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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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.
The authors declare no competing interests.
Peer review information Nature thanks the anonymous reviewers for their contribution to the peer review of this work.
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Extended data figures and tables
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.
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 (fx, fy, 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.
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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