Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Atomic Bose–Einstein condensate in twisted-bilayer optical lattices

Nature volume 615, pages 231–236 (2023)Cite this article

Subjects

Abstract

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.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on SpringerLink
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

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.

Similar content being viewed by others

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.

References

  1. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).

    Article  ADS  CAS  PubMed  Google Scholar 

  2. Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).

    Article  ADS  CAS  PubMed  Google Scholar 

  3. Yankowitz, M. et al. Tuning superconductivity in twisted bilayer graphene. Science 363, 1059–1064 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

  4. Lu, X. et al. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature 574, 653–657 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

  5. Wang, J., Mu, X., Wang, L. & Sun, M. Properties and applications of new superlattice: twisted bilayer graphene. Mater. Today Phys. 9, 100099 (2019).

    Article  Google Scholar 

  6. Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist. Nat. Mater. 19, 1265–1275 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  7. Balents, L., Dean, C. R., Efetov, D. K. & Young, A. F. Superconductivity and strong correlations in Moiré flat bands. Nat. Phys. 16, 725–733 (2020).

    Article  CAS  Google Scholar 

  8. Kennes, D. M. et al. Moiré heterostructures as a condensed-matter quantum simulator. Nat. Phys. 17, 155–163 (2021).

    Article  CAS  Google Scholar 

  9. Lopes dos Santos, J. M. B., Peres, N. M. R. & Castro Neto, A. H. Graphene bilayer with a twist: electronic structure. Phys. Rev. Lett. 99, 256802 (2007).

    Article  ADS  CAS  PubMed  Google Scholar 

  10. Mele, E. J. Commensuration and interlayer coherence in twisted bilayer graphene. Phys. Rev. B 81, 161405 (2010).

    Article  ADS  Google Scholar 

  11. Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA 108, 12233–12237 (2011).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  12. Moon, P. & Koshino, M. Energy spectrum and quantum Hall effect in twisted bilayer graphene. Phys. Rev. B 85, 195458 (2012).

    Article  ADS  Google Scholar 

  13. Wang, P. et al. Localization and delocalization of light in photonic moiré lattices. Nature 577, 42–46 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  14. Huang, C. et al. Localization-delocalization wavepacket transition in Pythagorean aperiodic potentials. Sci. Rep. 6, 32546 (2016).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  15. Fu, Q. et al. Optical soliton formation controlled by angle twisting in photonic Moiré lattices. Nat. Photon. 14, 663–668 (2020).

    Article  ADS  CAS  Google Scholar 

  16. Hu, G. et al. Topological polaritons and photonic magic angles in twisted α-MoO3 bilayers. Nature 582, 209–213 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  17. Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008).

    Article  ADS  CAS  Google Scholar 

  18. Lewenstein, M., Sanpera, A. & Ahufinger, V. Ultracold Atoms in Optical Lattices: Simulating Quantum Many-Body systems (Oxford Univ. Press, 2012).

  19. Windpassinger, P. & Sengstock, K. Engineering novel optical lattices. Rep. Prog. Phys. 76, 086401 (2013).

    Article  ADS  PubMed  Google Scholar 

  20. Soltan-Panahi, P. et al. Multi-component quantum gases in spin-dependent hexagonal lattices. Nat. Phys. 7, 434–440 (2011).

    Article  CAS  Google Scholar 

  21. Wirth, G., Ölschläger, M. & Hemmerich, A. Evidence for orbital superfluidity in the P-band of a bipartite optical square lattice. Nat. Phys. 7, 147–153 (2011).

    Article  CAS  Google Scholar 

  22. Tarruell, L., Greif, D., Uehlinger, T., Jotzu, G. & Esslinger, T. Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice. Nature 483, 302–305 (2012).

    Article  ADS  CAS  PubMed  Google Scholar 

  23. Jo, G. B. et al. Ultracold atoms in a tunable optical kagome lattice. Phys. Rev. Lett. 108, 045305 (2012).

    Article  ADS  PubMed  Google Scholar 

  24. Taie, S. et al. Coherent driving and freezing of bosonic matter wave in an optical Lieb lattice. Sci. Adv. 1, e1500854 (2015).

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  25. Gall, M., Wurz, N., Samland, J., Chan, C. F. & Köhl, M. Competing magnetic orders in a bilayer Hubbard model with ultracold atoms. Nature 589, 40 (2021).

    Article  ADS  CAS  PubMed  Google Scholar 

  26. González-Tudela, A. & Cirac, J. I. Cold atoms in twisted-bilayer optical potentials. Phys. Rev. A 100, 053604 (2019).

    Article  ADS  Google Scholar 

  27. Luo, X. & Zhang, C. Spin-twisted optical lattices: tunable flat bands and Larkin-Ovchinnikov superfluids. Phys. Rev. Lett. 126, 103201 (2021).

    Article  ADS  CAS  PubMed  Google Scholar 

  28. Graß, T., Chhajlany, R. W., Tarruell, L., Pellegrini, V. & Lewenstein, M. Proximity effects in cold atom artificial graphene. 2D Mater. 4, 015039 (2016).

    Article  Google Scholar 

  29. Salamon, T. et al. Simulating twistronics without a twist. Phys. Rev. Lett. 125, 030504 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  30. Boada, O., Celi, A., Latorre, J. I. & Lewenstein, M. Quantum simulation of an extra dimension. Phys. Rev. Lett. 108, 133001 (2012).

    Article  ADS  CAS  PubMed  Google Scholar 

  31. Celi, A. et al. Synthetic gauge fields in synthetic dimensions. Phys. Rev. Lett. 112, 043001 (2014).

    Article  ADS  CAS  PubMed  Google Scholar 

  32. Ozawa, T. & Price, H. M. Topological quantum matter in synthetic dimensions. Nat. Rev. Phys. 1, 349–357 (2019).

    Article  Google Scholar 

  33. LeBlanc, L. J. & Thywissen, J. H. Species-specific optical lattices. Phys. Rev. A 75, 053612 (2007).

    Article  ADS  Google Scholar 

  34. Arora, B., Safronova, M. S. & Clark, C. W. Tune-out wavelengths of alkali-metal atoms and their applications. Phys. Rev. A 84, 043401 (2011).

    Article  ADS  Google Scholar 

  35. Wen, K. et al. Experimental study of tune-out wavelengths for spin-dependent optical lattice in 87Rb Bose-Einstein condensation. J. Opt. Soc. Am. B 38, 3269 (2021).

    Article  ADS  Google Scholar 

  36. McDonald, M., Trisnadi, J., Yao, K. & Chin, C. Superresolution microscopy of cold atoms in an optical lattice. Phys. Rev. X 9, 021001 (2019).

    CAS  Google Scholar 

  37. Gerbier, F. et al. Interference pattern and visibility of a Mott insulator. Phys. Rev. A 72, 053606 (2005).

    Article  ADS  Google Scholar 

  38. Fisher, M. P. A., Weichman, P. B., Grinstein, G. & Fisher, D. S. Boson localization and the superfluid-insulator transition. Phys. Rev. B 40, 546–570 (1989).

    Article  ADS  CAS  Google Scholar 

  39. Krauth, W., Trivedi, N. & Ceperley, D. Superfluid-insulator transition in disordered boson systems. Phys. Rev. Lett. 67, 2307–2310 (1991).

    Article  ADS  CAS  PubMed  Google Scholar 

  40. Freericks, J. K. & Monien, H. Strong-coupling expansions for the pure and disordered Bose-Hubbard model. Phys. Rev. B 53, 2691–2700 (1996).

    Article  ADS  CAS  Google Scholar 

  41. Kariyado, T. & Vishwanath, A. Flat band in twisted bilayer Bravais lattices. Phys. Rev. Res. 1, 033076 (2019).

    Article  CAS  Google Scholar 

  42. Kennes, D. M., Xian, L., Claassen, M. & Rubio, A. One-dimensional flat bands in twisted bilayer germanium selenide. Nat. Commun. 11, 31124 (2020).

    Article  ADS  Google Scholar 

  43. Xiong, D., Wang, P., Fu, Z., Chai, S. & Zhang, J. Evaporative cooling of 87Rb atoms into Bose-Einstein condensate in an optical dipole trap. Chin. Opt. Lett. 8, 627–629 (2010).

    Article  CAS  Google Scholar 

  44. Greiner, M., Mandel, O., Esslinger, T., Hänsch, T. W. & Bloch, I. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415, 39 (2002).

    Article  ADS  CAS  PubMed  Google Scholar 

  45. Steck, D. A. Quantum and Atom Optics https://atomoptics.uoregon.edu/~dsteck/teaching/quantum-optics/ (2007).

  46. Zwerger, W. Mott Hubbard transition of cold atoms in optical lattices. J. Opt. B: Quantum Semiclass. Opt. 5, S9–S16 (2003).

    Article  ADS  CAS  Google Scholar 

  47. Krauth, W., Caffarel, M. & Bouchaud, J. P. Gutzwiller wave function for a model of strongly interacting bosons. Phys. Rev. B 45, 3137–3140 (1992).

    Article  ADS  CAS  Google Scholar 

  48. Sheshadri, K., Krishnamurthy, H. R., Pandit, R. & Ramakrishnan, T. V. Superfluid and insulating phases in an interacting-Boson model: mean-field theory and the RPA. Europhys. Lett. 22, 257–263 (1993).

    Article  ADS  CAS  Google Scholar 

  49. Jaksch, D., Bruder, C., Cirac, J. I., Gardiner, C. W. & Zoller, P. Cold Bosonic atoms in optical lattices. Phys. Rev. Lett. 81, 3108–3111 (1998).

    Article  ADS  CAS  Google Scholar 

Download references

Acknowledgements

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

Author notes

  1. These authors contributed equally: Zengming Meng, Liangwei Wang

Authors and Affiliations

  1. State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, P. R. China

    Zengming Meng, Liangwei Wang, Wei Han, Fangde Liu, Kai Wen, Pengjun Wang & Jing Zhang

  2. Department of Physics, Zhejiang Normal University, Jinhua, P. R. China

    Chao Gao

  3. James Franck Institute, Enrico Fermi Institute, Department of Physics, University of Chicago, Chicago, IL, USA

    Cheng Chin

  4. Hefei National Laboratory, Hefei, P. R. China

    Jing Zhang

Authors
  1. Zengming Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liangwei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Fangde Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Kai Wen
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Chao Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Pengjun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Cheng Chin
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Jing Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

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.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature thanks Richard Schmidt and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Extended data

is available for this paper at https://doi.org/10.1038/s41586-023-05695-4.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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.

a–b, 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. c–d, 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.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meng, Z., Wang, L., Han, W. et al. Atomic Bose–Einstein condensate in twisted-bilayer optical lattices. Nature 615, 231–236 (2023). https://doi.org/10.1038/s41586-023-05695-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41586-023-05695-4

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing