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Observation of vortices in a dipolar supersolid

Nature volume 635, pages 327–331 (2024)Cite this article

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Abstract

Supersolids are states of matter that spontaneously break two continuous symmetries: translational invariance owing to the appearance of a crystal structure and phase invariance owing to phase locking of single-particle wavefunctions, responsible for superfluid phenomena. Although originally predicted to be present in solid helium1,2,3,4,5, ultracold quantum gases provided a first platform to observe supersolids6,7,8,9,10, with particular success coming from dipolar atoms8,9,10,11,12. Phase locking in dipolar supersolids has been investigated through, for example, measurements of the phase coherence8,9,10 and gapless Goldstone modes13, but quantized vortices, a hydrodynamic fingerprint of superfluidity, have not yet been observed. Here, with the prerequisite pieces at our disposal, namely a method to generate vortices in dipolar gases14,15 and supersolids with two-dimensional crystalline order11,16,17, we report on the theoretical investigation and experimental observation of vortices in the supersolid phase (SSP). Our work reveals a fundamental difference in vortex seeding dynamics between unmodulated and modulated quantum fluids. This opens the door to study the hydrodynamic properties of exotic quantum systems with numerous spontaneously broken symmetries, in disparate domains such as quantum crystals and neutron stars18.

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Fig. 1: Simulation of vortex nucleation in a supersolid and unmodulated BEC.
Fig. 2: Magnetostirring of a 164Dy dipolar supersolid.
Fig. 3: Vortex nucleation in a dipolar supersolid and BEC.
Fig. 4: TOF interference pattern.

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Data availability

Data pertaining to this work can be found at https://doi.org/10.5281/zenodo.10695943 (ref. 72). Source data are provided with this paper.

Code availability

The codes that support the findings of this study are available from the corresponding author on reasonable request.

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Acknowledgements

We are indebted to J. Dalibard for inspiring discussions on the interference pattern of supersolids in the presence of a vortex. We thank W. Ketterle, S. Stringari, A. Recati and G. Lamporesi for discussions. This work was supported by the European Research Council through the Advanced Grant DyMETEr (no. 101054500), the QuantERA grant MAQS by the Austrian Science Fund (FWF) (no. I4391-N), a joint project grant from the Austrian Science Fund (FWF) (no. I-4426), a NextGenerationEU grant AQuSIM by the Austrian Research Promotion Agency (FFG) (no. FO999896041) and by the Austrian Science Fund (FWF) Cluster of Excellence quantA (10.55776/COE1). A.L. acknowledges financial support through the Disruptive Innovation – Early Career Seed Money grant by the Austrian Science Fund (FWF) and Austrian Academy of Sciences (ÖAW). E.P. acknowledges support by the Austrian Science Fund (FWF) within the DK-ALM (no. W1259-N27). T.B. acknowledges financial support through an ESQ Discovery grant by the Austrian Academy of Sciences.

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  1. Claudia Politi

    Present address: Institute for Quantum Electronics, ETH Zürich, Zürich, Switzerland

  2. These authors contributed equally: Eva Casotti, Elena Poli

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  1. Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Innsbruck, Austria

    Eva Casotti, Lauritz Klaus, Andrea Litvinov, Clemens Ulm, Claudia Politi, Manfred J. Mark & Francesca Ferlaino

  2. Universität Innsbruck, Fakultät für Mathematik, Informatik und Physik, Institut für Experimentalphysik, Innsbruck, Austria

    Eva Casotti, Elena Poli, Lauritz Klaus, Clemens Ulm, Claudia Politi, Manfred J. Mark, Thomas Bland & Francesca Ferlaino

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Contributions

E.C., L.K., A.L., C.U., C.P., M.J.M. and F.F. performed the experimental work and data analysis. E.P. and T.B. performed the theoretical work. All authors contributed to the interpretation of the results and the preparation of the manuscript.

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Correspondence to Francesca Ferlaino.

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Extended data figures and tables

Extended Data Fig. 1 Loss spectrum of 164Dy.

The spectrum is obtained from horizontal absorption imaging by varying the magnetic field at which the evaporative cooling (T ≈ 500 nK) is conducted, with a step size of 20 mG. The magnetic-field values used are highlighted in red (SSP) and green (BEC). Error bars represent the standard error.

Extended Data Fig. 2 Ground-state phase diagram obtained by varying the atom number and the scattering length.

The results are obtained from eGPE calculations with (ω⊥, ωz) = 2π × [50, 103] Hz. The identified phases are: BEC, SS (supersolid), SD (single droplet) and ID (isolated droplets). On the sides are exemplar ground states extracted from the phase diagram.

Extended Data Fig. 3 Vortex nucleation in a dipolar BEC and supersolid for different parameters.

Vortex nucleation in a dipolar BEC (a) and in a supersolid (b) for different atom number and different scattering length as. c, Vortex nucleation for initial noise with different temperatures. All of the results are obtained from eGPE calculations with (ω⊥, ωz) = 2π × [50, 103] Hz, magnetic-field angle from the z axis θ = 30° and three-body recombination losses are included.

Extended Data Fig. 4 Phase-coherence measurement of the initial four-droplet state before rotation, after 36-ms TOF.

The lower (right) figure shows the horizontal (vertical) integrated density. The modulation and central interference peak are present on single images (grey lines) and remain after averaging over 173 images (black line).

Extended Data Fig. 5 Vortex number and expectation value of the angular momentum.

Left, vortex number after 1 s of rotation. Right, expectation value of the angular-momentum operator also after 1 s of rotation. The other parameters are the same as in Fig. 1.

Extended Data Fig. 6 TOF predictions from the Gaussian toy model.

Longer TOF density profiles for the solution shown in Fig. 4. The inset of the first figure shows the initial condition for all states. After 10 ms, the density pattern has frozen into the momentum distribution of the initial cloud. The grey lines show the axis centre (0, 0), highlighting the immediate difference between a no-vortex and vortex expansion from the central density.

Extended Data Fig. 7 Comparison of different vortex-detection methods applied to the theoretical data.

Each point is obtained by applying the experimental vortex-detection algorithm to the states of Fig. 3 and averaging over time. For the SSP, the scattering length is ramped from as = 93a0 to as = 104a0 in 1 ms and the state is expanded for 3 ms, before applying the algorithm. The results are shown for different sizes of Gaussian filter σ and compared with the standard method of counting the 2π phase windings (black line) and the experimental data, in green (red) for the BEC (SSP). The shaded area indicates the error on the mean.

Extended Data Fig. 8 Image processing for the detection of vortices.

Each row indicates different rotation frequency and duration parameters (indicated on the left), for which images are taken following an interaction quench from the supersolid to unmodulated BEC phase. Each column is a step of the processing protocol that proceeds as follows. The data (column 1) are normalized and denoised with a Gaussian filter of size σ = 1 (column 2) and a sharpening mask is applied to magnify the presence of vortices (column 3). The reference image is built from the data image, in which all density variations are eliminated with a Gaussian filter of size σ = 3 (column 4). The residuals (column 5) are obtained from the subtraction of the data to the reference, converting the density depletions to a positive signal. The vortices (black circles) are detected with a peak-detection algorithm with threshold 0.38. The last column shows the location of the vortices on the original image data. Varying the threshold value modifies the absolute vortex count of each individual image but not the overall qualitative result (see Extended Data Fig. 9).

Extended Data Fig. 9 Experimental vortex detection as a function of the threshold parameter.

Normalized vortex occurrence integrated over 1 s of rotation in the BEC phase (a) and in the SSP (b) as a function of the rotation frequency, for varying contrast threshold between 0.34 and 0.42 (see Extended Data Fig. 8). The shaded areas indicate the error on the mean, that is, the standard deviation divided by the square root of the number of points (8). The solid lines are guides to the eye. The results of the eGPE simulations (see Fig. 3) are plotted in thick solid lines as a comparison.

Extended Data Fig. 10 Probability of detecting a vortex as a function of the rotation frequency.

a, Cumulative distribution function obtained from the calculated sum squared differences over the whole dataset, 83 images per frequency, with each of the vortex (solid line) and vortex-free (dashed line) references (see inset images). b, With a defined threshold X (dashed-dotted lines in a) on the cumulative distribution function. Each image is assigned to a category: vortex (red empty circles), vortex-free (blue filled circles) or no classification (grey filled circles). c, Probability of detecting a vortex signal and vortex-free signal out of the selected images in b. The error bars indicate the Clopper–Pearson uncertainty associated with image classification. Top and bottom rows show the classification result for respective thresholds 0.15 and 0.30 on the cumulative distribution function, showing the independence of the signal from the threshold.

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Casotti, E., Poli, E., Klaus, L. et al. Observation of vortices in a dipolar supersolid. Nature 635, 327–331 (2024). https://doi.org/10.1038/s41586-024-08149-7

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