Interaction-Driven Shift and Distortion of a Flat Band in an Optical Lieb Lattice

We report the momentum-resolved measurement of Bloch bands in an optical Lieb lattice for a Bose-Einstein condensate (BEC). A BEC in the lattice is transported to a desired quasimomentum by applying a constant force. The energy dispersion of the lowest band is obtained by integrating measured group velocities. We also measure the gap from the lowest band to the higher bands with the same quasimomentum, which can be extracted from the oscillation of the sublattice populations after preparing a superposition of the band eigenstates. We show that the experimental results agree with a band calculation based on the Bogoliubov approximation. It is revealed that the second band, which should be flat in a single-particle description, is shifted and, in particular, distorted around the Brillouin zone edge as the interaction strength increases.

Figure 1

(a) Schematic of Lieb lattice. In our system, the atoms are weakly trapped along the $y$ direction, and distribute like tubes as shown in gray in the figure. (b) Band structure of Lieb lattice in a single-particle description. (c) Method for measuring dispersion of the lowest band. Red circles indicate the atomic cloud. After transporting a BEC to a desired quasimomentum, we measure the group velocity and integrate the results. (d) Method for measuring a band gap in a momentum-resolved manner. We prepare the superposition of band eigenstates, transport the atoms to a desired quasimomentum, and measure an oscillation frequency of the sublattice population.

Figure 2

(a),(b) Absorption images at quasimomentum $(q_x,q_z)=(0,0)$ and $(0.5,0.5)k_{\mathrm{BZ}}$, respectively, where $k_{\mathrm{BZ}}$ corresponds to the quasimomentum at the BZ edge. The images are taken after a TOF of 14 ms. We restrict the integration region within the white squares in which the atoms are mostly detected. (c) Dispersion of the 1st band of optical Lieb lattice $(s_{\text{long}},s_{\text{short}},s_{\mathrm{diag}})=(13,13,15.5)$. The experimental data are denoted as green circles. The inset shows the first BZ. The dashed black line shows single-particle theory. The dotted blue line is the calculation for half of the maximum number density. The solid yellow line is for the maximum number density. The vertical axis shows the energy difference from the 1st band energy at $\mathrm{\Gamma }$ for each interaction strength. The error bar indicates the standard deviation of three independent scans.

Figure 3

(a),(b) Oscillation of the sublattice population at the $\mathrm{\Gamma }$ point according to (a)$E_{3-1}$ and (b) $E_{2-1}$, respectively. First, we load a BEC into an optical lattice the configuration of which is different from that of a Lieb lattice. In this way, we create a superposition of (a) the 1st and 3rd or (b) the 1st and 2nd band eigenstates of the Lieb lattice. Next, we suddenly change the lattice configuration into the Lieb lattice, and take various hold time. We observe the temporal change of the sublattice populations obtained by the projection measurement described in the text. Each sublattice occupancy is normalized by the summation over all of the sublattice occupancy. Green, blue, and red circles show the $A$-, $B$-, and $C$-sublattice population, respectively. The error bar shows the standard deviation of three independent scans. Solid lines are fits to the data with damped sine functions(4). (c) The lowest three bands in the optical Lieb lattice of $(s_{\text{long}},s_{\text{short}},s_{\mathrm{diag}})=(13,13,15.5)$. Red and blue circles are the reconstructed 2nd and 3rd band energies, respectively. Error bars mean the fitting errors. Dashed lines are the predictions based on a single-particle theory. The dotted and solid lines are the calculations including the interaction based on the BdGE with the half and maximum densities, respectively.

Figure 4

(a) Band gaps from the lowest band to the 2nd band. Dashed black line is the prediction on the single particle theory of the Lieb lattice potential $(s_{\text{long}},s_{\text{short}},s_{\mathrm{diag}})=(13,13,15.5)$. Solid yellow line is the calculation including the interaction based on the BdGE. Thick red, green, and blue lines, respectively, show the quasimomenta $q_z=0.74k_{\mathrm{BZ}}$, $0.37k_{\mathrm{BZ}}$, and 0 at which we have investigated the density dependence of the band gaps. (b),(c),(d) Band gap versus the atom number for $q_z=0.74k_{\mathrm{BZ}}$, $0.37k_{\mathrm{BZ}}$, 0, respectively. Solid and dotted lines show the calculations including the interaction based on the BdGE for the maximum and half densities, respectively. Error bars indicate the fitting errors.