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Supersolid formation in a quantum gas breaking a continuous translational symmetry

Abstract

The concept of a supersolid state combines the crystallization of a many-body system with dissipationless flow of the atoms from which it is built. This quantum phase requires the breaking of two continuous symmetries: the phase invariance of a superfluid and the continuous translational invariance to form the crystal1,2. Despite having been proposed for helium almost 50 years ago3,4, experimental verification of supersolidity remains elusive5,6. A variant with only discrete translational symmetry breaking on a preimposed lattice structure—the ‘lattice supersolid’7—has been realized, based on self-organization of a Bose–Einstein condensate8,9. However, lattice supersolids do not feature the continuous ground-state degeneracy that characterizes the supersolid state as originally proposed. Here we report the realization of a supersolid with continuous translational symmetry breaking along one direction in a quantum gas. The continuous symmetry that is broken emerges from two discrete spatial symmetries by symmetrically coupling a Bose–Einstein condensate to the modes of two optical cavities. We establish the phase coherence of the supersolid and find a high ground-state degeneracy by measuring the crystal position over many realizations through the light fields that leak from the cavities. These light fields are also used to monitor the position fluctuations in real time. Our concept provides a route to creating and studying glassy many-body systems with controllably lifted ground-state degeneracies, such as supersolids in the presence of disorder.

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Figure 1: Breaking continuous translational symmetry in a superfluid quantum gas.
Figure 2: Self-organized phases in the two-cavity system.
Figure 3: Emergence of a doubly self-organized phase.
Figure 4: Breaking continuous translational symmetry.

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Acknowledgements

We thank E. Demler, J. Larson and B. P. Venkatesh for discussions on U(1) symmetries in optical cavities and M. Lee for contributions in the building stage of the experimental apparatus. We acknowledge funding from Synthetic Quantum Many-Body Systems (European Research Council advanced grant) and the EU Collaborative Project TherMiQ (Grant Agreement 618074), and also SBFI support for Horizon2020 project QUIC and SNF support for NCCR QSIT and DACH project ‘Quantum Crystals of Matter and Light’.

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All authors contributed extensively to the work presented here.

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Correspondence to Tilman Esslinger.

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The authors declare no competing financial interests.

Additional information

Reviewer Information Nature thanks K. Hazzard and D. Huse for their contribution to the peer review of this work.

Extended data figures and tables

Extended Data Figure 1 Mean-field solution of the phase diagram.

Order parameters α1 and α2 as a function of the pump–cavity detunings Δ1 and Δ2. The Raman coupling λ is fixed by the experimental value of the transverse pump lattice depth of 38ħωrec. The mean-field model includes cavity decay, different vacuum Rabi frequencies gi and the transverse pump potential.

Source data

Extended Data Figure 2 Lattice geometries for different choices of the phase ϕ of the transverse pump field for balanced cavity fields.

The atoms are cut into one-dimensional lines by the strong transverse pump field. On top, spatial distributions between triangular (ϕ = 0) and hexagonal (ϕ = π/2) can form through the interference between the cavity light fields and the transverse pump, depending on the phase ϕ of the latter.

Extended Data Figure 3 Dependence of the lattice structure on the cavity field amplitudes.

The ground-state manifold for equal couplings and detunings in is a circle in the space of the cavity fields α1 and α2. For each combination of fields, the interference potential in equation (3) between the transverse pump and cavity fields for ϕ = π/2 will have its minima at different positions. Following the circle clockwise, every second line moves left (top highlighted line) while the others move right (bottom highlighted line).

Extended Data Figure 4 Transition between discrete and continuous symmetries.

ac, Histograms (normalized to unity area) of the angles θ describing the position of the crystal lattice, measured in the same way as the data in Fig. 4, but at Δ1 = −2.1 MHz (a), Δ1 = −2.2 MHz (b) and Δ1 = −2.3 MHz (c). The grey lines show kernel density estimation analyses with a Gaussian kernel, whose bandwidth of 0.13π/2 was determined from a cross-validation maximum likelihood reconstruction. Each histogram consists of 19–49 realizations.

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Léonard, J., Morales, A., Zupancic, P. et al. Supersolid formation in a quantum gas breaking a continuous translational symmetry. Nature 543, 87–90 (2017). https://doi.org/10.1038/nature21067

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