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Two-dimensional supersolidity in a dipolar quantum gas

Abstract

Supersolid states simultaneously feature properties typically associated with a solid and with a superfluid. Like a solid, they possess crystalline order, manifesting as a periodic modulation of the particle density; but unlike a typical solid, they also have superfluid properties, resulting from coherent particle delocalization across the system. Such states were initially envisioned in the context of bulk solid helium, as a possible answer to the question of whether a solid could have superfluid properties1,2,3,4,5. Although supersolidity has not been observed in solid helium (despite much effort)6, ultracold atomic gases provide an alternative approach, recently enabling the observation and study of supersolids with dipolar atoms7,8,9,10,11,12,13,14,15,16. However, unlike the proposed phenomena in helium, these gaseous systems have so far only shown supersolidity along a single direction. Here we demonstrate the extension of supersolid properties into two dimensions by preparing a supersolid quantum gas of dysprosium atoms on both sides of a structural phase transition similar to those occurring in ionic chains17,18,19,20, quantum wires21,22 and theoretically in chains of individual dipolar particles23,24. This opens the possibility of studying rich excitation properties25,26,27,28, including vortex formation29,30,31, and ground-state phases with varied geometrical structure7,32 in a highly flexible and controllable system.

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Fig. 1: Calculated phases of dipolar droplet array.
Fig. 2: Linear to zigzag transition in an anisotropic trap.
Fig. 3: Coherence in linear and zigzag states.

Data availability

Data pertaining to this work can be found at https://doi.org/10.5281/zenodo.4729519.

Code availability

Code used for this work is available from the corresponding author upon reasonable request.

References

  1. 1.

    Gross, E. P. Unified theory of interacting bosons. Phys. Rev. 106, 161–162 (1957).

    ADS  CAS  MATH  Article  Google Scholar 

  2. 2.

    Gross, E. P. Classical theory of boson wave fields. Ann. Phys. 4, 57–74 (1958).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Andreev, A. F. & Lifshitz, I. M. Quantum theory of defects in crystals. Sov. Phys. JETP 29, 1107–1114 (1969).

    ADS  Google Scholar 

  4. 4.

    Chester, G. V. Speculations on Bose–Einstein condensation and quantum crystals. Phys. Rev. A 2, 256–258 (1970).

    ADS  Article  Google Scholar 

  5. 5.

    Leggett, A. J. Can a solid be “superfluid”? Phys. Rev. Lett. 25, 1543–1546 (1970).

    ADS  CAS  Article  Google Scholar 

  6. 6.

    Chan, M. H.-W., Hallock, R. & Reatto, L. Overview on solid 4He and the issue of supersolidity. J. Low Temp. Phys. 172, 317–363 (2013).

    ADS  CAS  Article  Google Scholar 

  7. 7.

    Lu, Z.-K., Li, Y., Petrov, D. S. & Shlyapnikov, G. V. Stable dilute supersolid of two-dimensional dipolar bosons. Phys. Rev. Lett. 115, 075303 (2015).

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

  8. 8.

    Baillie, D. & Blakie, P. B. Droplet crystal ground states of a dipolar Bose gas. Phys. Rev. Lett. 121, 195301 (2018).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  9. 9.

    Roccuzzo, S. M. & Ancilotto, F. Supersolid behavior of a dipolar Bose–Einstein condensate confined in a tube. Phys. Rev. A 99, 041601 (2019).

    ADS  CAS  Article  Google Scholar 

  10. 10.

    Boninsegni, M. & Prokof’ev, N. V. Colloquium: Super-solids: what and where are they? Rev. Mod. Phys. 84, 759–776 (2012).

    ADS  CAS  Article  Google Scholar 

  11. 11.

    Tanzi, L. et al. Observation of a dipolar quantum gas with metastable supersolid properties. Phys. Rev. Lett. 122, 130405 (2019).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  12. 12.

    Böttcher, F. et al. Transient supersolid properties in an array of dipolar quantum droplets. Phys. Rev. X 9, 011051 (2019).

    Google Scholar 

  13. 13.

    Chomaz, L. et al. Long-lived and transient supersolid behaviors in dipolar quantum gases. Phys. Rev. X 9, 021012 (2019).

    CAS  Google Scholar 

  14. 14.

    Guo, M. et al. The low-energy Goldstone mode in a trapped dipolar super-solid. Nature 574, 386–389 (2019).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  15. 15.

    Natale, G. et al. Excitation spectrum of a trapped dipolar supersolid and its experimental evidence. Phys. Rev. Lett. 123, 050402 (2019).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  16. 16.

    Tanzi, L. et al. Supersolid symmetry breaking from compressional oscillations in a dipolar quantum gas. Nature 574, 382–385 (2019).

    ADS  CAS  PubMed  Article  Google Scholar 

  17. 17.

    Birkl, G., Kassner, S. & Walther, H. Multiple-shell structures of laser-cooled 24Mg+ ions in a quadrupole storage ring. Nature 357, 310–313 (1992).

    ADS  CAS  Article  Google Scholar 

  18. 18.

    Raizen, M. G., Gilligan, J. M., Bergquist, J. C., Itano, W. M. & Wineland, D. J. Ionic crystals in a linear Paul trap. Phys. Rev. A 45, 6493–6501 (1992).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  19. 19.

    Fishman, S., De Chiara, G., Calarco, T. & Morigi, G. Structural phase transitions in low-dimensional ion crystals. Phys. Rev. B 77, 064111 (2008).

    ADS  Article  CAS  Google Scholar 

  20. 20.

    Shimshoni, E., Morigi, G. & Fishman, S. Quantum zigzag transition in ion chains. Phys. Rev. Lett. 106, 010401 (2011).

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

  21. 21.

    Hew, W. K. et al. Incipient formation of an electron lattice in a weakly confined quantum wire. Phys. Rev. Lett. 102, 056804 (2009).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  22. 22.

    Mehta, A. C., Umrigar, C. J., Meyer, J. S. & Baranger, H. U. Zigzag phase transition in quantum wires. Phys. Rev. Lett. 110, 246802 (2013).

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

  23. 23.

    Astrakharchik, G. E., Morigi, G., De Chiara, G. & Boronat, J. Ground state of low-dimensional dipolar gases: linear and zigzag chains. Phys. Rev. A 78, 063622 (2008).

    ADS  Article  CAS  Google Scholar 

  24. 24.

    Ruhman, J., Dalla Torre, E. G., Huber, S. D. & Altman, E. Nonlocal order in elongated dipolar gases. Phys. Rev. B 85, 125121 (2012).

    ADS  Article  CAS  Google Scholar 

  25. 25.

    Santos, L., Shlyapnikov, G. V. & Lewenstein, M. Roton-maxon spectrum and stability of trapped dipolar Bose–Einstein condensates. Phys. Rev. Lett. 90, 250403 (2003).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  26. 26.

    Ronen, S., Bortolotti, D. C. E. & Bohn, J. L. Radial and angular rotons in trapped dipolar gases. Phys. Rev. Lett. 98, 030406 (2007).

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

  27. 27.

    Wilson, R. M., Ronen, S., Bohn, J. L. & Pu, H. Manifestations of the roton mode in dipolar Bose–Einstein condensates. Phys. Rev. Lett. 100, 245302 (2008).

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

  28. 28.

    Bisset, R. N., Baillie, D. & Blakie, P. B. Roton excitations in a trapped dipolar Bose–Einstein condensate. Phys. Rev. A 88, 043606 (2013).

    ADS  Article  CAS  Google Scholar 

  29. 29.

    Gallemí, A., Roccuzzo, S. M., Stringari, S. & Recati, A. Quantized vortices in dipolar supersolid Bose–Einstein-condensed gases. Phys. Rev. A 102, 023322 (2020).

    ADS  Article  Google Scholar 

  30. 30.

    Roccuzzo, S. M., Gallemí, A., Recati, A. & Stringari, S. Rotating a supersolid dipolar gas. Phys. Rev. Lett. 124, 045702 (2020).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  31. 31.

    Ancilotto, F., Barranco, M., Pi, M. & Reatto, L. Vortex properties in the extended supersolid phase of dipolar Bose–Einstein condensates. Phys. Rev. A 103, 033314 (2021).

    ADS  CAS  Article  Google Scholar 

  32. 32.

    Zhang, Y.-C., Maucher, F. & Pohl, T. Supersolidity around a critical point in dipolar Bose–Einstein condensates. Phys. Rev. Lett. 123, 015301 (2019).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  33. 33.

    Li, J.-R. et al. A stripe phase with supersolid properties in spin–orbit-coupled Bose–Einstein condensates. Nature 543, 91–94 (2017).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  34. 34.

    Léonard, J., Morales, A., Zupancic, P., Esslinger, T. & Donner, T. Supersolid formation in a quantum gas breaking a continuous translational symmetry. Nature 543, 87–90 (2017).

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

  35. 35.

    Kadau, H. et al. Observing the Rosensweig instability of a quantum ferrofluid. Nature 530, 194–197 (2016).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  36. 36.

    Ferrier-Barbut, I., Kadau, H., Schmitt, M., Wenzel, M. & Pfau, T. Observation of quantum droplets in a strongly dipolar Bose gas. Phys. Rev. Lett. 116, 215301 (2016).

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

  37. 37.

    Chomaz, L. et al. Quantum-fluctuation-driven crossover from a dilute Bose–Einstein condensate to a macrodroplet in a dipolar quantum fluid. Phys. Rev. X 6, 041039 (2016).

    Google Scholar 

  38. 38.

    Wächtler, F. & Santos, L. Quantum filaments in dipolar Bose–Einstein condensates. Phys. Rev. A 93, 061603 (2016).

    ADS  Article  CAS  Google Scholar 

  39. 39.

    Bisset, R. N., Wilson, R. M., Baillie, D. & Blakie, P. B. Ground-state phase diagram of a dipolar condensate with quantum fluctuations. Phys. Rev. A 94, 033619 (2016).

    ADS  Article  CAS  Google Scholar 

  40. 40.

    Lavoine, L. & Bourdel, T. Beyond-mean-field crossover from one dimension to three dimensions in quantum droplets of binary mixtures. Phys. Rev. A 103, 033312 (2021).

    ADS  CAS  Article  Google Scholar 

  41. 41.

    Sohmen, M. et al. Birth, life, and death of a dipolar supersolid. Phys. Rev. Lett. 126, 233401 (2021).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  42. 42.

    Hadzibabic, Z., Stock, S., Battelier, B., Bretin, V. & Dalibard, J. Interference of an array of independent Bose–Einstein condensates. Phys. Rev. Lett. 93, 180403 (2004).

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

  43. 43.

    Schmidt, J.-N. et al. Roton excitations in an oblate dipolar quantum gas. Phys. Rev. Lett. 126, 193002 (2021).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  44. 44.

    Pyka, K. et al. Topological defect formation and spontaneous symmetry breaking in ion Coulomb crystals. Nat. Commun. 4, 2291 (2013).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  45. 45.

    Ulm, S. et al. Observation of the Kibble–Zurek scaling law for defect formation in ion crystals. Nat. Commun. 4, 2290 (2013).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  46. 46.

    Trautmann, A. et al. Dipolar quantum mixtures of erbium and dysprosium atoms. Phys. Rev. Lett. 121, 213601 (2018).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  47. 47.

    Chomaz, L. et al. Observation of roton mode population in a dipolar quantum gas. Nat. Phys. 14, 442–446 (2018).

    PubMed  PubMed Central  Article  CAS  Google Scholar 

  48. 48.

    Lima, A. R. P. & Pelster, A. Quantum fluctuations in dipolar Bose gases. Phys. Rev. A 84, 041604 (2011).

    ADS  Article  CAS  Google Scholar 

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Acknowledgements

We thank the Innsbruck Erbium team, T. Bland, G. Morigi and B. Blakie for discussions. We acknowledge R. M. W. van Bijnen for developing the code for our eGPE ground-state simulations. The experimental team is financially supported through an ERC Consolidator Grant (RARE, number 681432), an NFRI grant (MIRARE, number ÖAW0600) of the Austrian Academy of Science, the QuantERA grant MAQS by the Austrian Science Fund FWF number I4391-N. L.S. and F.F. acknowledge the DFG/FWF via FOR 2247/PI2790. M.S. acknowledges support by the Austrian Science Fund FWF within the DK-ALM (number W1259-N27). L.S. thanks the funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC-2123 QuantumFrontiers - 390837967. M.A.N. has received funding as an ESQ Postdoctoral Fellow from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement number 801110 and the Austrian Federal Ministry of Education, Science and Research (BMBWF). M.J.M. acknowledges support through an ESQ Discovery Grant by the Austrian Academy of Sciences. We also acknowledge the Innsbruck Laser Core Facility, financed by the Austrian Federal Ministry of Science, Research and Economy. Part of the computational results presented have been achieved using the HPC infrastructure LEO of the University of Innsbruck.

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Contributions

M.A.N., C.P., L.K., M.S., M.J.M. and F.F. contributed experimental work. E.P. and R.N.B. performed eGPE calculations. L.S. contributed variational model. All authors contributed to interpretation of results and preparation of manuscript.

Corresponding author

Correspondence to Francesca Ferlaino.

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Extended data figures and tables

Extended Data Fig. 1 Fourier transforms of in-trap images.

The upper row shows individual in-trap images for different trap aspect ratios, as shown in Fig. 2b. The lower row shows the data for the same parameters in the Fourier domain, with k the associated wavenumber. As the trap aspect ratio is increased, the modulation goes from being present along a single direction to two, and a clear hexagonal pattern is visible.

Extended Data Fig. 2 Supersolid droplet array with more than two rows.

a, In-trap image of a droplet array with more than two rows. b, Averaged Fourier transform of 309 images in conditions of a, showing that a regular modulated structure persists in the more extended system. c, Calculated ground state from the eGPE for trap parameters (fxfy, fz) = (22, 55, 140) Hz, and N = 60,000 atoms in the droplets, representative of the experimental conditions in a, b. d, Averaged TOF interference pattern for the conditions of a, b. The inset shows the measured 2D density profile and the main panel shows a radially averaged density, normalized to the peak density of the averaged image. The grey lines represent individual trials and the red line is the average. The repeatability of the modulation indicates the presence of phase coherence between droplets.

Extended Data Fig. 3 Prospects for larger and isotropic droplet arrays.

The panels show eGPE-calculated ground-state density profiles with fixed average atomic density (see text) and either fixed atom number and trap volume (upper row) or fixed fx (lower row). Here N refers to the total number of atoms in the simulation (droplets plus halo), in contrast to the definition used elsewhere to compare with experimental conditions (droplets only).

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Norcia, M.A., Politi, C., Klaus, L. et al. Two-dimensional supersolidity in a dipolar quantum gas. Nature 596, 357–361 (2021). https://doi.org/10.1038/s41586-021-03725-7

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