $M$ to $K$) bands. The orange trajectory indicates the range of wave vector $k$ atoms are accelerated to in experiments reported in Fig. 2. (d) Noninteracting band structure of the optical kagome lattice. Solid black lines: (${\mathrm{V}}_{\mathrm{SW}}$, ${\mathrm{V}}_{\mathrm{LW}}$)=$h\times$ (25, 15) kHz. Dashed gray lines: zero lattice depth. The yellow trajectory matches that in (c). Circled labels are referenced in Fig. 2. The recoil energy $E_R=h\times 2.0\mathrm{kHz}$ in the kagome lattice.

Figure 2

Measured group velocity in the kagome lattice. Momentum distributions are shown [(a) noninteracting Bloch theory; (b) measured] at four representative initial $\mathbf{k}$ [indicated by dashed circles in (b)] with labels corresponding to those in Fig. 1. Basis reciprocal lattice vectors ${\mathbf{g}}_{1,2}$ are shown. (c) Measured $v_g={\mathbf{v}}_g·\mathbf{y}$. The initial wave vector $\mathbf{k}$ is measured for each experimental repetition. Data with $k_y$ within a binning range (blue shaded bars) are averaged, with 3–8 measurements per bin. Error bars are standard mean errors. Data agree with calculations that include effects of interactions at a peak density of $n_0=5.4(5)\times {10}^{13}{\mathrm{cm}}^{-3}$ (solid line, gray region indicates effect of density uncertainty), and disagree with noninteracting band-theory predictions (dashed line). Final lattice depths are $(V_{\mathrm{SW}},V_{\mathrm{LW}})=h\times (25,15)\mathrm{kHz}$.

Figure 3

Dependence of $v_g$ on lattice depths and densities measured at a fixed initial wave vector of $k_y=1.25q_K$. (a) $v_g$ measured with a peak density $n_0=6.2(6)\times {10}^{13}{\mathrm{cm}}^{-3}$ and different lattice depths ($V_{\mathrm{SW}}/V_{\mathrm{LW}}$ constant). While band theory (dashed line) predicts that $v_g$ is suppressed quickly at increasing lattice depths, data (each point is average of 4–7 measurements) show a significantly smaller rate of suppression, in agreement with Gross-Pitaevskii equation predictions (solid line; gray region indicates effects of density uncertainty). (b) $v_g$ measured at $(V_{\mathrm{SW}},V_{\mathrm{LW}})=h\times (20,10)\mathrm{kHz}$ increases monotonically with number density as predicted by the Gross-Pitaevskii equation (black line). Data within a small binning range (blue shaded bars) of densities, determined up to 10% systematic uncertainty, are averaged, with 3–8 measurements per bin. Error bars are standard mean errors. Insets are single-shot images taken at indicated settings.

Figure 4

Real-space distribution of atoms in the $n=3$, $k_y=1.25q_K$ Bloch state. (a) Calculated fractional population in the four kagome-lattice sites in a unit cell as a function of lattice depths with $V_{\mathrm{SW}}/V_{\mathrm{LW}}=2$ (dashed line: band theory; solid line: Gross-Pitaevskii equation). (b) Real-space distribution of atoms in a lattice at $(V_{\mathrm{SW}},V_{\mathrm{LW}})=h\times (20,10)\mathrm{kHz}$ calculated by solving the Gross-Pitaevskii equation with $n_0=6.2\times {10}^{13}{\mathrm{cm}}^{-3}$.