Unconventional Superfluid Order in the $F$ Band of a Bipartite Optical Square Lattice

We report on the first observation of bosons condensed into the energy minima of an $F$ band of a bipartite square optical lattice. Momentum spectra indicate that a truly complex-valued staggered angular momentum superfluid order is established. The corresponding wave function is composed of alternating local $F_{2x^3-3x}\pm i{F}_{2y^3-3y}$ orbits and local $S$ orbits residing in the deep and shallow wells of the lattice, which are arranged as the black and white areas of a checkerboard. A pattern of staggered vortical currents arises, which breaks time-reversal symmetry and the translational symmetry of the lattice potential. We have measured the populations of higher order Bragg peaks in the momentum spectra for varying relative depths of the shallow and deep lattice wells and find remarkable agreement with band calculations.

Figure 1

(a) The bipartite lattice comprises two classes of lattice sites denoted by $\mathcal{A}$ and $\mathcal{B}$. The grey area shows the Wigner-Seitz unit cell of the $\mathcal{A}$ sublattice. (b), (c), and (d) Observed populations of Brillouin zones (BZs) after applying the population swapping procedure with final values of $(\theta /\pi ,V_0/E_{\mathrm{rec}})=(0.61,5.8)$, (0.66, 8.3), (0.69, 13.3), respectively. (e) Schematic of the 6th [dark (blue and purple) areas] and 7th [light (orange) areas] BZs. The eight open (red) circles mark the points, where the atoms are observed to gather in (c). In the center, the first BZ is reconstructed by translations of subsets of the 6th BZ via reciprocal lattice vectors indicated by the (red) dashed arrows. (f) 3D representation of the image in (c). (g) energy surface of the 7th band with degenerate minima indicated by filled black circles corresponding to those in (e). (h) Band energies $E$ at ${\mathbf{K}}_{(1,1)}$ versus the well depth $V_0$. The black circle indicates a crossing of the 7th [dotted (red) line] and the 6th [dashed (blue) line] bands.

Figure 2

In (a), (b), and (c) momentum spectra are shown for a hold time of 1 ms, $V_0/E_{\mathrm{rec}}=8.3$ and $\theta /\pi =0.66$, 0.70, 0.75, respectively. Observations (calculations) are shown in the upper (lower) row. (d) Population ratio for higher order and zero order Bragg peaks plotted versus $\theta$. Observations (calculations) are shown by the symbols (solid lines). (e) Temporal evolution of the populations of Bragg peaks. The solid lines are exponential fits applied in the wings of the graphs to determine relaxations times. (f) Temporal evolution of band populations for $V_0/E_{\mathrm{rec}}=8.3$ and $\theta /\pi =0.67$. The solid lines are exponential fits. (g) $1/e$ times for depopulation of the 6th and 7th BZ and repopulation of the 1st BZ plotted versus $\theta$.

Figure 3

Orbit configuration of order parameter ${\Psi }_F$. The grey areas characterize the antinodal structure of the orbits, the (colored) numbers indicate the local phases.