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Dipolar quantum solids emerging in a Hubbard quantum simulator


In quantum mechanical many-body systems, long-range and anisotropic interactions promote rich spatial structure and can lead to quantum frustration, giving rise to a wealth of complex, strongly correlated quantum phases1. Long-range interactions play an important role in nature; however, quantum simulations of lattice systems have largely not been able to realize such interactions. A wide range of efforts are underway to explore long-range interacting lattice systems using polar molecules2,3,4,5, Rydberg atoms2,6,7,8, optical cavities9,10,11 or magnetic atoms12,13,14,15. Here we realize novel quantum phases in a strongly correlated lattice system with long-range dipolar interactions using ultracold magnetic erbium atoms. As we tune the dipolar interaction to be the dominant energy scale in our system, we observe quantum phase transitions from a superfluid into dipolar quantum solids, which we directly detect using quantum gas microscopy with accordion lattices. Controlling the interaction anisotropy by orienting the dipoles enables us to realize a variety of stripe-ordered states. Furthermore, by transitioning non-adiabatically through the strongly correlated regime, we observe the emergence of a range of metastable stripe-ordered states. This work demonstrates that novel strongly correlated quantum phases can be realized using long-range dipolar interactions in optical lattices, opening the door to quantum simulations of a wide range of lattice models with long-range and anisotropic interactions.

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Fig. 1: Experimental set-up.
Fig. 2: Dipolar quantum solids.
Fig. 3: Solids and global phase separation with spatial anisotropy.
Fig. 4: Out-of-equilibrium dynamics.

Data availability

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


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We wish to acknowledge V. Kaxiras, A. Kale, M. Xu, M. Sohmen, M. Mark and Y. Bao for help on building the experiment. We wish to acknowledge R. Sahay, B. Capogrosso-Sansone, E.-A. Kim, L. Homeier, A. Bohrdt, F. Grusdt, M. Lebrat, T. Esslinger, M. Kebric, S. Sachdev and C. A. R. Sa de Melo for helpful discussions. We are supported by the US Department of Energy Quantum Systems Accelerator (grant no. DE-AC02-05CH11231), the National Science Foundation (NSF) Center for Ultracold Atoms (grant no. PHY-1734011), the Army Research Office Defense University Research Instrumentation Program (W911NF2010104), the Office of Naval Research Vannevar Bush Faculty Fellowship (N00014-18-1-2863) and the Defense Advanced Research Projects Agency Optimization with Noisy Intermediate-Scale Quantum devices (W911NF-20-1-0021). A.D. acknowledges support from the NSF Graduate Research Fellowship Program (grant no. DGE2140743). The computations in this paper were run on the FASRC Cannon cluster supported by the FAS Division of Science Research Computing Group at Harvard University.

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Authors and Affiliations



L.S., A.D., M.S., R.G., S.F.O., A.K., A.H.H., G.A.P., S.E., S.D. and O.M. contributed to building the experiment set-up. L.S., A.D., M.S. and O.M. performed the measurements and analysed the data. A.D. performed the theoretical analysis. M.G. conceived the experiment in collaboration with F.F. M.G. supervised all works. All authors discussed the results. L.S., A.D., M.S., R.G., S.F.O., F.F., O.M. and M.G. contributed to the manuscript.

Corresponding authors

Correspondence to Lin Su or Markus Greiner.

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M.G. is a cofounder and shareholder of QuEra Computing. All other authors declare no competing interests.

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Extended data figures and tables

Extended Data Fig. 1 Adiabaticity of the lattice ramp.

We probe the adiabaticity of the lattice ramp by varying the duration as we follow the solid and dashed arrow to return to the star position at 7 ER in Fig. 1d. When the ramp duration is very short at 1 ms (a), we see sharp coherence peaks in the time-of-flight image. As we slow down the ramp to a duration of 9 ms (b), the coherence peaks in the time-of-flight image are the least resolved. Further increasing the ramp duration up to 438 ms (c), we observe well-resolved coherence peaks again. The peaks are less sharp compared to those in a, possibly due to the decoherence and atom loss during the ramps, which in total takes almost 900 ms. These averaged images demonstrate that, with the ramp duration on the order of 100 ms in this paper, the system is in the adiabatic regime.

Extended Data Fig. 2 Histogram for digitization of occupation number.

We perform high-fidelity site-resolved imaging after expanding the 2D accordion lattice to 3 μm spacing. a, half-filling histogram. The fidelity to distinguish between 0 and 1 filling per site after expanding the 2D accordion lattice is more than 99%. b, unity-filling histogram. The efficiency of transferring atoms to the 2D accordion lattice and expanding is more than 98%.

Extended Data Fig. 3 Stripe overlap in a 2 by 6 box.

We demonstrate the bimodal distribution of the macrostate that is temporally uncorrelated. a, histogram of the overlap of the stripe order with the single shot data (blue) and simulation of the infinite temperature state (orange). b, Auto-correlation of the stripe overlap data.

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Su, L., Douglas, A., Szurek, M. et al. Dipolar quantum solids emerging in a Hubbard quantum simulator. Nature 622, 724–729 (2023).

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