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The low-energy Goldstone mode in a trapped dipolar supersolid


A supersolid is a counter-intuitive state of matter that combines the frictionless flow of a superfluid with the crystal-like periodic density modulation of a solid1,2. Since the first prediction3 in the 1950s, experimental efforts to realize this state have focused mainly on helium, in which supersolidity remains unobserved4. Recently, supersolidity has also been studied in ultracold quantum gases, and some of its defining properties have been induced in spin–orbit-coupled Bose–Einstein condensates (BECs)5,6 and BECs coupled to two crossed optical cavities7,8. However, no propagating phonon modes have been observed in either system. Recently, two of the three hallmark properties of a supersolid—periodic density modulation and simultaneous global phase coherence—have been observed in arrays of dipolar quantum droplets9,10,11, where the crystallization happens in a self-organized manner owing to intrinsic interactions. Here we directly observe the low-energy Goldstone mode, revealing the phase rigidity of the system and thus proving that these droplet arrays are truly supersolid. The dynamics of this mode is reminiscent of the effect of second sound in other superfluid systems12,13 and features an out-of-phase oscillation of the crystal array and the superfluid density. This mode exists only as a result of the phase rigidity of the experimentally realized state, and therefore confirms the superfluidity of the supersolid.

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Fig. 1: Goldstone and Higgs modes arising from the broken continuous translational symmetry in an infinite supersolid.
Fig. 2: Goldstone modes in a harmonic potential.
Fig. 3: Experimental correlation between η and ∆x.
Fig. 4: The η–∆x correlation across the phase diagram.

Data availability

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


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We acknowledge discussions with the groups of F. Ferlaino, G. Modugno, L. Santos and T. Pohl, as well as with A. Pelster and A. Balaz. This work is supported by the German Research Foundation (DFG) within FOR2247 under Pf381/16-1 and Bu2247/1, Pf381/20-1 and FUGG INST41/1056-1. T.L. acknowledges support from the EU within Horizon2020 Marie Skłodowska Curie IF (grant numbers 746525 coolDips).

Author information




M.G., F.B. and J.-N.S. performed the experiment and analysed the data. J.H. and M.W. performed the numerical analysis. H.P.B., T.L. and T.P. provided scientific guidance in experimental and theoretical questions. All authors contributed to the interpretation of the data and the writing of the manuscript.

Corresponding author

Correspondence to Tilman Pfau.

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The authors declare no competing interests.

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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 Theoretical phase boundaries.

As an indicator of the phase coherence, we show the ratio of the first minimum in the density compared to the central droplet peak density as an indicator of the overlap between the droplets43 of the calculated density profile of the ground state in red. The shaded area is the coherent region determined from the overlap. As in the experiment, three regions are identified and the coherent region locates between 94a0 and 97a0, shifting approximately 3a0 from the experimentally obtained phase diagram. These simulations are done for an atom number of 30 × 103. The black points indicate the measured average atom number in the experiment, with the error bars representing standard deviations for more than ten shots at each scattering length. (a.u., arbitrary units.).

Extended Data Fig. 2 Dynamic structure factor of the collective excitations.

Calculated structure factor S(ωq) in the BEC phase for the scattering length as = 100a0 (a) and as = 98a0 (b) and in the supersolid droplet arrays (c) for as = 96a0. Owing to the finite size of the system, long-wavelength modes become discrete and the lowest possible excitation energy is set by the trap frequency (white horizontal line). For decreasing contact interaction strengths we can observe the roton minimum emerging, until finally its gap closes and the supersolid appears. In the supersolid regime we can clearly see the low-energy out-of-phase Goldstone mode and the large gap to all the other modes above the trap frequency of 30 Hz. The colour scale is normalized to the mode with the highest response across the scattering lengths shown.

Extended Data Fig. 3 Phase pattern of the Goldstone mode.

Shown are the density cut through the three-droplet ground state (black) along the x axis and the phase pattern (red) corresponding to the low-energy Goldstone mode.

Extended Data Fig. 4 Large-amplitude dynamics of the out-of-phase Goldstone mode.

Starting from an array of three droplets, the state can change to a four-droplet state for large excitation amplitudes. From there it either oscillates back and forth between the two (a) or the excitation amplitude is so large that we find that the motion is no longer oscillatory, with the excitation instead travelling only in one direction (b). Similar to Fig. 2, n(x) is the normalized line density along the x axis.

Extended Data Fig. 5 Simulated η–∆x correlation of the four-droplet dynamic states.

Numerically predicted correlation between the imbalance η and droplet displacement ∆x for the four-droplet states appearing at large excitation amplitudes of the low-energy Goldstone mode (blue points, with error bars indicating the uncertainty of the fit used to extract η and ∆x, as in Fig. 2). The red line is a linear fit.

Extended Data Fig. 6 Experimental correlation of the observed four-droplet states.

Similar to Fig. 3, we show the experimental correlation in the supersolid (97.6a0, a) and isolated droplet (91.2a0, b) regions, as well as example in situ images of four-droplet states, with n(x,y) the normalized density (c). For the four-droplet states we observe a clear correlation of imbalance and displacement throughout the supersolid region.

Extended Data Fig. 7 Variance of the four-droplet data with respect to the theoretical correlation.

Similar to Fig. 4b, we find that the variance of the four-droplet data with respect to the theoretical correlation curve (Extended Data Fig. 6) is lowest in the supersolid region. As in Fig. 4a, the shaded area is the coherent region determined experimentally. The error bars indicate one standard error of the experimental data with respect to the theoretical correlation.

Extended Data Fig. 8 Theory-independent evaluation.

Intersection point of the fit with the displacement axis for the three-droplet states (a) and the four-droplet states (b) obtained from a linear fit to the η–∆x correlation data across the part of the phase diagram we explored. An intersection point close to zero indicates the presence of a superfluid flow that can compensate fluctuations during the formation process. The error bars shown represent the standard error of the fitted intersections. As in Fig. 4a, the shaded area is the coherent region determined experimentally.

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Guo, M., Böttcher, F., Hertkorn, J. et al. The low-energy Goldstone mode in a trapped dipolar supersolid. Nature 574, 386–389 (2019).

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