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Phase coherence in out-of-equilibrium supersolid states of ultracold dipolar atoms

Nature Physics volume 17, pages 356–361 (2021)Cite this article

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Abstract

A supersolid is a counterintuitive phase of matter that combines the global phase coherence of a superfluid with a crystal-like self-modulation in space. Recently, such states have been experimentally realized using dipolar quantum gases. Here we investigate the response of a dipolar supersolid to an interaction quench that shatters the global phase coherence. We identify a parameter regime in which this out-of-equilibrium state rephases, indicating superfluid flow across the sample as well as an efficient dissipation mechanism. We find a crossover to a regime where the tendency to rephase gradually decreases until the system relaxes into an incoherent droplet array. Although a dipolar supersolid is, by its nature, ‘soft’, we capture the essential behaviour of the de- and rephasing process within a rigid Josephson junction array model. Yet, both experiment and simulation indicate that the interaction quench causes substantial collective mode excitations that connect to phonons in solids and affect the phase dynamics.

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Fig. 1: Phase diagram, experimental sequence and starting condition.
Fig. 2: Phase scrambling.
Fig. 3: Rephasing.
Fig. 4: Out-of-equilibrium dynamics.
Fig. 5: Density-link dependence of rephasing dynamics.

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Data availability

Source data are available for this paper51. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

We are grateful to S. Erne, J. Schmiedmayer and the ERBIUM team for insightful discussions and M. A. Norcia for careful reading the manuscript. We acknowledge R. M. W. van Bijnen for developing the code for our eGPE ground-state simulations. G.M. and T.G. thank N. Caballero for insightful discussions on the numerical solution of the Langevin equation. This work is financially supported through an ERC Consolidator grant (RARE, no. 681432), an NFRI grant (MIRARE, no. ÖAW0600) from the Austrian Academy of Science and DFG/FWF (FOR 2247/PI2790) and by the Swiss National Science Foundation under Division II. M.S. and G.D. acknowledge support by the Austrian Science Fund FWF within the DK-ALM (no. W1259-N27). A.T. acknowledges support by the Austrian Science Fund FWF within the Lise Meitner programme (no. M2683-N36). L.C. acknowledges support through the FWF Elise Richter Fellowship no. V792. 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 at the University of Innsbruck.

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  1. These authors contributed equally: P. Ilzhöfer, M. Sohmen.

Authors and Affiliations

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

    P. Ilzhöfer, M. Sohmen, G. Durastante, C. Politi, A. Trautmann, M. J. Mark & F. Ferlaino

  2. Institut für Experimentalphysik und Zentrum für Quantenoptik, Universität Innsbruck, Innsbruck, Austria

    M. Sohmen, G. Durastante, G. Natale, L. Chomaz, M. J. Mark & F. Ferlaino

  3. DQMP, University of Geneva, Geneva, Switzerland

    G. Morpurgo & T. Giamarchi

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Contributions

P.I., G.D. and A.T. conducted the experiment and collected the experimental data. M.S. and C.P. analysed the data. G.N., M.J.M., L.C., M.S. and C.P. performed and analysed the eGPE simulations. G.M. and T.G. developed the JJA model and performed the corresponding simulations. All the authors contributed to the writing of the paper. F.F. supervised the project.

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Correspondence to F. Ferlaino.

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Extended data

Extended Data Fig. 1 Wavelength of the modulation and finite-sampling effect.

a, Difference between incoherent and coherent mean of the density profiles in the ID regime (1.65 G), peaking at the modulation wavelength d ≃ ± 2 μm (dashed lines). b, Histograms of 106 realisations (each) for calculations of ΔqΦ from uniformly random phases Φi, for q = 35 (green) and q = 100 (yellow) draws, respectively. The dashed vertical lines reflect the confidence interval enclosing 68.3 % (‘one σ’) of the calculated values. The solid lines depict a Beta distribution with same mean and variance as the drawn distribution of ΔqΦ (no free fit parameters).

Extended Data Fig. 2 Estimated scattering length.

Calculated B-to-as conversion for 164Dy. Red and blue shaded areas indicate the SSP and the ID region, respectively. The grey area indicates the BEC region, while the yellow areas indicate regions around the two narrow Feshbach resonances located at 2.174 G and 2.336 G where we observe increased atom loss. We estimate as,SSP = 88 a0 in the SSP at 2.43 G and as,ID = 76.9 a0 in the ID at 1.65 G.

Extended Data Fig. 3 Temporal evolution of the atom numbers and temperature in the ID regime and SSP.

a, Total atom number Ntotal, b, temperature T and atom numbers of c, the thermal and d, the coherent part, Nthermal and Ncoherent, as a function of the hold time th. The data sets at 1.65 G (blue) and 2.43 G (red) correspond to the ID regime and the SSP, respectively, whereas the one at 2 G (light blue) corresponds to the intermediate regime.

Extended Data Fig. 4 The global phase variation α from the RTE simulation of a scramble-and-rephase protocol.

a, Evolution of α over the hold time th. The solid orange line depicts an exponential fit to the data. In the inset, the integrated density n and the phase profile θ are exemplarily shown for t* = [3.5, 9.5, 60.5] ms (note the corresponding color filling of the plot markers). b, Residuals from the exponential fit to α.

Extended Data Fig. 5 Dependence of experimental rephasing dynamics on the density link strength \({\mathscr{L}}\).

a, Temporal evolution of ΔΦ (color map) at different \({\mathscr{L}}\) starting from phases scrambled in the ID regime. For each th we record q≥35 individual experimental realizations. For large \({\mathscr{L}}\) the system recovers its global phase coherence (ΔΦ ≃ 0), whereas for small \({\mathscr{L}}\) it does not (ΔΦ ≃ 1). b, AM (circles) and AΦ (diamonds) for the same data set at long hold time, th = 100 ms. The error bars (partly covered by plot markers) are statistical standard errors of AM and AΦ. The red filled pair of symbols corresponds to the data set presented in Fig. 3.

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Ilzhöfer, P., Sohmen, M., Durastante, G. et al. Phase coherence in out-of-equilibrium supersolid states of ultracold dipolar atoms. Nat. Phys. 17, 356–361 (2021). https://doi.org/10.1038/s41567-020-01100-3

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