Observation of a supersolid stripe state in two-dimensional dipolar gases

Fluctuations typically destroy long-range order in two-dimensional (2D) systems, posing a fundamental challenge to the existence of exotic states like supersolids, which paradoxically combine solid-like structure with frictionless superfluid flow. While long-predicted, the definitive observation of a 2D supersolid has remained an outstanding experimental goal. Here, we report the observation of a supersolid stripe phase in a strongly dipolar quantum gas of erbium atoms confined to 2D. We directly image the periodic density modulation, confirming its global phase coherence through matter-wave interference and demonstrating its phase rigidity relevant to the low-energy Goldstone mode, consistent with numerical calculations. Through collective excitation measurements, we demonstrate the hydrodynamic behavior of the supersolid. This work highlights a novel mechanism for supersolid formation in low dimensions, and opens the door for future research on the intricate interplay between temperature, supersolidity, and dimensionality.

💡 Research Summary

The authors report the first unambiguous observation of a supersolid stripe phase in a genuinely two‑dimensional (2D) dipolar quantum gas. Using a quasi‑2D trap for ^166Er atoms, they exploit the large magnetic dipole moment (μ = 7 μ_B) and a tunable magnetic field to control both the contact scattering length a_s (via a Feshbach resonance) and the dipole tilt angle θ with respect to the confinement axis. By linearly ramping the magnetic field within 10 ms they increase the relative dipolar strength ε_dd = a_dd/a_s to ≈ 1.46, driving the system from a stable superfluid into either a roton‑instability (RI) regime (θ ≈ 70°) or a phonon‑instability (PI) regime (θ ≈ 80°).

Theoretical analysis starts from the Bogoliubov dispersion for a homogeneous 2D dipolar condensate, Eq. (1), which includes the anisotropic momentum‑dependent interaction V(k). The effective contact coupling g_eff = g_s + g_d(3 cos²θ − 1) determines the stability diagram shown in Fig. 1a. In the RI region g_eff > 0 but the excitation spectrum acquires an imaginary part at a finite wavevector k_rot, signalling a roton softening that favours density modulation along a single direction. Mean‑field theory predicts only transient stripes; however, inclusion of the Lee‑Huang‑Yang (LHY) quantum‑fluctuation correction within an extended Gross‑Pitaevskii equation (teGPE) stabilises the stripe crystal as a metastable state. The LHY term effectively provides a repulsive pressure that balances the attractive dipolar forces at the roton minimum, allowing the stripe pattern to persist for tens of milliseconds.

Experimentally, about 3.2 × 10⁴ atoms are prepared at T ≈ 30 nK in a trap with frequencies (ω_x, ω_y, ω_z) = 2π × (14.3, 15.9, 820) Hz, ensuring k_B T ≪ ℏ ω_z and thus a quasi‑2D geometry (ℓ_z ≈ 272 nm). High‑intensity absorption imaging along the tightly confined z‑axis provides an in‑situ spatial resolution of ~1 µm. After the ramp into the unstable regime, the system is held for a typical evolution time t_h ≈ 15 ms, during which clear stripe patterns appear. In‑situ images (Fig. 2a) show high‑contrast, periodic density modulations for both θ = 70° (RI) and θ = 80° (PI). Numerical teGPE simulations reproduce the observed structures: in the RI case the stripes remain connected by a superfluid background (superfluid fraction f_s ≈ 0.90), whereas in the PI case the stripes become isolated droplets, indicating loss of global phase coherence.

The static structure factor S(k) is extracted from the Fourier transform of density fluctuations δn(r) over 50 experimental realizations. For θ = 70°, S(k) exhibits sharp peaks at (±k_rot, 0), confirming a well‑defined lattice spacing; for θ = 80°, the peaks broaden into a diffuse band, reflecting the absence of a fixed period. The integrated weight S_W, defined as the sum of S(k) over a 1 µm × 1 µm region around the peaks, grows sharply once ε_dd exceeds the critical value predicted by teGPE, providing quantitative agreement between theory and experiment.

To probe global phase coherence, the authors perform time‑of‑flight (TOF) interference measurements. The trap is switched off abruptly, and within the first 0.5 ms a 1 G magnetic field is applied along z to accelerate expansion of the stripe pattern. After 8.5 ms (and also after 16 ms) the expanded cloud exhibits interference fringes. For θ = 70°, the fringes survive averaging over > 50 shots, demonstrating that all stripes share a common phase—a hallmark of supersolidity. In contrast, for θ = 80° the interference pattern fluctuates strongly from shot to shot and averages out, indicating that each droplet evolves independently.

A further hallmark of supersolidity is the presence of a low‑energy Goldstone mode coupling the superfluid phase to the lattice displacement. The authors analyse the correlation between the centre‑of‑mass displacement Δx of the three dominant stripes and the population imbalance η among them. By fitting the in‑situ density profile with a sum of Gaussians they extract Δx and η for each realization. The data for θ = 70° display a linear Δx–η relationship, precisely as predicted by the teGPE simulations and consistent with a phonon‑like Goldstone mode that restores the broken translational symmetry via superfluid flow. No such correlation is observed for θ = 80°.

Finally, collective excitation spectroscopy reveals hydrodynamic behaviour characteristic of a supersolid. By modulating the trap frequency and monitoring the response, the authors measure both compressional and shear modes. Their frequencies and damping rates match those expected for a system that simultaneously supports superfluid flow and a crystalline lattice, further confirming the supersolid nature of the stripe phase.

In summary, the work establishes a clear pathway to realize and diagnose a 2D supersolid: (i) engineering an anisotropic roton instability through dipole tilt, (ii) stabilising the resulting stripe crystal with beyond‑mean‑field LHY corrections, (iii) directly imaging the density modulation, (iv) confirming global phase coherence via TOF interference, (v) demonstrating phase rigidity through displacement‑imbalance correlations (Goldstone mode), and (vi) verifying supersolid hydrodynamics through collective mode spectroscopy. This achievement overcomes the longstanding obstacle posed by the Mermin‑Wagner theorem and the Berezinskii‑Kosterlitz‑Thouless physics in two dimensions, opening a versatile platform for exploring low‑dimensional quantum phases, the interplay of temperature, interactions, and dimensionality, and for probing exotic excitations such as supersolid vortices and topological defects.