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Atomic Bose–Einstein condensate in twisted-bilayer optical lattices


Observation of strong correlations and superconductivity in twisted-bilayer graphene1,2,3,4 has stimulated tremendous interest in fundamental and applied physics5,6,7,8. In this system, the superposition of two twisted honeycomb lattices, generating a moiré pattern, is the key to the observed flat electronic bands, slow electron velocity and large density of states9,10,11,12. Extension of the twisted-bilayer system to new configurations is highly desired, which can provide exciting prospects to investigate twistronics beyond bilayer graphene. Here we demonstrate a quantum simulation of superfluid to Mott insulator transition in twisted-bilayer square lattices based on atomic Bose–Einstein condensates loaded into spin-dependent optical lattices. The lattices are made of two sets of laser beams that independently address atoms in different spin states, which form the synthetic dimension accommodating the two layers. The interlayer coupling is highly controllable by a microwave field, which enables the occurrence of a lowest flat band and new correlated phases in the strong coupling limit. We directly observe the spatial moiré pattern and the momentum diffraction, which confirm the presence of two forms of superfluid and a modified superfluid to insulator transition in twisted-bilayer lattices. Our scheme is generic and can be applied to different lattice geometries and for both boson and fermion systems. This opens up a new direction for exploring moiré physics in ultracold atoms with highly controllable optical lattices.

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Fig. 1: Simulation of twisted-bilayer systems based on atoms in spin-dependent optical lattices.
Fig. 2: Independent diffraction of atoms in different spin states by the twisted-bilayer optical lattices.
Fig. 3: Interlayer coupling in twisted-bilayer optical lattices.
Fig. 4: Moiré pattern and superfluid ground state in twisted-bilayer optical lattices.
Fig. 5: Phase transition for the twisted-bilayer optical lattice.

Data availability

All data generated or analysed during this study are included in this published article. Further data are also available from the corresponding authors upon reasonable request.


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This research is supported by Innovation Program for Quantum Science and Technology (grant no. 2021ZD0302003), National Key Research and Development Program of China (grant nos. 2016YFA0301602, 2018YFA0307601 and 2022YFA1404101), NSFC (grant nos. 12034011, 12022406, 12074342 and 11804203), the Fund for Shanxi ‘1331 Project’ Key Subjects Construction and Tencent (Xplorer Prize). C.C. acknowledges support by the National Science Foundation (grant no. PHY-2103542) and the Army Research Office STIR (grant no. W911NF2110108).

Author information

Authors and Affiliations



J.Z. conceived the idea and performed the experimental designs. L.W., Z.M., F.L., K.W., P.W. and J.Z. performed the experiments. C.C., Z.M., L.W., F.L., W.H. and J.Z. analysed the data and all authors discussed the results. W.H., C.G. and J.Z. performed the simulation. Z.M. plotted the figures. J.Z. and C.C. wrote the manuscript. All authors interpreted the results and reviewed the manuscript. J.Z. designed and supervised the project.

Corresponding author

Correspondence to Jing Zhang.

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Extended data figures and tables

Extended Data Fig. 1 Coherence in the SF-MI transition.

a, The initial BEC in 2D pancake-like potential. b, Absorption image after atoms are released abruptly from an optical lattice potential V1 (or V2) with a potential depth 24Er. c, Absorption image when the lattice is ramped up to the lattice depth 24Er and then ramp down to zero. The images are obtained after 18 ms free space expansion.

Extended Data Fig. 2 Determination of tune-out wavelengths.

ab, The lattice depth V1x (blue) and V1y (red) as a function of wavelength λ for the two different hyperfine states \(\left|F=1,{m}_{F}=1\right\rangle \) and \(\left|F=2,{m}_{F}=0\right\rangle \). The angles between V1x, V1y and B0 are 39.79° and 50.21° respectively. cd, The potential depth V2x (blue) and V2y (red) as a function of wavelength λ for the two different hyperfine states \(\left|F=1,{m}_{F}=1\right\rangle \) and \(\left|F=2,{m}_{F}=0\right\rangle \). e, Theoretical light shift of V1x, V1y for \(\left|1,1\right\rangle \) and \(\left|2,0\right\rangle \). f, Theoretical lattice depth of V2x, V2y for \(\left|1,1\right\rangle \) and \(\left|2,0\right\rangle \). The bias magnetic field of 10 Gauss is applied along the 45° diagonal line of the square lattice V2.

Extended Data Fig. 3 Band structure of the twisted-bilayer optical lattices.

The twist angle of the commensurate optical lattice is \(\theta =2\,\arctan \,(1/22)\), whose band structure is regarded as an approximation of the experimental case θ = 5.21°. a, b and c show the band structures for the interlayer coupling strength ΩR = 0Er, 0.1Er and 1Er respectively. a also gives the band structure without the interlayer coupling in the form of the superlattice minibands within the same reduced Brillouin zone. d,e and f are the enlargement of the lowest bands of a,b, and c, respectively. g,h and i are the further enlargement of the lowest bands of d,e and f, respectively. Here, the MW detuning is Δ = 0, V0= 4Er and E0 corresponds to the energy of the lowest band.

Extended Data Fig. 4 Characteristics of the different phases.

a, Phase diagram, where SF, SF-II, MI, and I refer to superfluid, superfluid only with short-range coherence, Mott insulator, and insulator. b, Table shows the features of the different phases. c, Plots of the order parameter \(\langle {\hat{b}}_{i}\rangle \) and the filling of the atoms on the site n for the different phases. Parameters (V/Er, ΩR/Er) are (10,0.6), (15,0.6), (23,0.3) and (23,1.1) for the plots from left to right respectively. The chemical potential μ/U = 1 is considered.

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Meng, Z., Wang, L., Han, W. et al. Atomic Bose–Einstein condensate in twisted-bilayer optical lattices. Nature 615, 231–236 (2023).

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