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# Supersolid symmetry breaking from compressional oscillations in a dipolar quantum gas

## Abstract

Supersolids are exotic materials combining the frictionless flow of a superfluid with the crystal-like periodic density modulation of a solid. The supersolid phase of matter was predicted 50 years ago1,2,3 for solid helium4,5,6,7,8. Ultracold quantum gases have recently been made to exhibit periodic order typical of a crystal, owing to various types of controllable interaction9,10,11,12,13. A crucial feature of a D-dimensional supersolid is the occurrence of D + 1 gapless excitations, reflecting the Goldstone modes associated with the spontaneous breaking of two continuous symmetries: the breaking of phase invariance, corresponding to the locking of the phase of the atomic wave functions at the origin of superfluid phenomena, and the breaking of translational invariance due to the lattice structure of the system. Such modes have been the object of intense theoretical investigations1,14,15,16,17,18, but they have not yet been observed experimentally. Here we demonstrate supersolid symmetry breaking through the appearance of two distinct compressional oscillation modes in a harmonically trapped dipolar Bose–Einstein condensate, reflecting the gapless Goldstone excitations of the homogeneous system. We observe that the higher-frequency mode is associated with an oscillation of the periodicity of the emergent lattice and the lower-frequency mode characterizes the superfluid oscillations. This work also suggests the presence of two separate quantum phase transitions between the superfluid, supersolid and solid-like configurations.

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## Data availability

The datasets that support the findings of this study are available from the corresponding authors upon reasonable request.

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## Acknowledgements

This work received funding from the EC-H2020 research and innovation program (grant number 641122-QUIC). We acknowledge discussions with R. Citro and J. G. Maloberti and technical assistance from A. Barbini, F. Pardini, M. Tagliaferri and M. Voliani. S.M.R., A.R. and S.S. acknowledge funding from Provincia Autonoma di Trento and the Q@TN initiative. We acknowledge discussions with the participants of the Stuttgart meeting on ‘Perspectives for supersolidity in dipolar droplet arrays’.

## Author information

Authors

### Contributions

L.T., E.L., F.F., A.F., C.G. and G.M. conceived the experimental investigation, performed the measurements and the experimental data analysis. S.M.R., A.R. and S.S. conceived the theoretical investigation, performed the simulations and the theoretical data analysis. All authors contributed to discussions and writing of the paper.

### Corresponding authors

Correspondence to G. Modugno or A. Recati.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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

Peer review information Nature thanks Sean Mossman, Georgy Shlyapnikov and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

## Extended data figures and tables

### Extended Data Fig. 1 Peak amplitude and relative distance.

In the supersolid regime, the axial breathing mode bifurcates into higher- and lower-frequency modes, mainly coupled to the relative distance between the density peaks and to their amplitude, respectively. a, b, Time evolution of the relative distance d(t) between two density peaks (a) and its Fourier transform, dominated by a peak at the higher frequency (red arrow) (b). c, d, Time evolution of the peak density amplitude A(t) (c) and its Fourier transform, dominated by the lower-frequency mode (blue arrow) (d).

### Extended Data Fig. 2 Examples of small- and large-amplitude oscillations.

a, b, Typical false-colour experimental distributions in the $$({k}_{x},{k}_{y})$$ plane for the BEC regime close to the roton instability, $${{\epsilon }}_{{\rm{dd}}}$$ = 1.35(3) (a) and the droplet regime, $${{\epsilon }}_{{\rm{dd}}}$$ = 1.50(5) (b). c, d, Small-amplitude oscillation of $${\sigma }_{k}(t)$$ under the same conditions, $${{\epsilon }}_{{\rm{dd}}}$$ = 1.35(3) (c) and $${{\epsilon }}_{{\rm{dd}}}$$ = 1.50(5) (d). e, f, Large-amplitude oscillation of $${\sigma }_{k}(t)$$ for the BEC regime, $${{\epsilon }}_{{\rm{dd}}}$$ = 1.21(2) (e) and the supersolid regime, $${{\epsilon }}_{{\rm{dd}}}$$ = 1.38(2) (f). In the supersolid regime, we observe a small frequency shift in the oscillation of $${\sigma }_{k}(t)$$ over time: we fit ω/ωx = 1.43(1) up to about 150 ms (blue line), and ω/ωx = 1.48(2) from 150 ms onwards (red line). Error bars represent the standard deviation of 4–8 measurements.

### Extended Data Fig. 3 Large temperature measurements in the BEC regime.

a, b, Measured frequency (a) and measured damping time (b) of the axial breathing mode at $${{\epsilon }}_{{\rm{dd}}}$$ ≈ 1.2 for increasing temperature. The measured frequency for a thermal gas (red dot) is compatible with ω/ωx = 2, demonstrating the collisionless nature of the system and the breaking of phase invariance for the BEC. Error bars represent the standard deviation of 4–8 measurements.

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Tanzi, L., Roccuzzo, S.M., Lucioni, E. et al. Supersolid symmetry breaking from compressional oscillations in a dipolar quantum gas. Nature 574, 382–385 (2019). https://doi.org/10.1038/s41586-019-1568-6

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• DOI: https://doi.org/10.1038/s41586-019-1568-6

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