Evidence for Quantum Stripe Ordering in a Triangular Optical Lattice

Understanding strongly correlated quantum materials, such as high-$T_{\mathrm{c}}$ superconductors, iron-based superconductors, and twisted bilayer graphene systems, remains as one of the outstanding challenges in condensed matter physics. Quantum simulation with ultracold atoms in particular optical lattices, which provide orbital degrees of freedom, is a powerful tool to contribute new insights to this endeavor. Here, we report the experimental realization of an unconventional Bose-Einstein condensate of ${}^{87}\mathrm{Rb}$ atoms populating degenerate $p$ orbitals in a triangular optical lattice, exhibiting remarkably long coherence times. Using time-of-flight spectroscopy, we observe that this state spontaneously breaks the rotational symmetry and its momentum spectrum agrees with the theoretically predicted coexistence of exotic stripe and loop-current orders. Like certain strongly correlated electronic systems with intertwined orders, such as high-$T_{\mathrm{c}}$ cuprate superconductors, twisted bilayer graphene, and the recently discovered chiral density-wave state in kagome superconductors $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ ($A=\mathrm{K}$, Rb, Cs), the newly demonstrated quantum state, in spite of its markedly different energy scale and the bosonic quantum statistics, exhibits multiple symmetry breakings at ultralow temperatures. These findings hold the potential to enhance our comprehension of the fundamental physics governing these intricate quantum materials.

Figure 1

Quantum stripe ordering for $p$-orbital bosons in a triangular optical lattice. (a) Lattice potential of the triangular optical lattice with maxima and minima shown by red and blue color, respectively. (b) The left-hand panel shows a contour plot of the second and third Bloch bands across the second and third Brillouin zones, respectively. The energy dispersion along the high symmetry lines forming the dashed red triangle is shown for both bands in the right-hand panel. The three inequivalent $M$ points provide degenerate minima protected by the discrete rotational symmetry of the lattice. (c) Schematic order parameter of one of six possible implementations of the ground states within the second band. The gray hexagon denotes the unit cell of the order parameter. Each site contains an equal hybridization of two nonorthogonal $p$ orbitals with a relative phase difference $\pm \pi /2$. The circular arrows illustrate the on-site orbital angular momentum. The solid blue lines with arrows denote the bond currents between nearest neighbor sites. (d) The corresponding mass current of the ground state. The amplitude (direction) of the current is shown by a color code (black arrows). (e) Momentum distributions of three representative ground states predicted theoretically. Solid black hexagons denotes the first Brillouin zone.

Figure 2

Quantum phase transition induced by fine-tuning the energy imbalance. (a) The momentum distribution for the initial state, 0.1 ms after preparation of a condensate at $M_1$. (b) A distortion of the triangular lattice is applied to adjust the energy imbalance $\mathrm{\Delta }E\equiv E_2(M_1)-[E_2(M_2)+E_2(M_3)]/2$ with equal energies $E_2(M_2)=E_2(M_3)$ maintained. The $M$-point populations after 130.1 ms evolution time, averaged over 111 experimental runs, is plotted versus $\mathrm{\Delta }E$. Error bars denote standard deviations. The insets illustrate the two characteristic configurations with respect to the $M$-point energies for $\mathrm{\Delta }E<0$ and $\mathrm{\Delta }E>0$. (c) Exemplary final state momentum distributions, recorded by time-of-flight measurements with $\mathrm{\Delta }E\approx -0.5\mathrm{nK}$ as indicated in the figure. (d) Same as (c), but with $\mathrm{\Delta }E\approx 0.8\mathrm{nK}$. (e),(f) Histograms for the occurrence frequencies of the single-run values of ${\chi }_{23}$, recorded after 140.1 ms holding time, for more than 600 runs with the same values of $\mathrm{\Delta }E$ used in (c) and (d), respectively. The red solid line in (f) shows a Gaussian fit.

Figure 3

Statistical analysis of the quantum stripe phase. (a) Statistical distribution of the final states at 140.1 ms holding time for 636 experimental runs. Each dark red dot shows the tuple $(n_1,n_2)$ of populations of the points $M_1$ and $M_2$ using an oblique coordinate system. The mean-field energy per atom calculated via choosing optimized relative phases between condensates at three $M$ points is shown by the color code and contour lines. Three red squares indicate the locations of the theoretically predicted ground states. The corresponding energy scale for the tunneling between two nearest neighbor $p$ orbitals is about 8 nK. (b) The kernel density estimation of the data shown in (a), where three local maxima appear close to the predicted ground states, respectively. (c) Momentum distribution for the final states observed close to the three local maxima in the kernel density distribution shown in (b).