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

Abstract

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.

Figures (4)

• Figure 1: Stability diagram of 2D dipolar condensates.a. Different colors denote different regimes calculated with a homogeneous 2D density $n_{2D}=100~\mu m^{-2}$ and $l_z=272$ nm by Eq. \ref{['Eq1']} for erbium atoms. In the stable regime (blue), $g_{\text{eff}}$ is well above zero and the excitation energy is always real. In the roton instability (RI) regime (yellow), an imaginary part emerges at finite momentum, while $g_{\text{eff}}$ is still positive. In the phonon instability (PI) regime (red), $g_{\text{eff}}<0$ so no stable phonon exists. Black triangles are the critical values for the formation of stripes, obtained by the teGPE simulations with our experimental sequences. A stripe state appears below the black triangles. b. Time sequence for sample preparation in unstable regimes. We first prepare stable superfluid samples with different dipolar angle $\theta$ close to the instability boundary denoted by open symbols in a, then linearly ramp the magnetic field in 10 ms to the final value denoted by solid symbols to enter unstable regimes, meanwhile keeping $\theta$ constant. c. Schematic 2D excitation spectrum in different regimes calculated from Eq. \ref{['Eq1']}. Two roton minima are clearly visualized at $(\pm k_{\text{rot}},0)$ in the RI regime. The anisotropic roton triggers the crystallization along the $x$-axis and so the formation of stripes.
• Figure 2: In-situ measurements of crystallization of a 2D dipolar superfluid.a. Upper row: exemplary In-situ images of samples at 180 mG ($\epsilon_{dd}=1.46$), $t_h=15$ ms with $\theta=60^\circ,70^\circ,80^\circ$ respectively. The interaction parameters correspond to the bottom symbols in Fig. \ref{['fig1']}a. Bottom row: 2D density profiles given by teGPE simulation. The saturating density in the color scale is adjusted to be better compared with the experimental data. b. Static structure factor measured by density fluctuations. c. SW as a function of final $\epsilon_{dd}$, showing that 2D dipolar superfluid start to crystallize after entering unstable regime. The vertical dotted (dashed) line denote the critical value in Fig. \ref{['fig1']}a for $\theta=70^\circ$ ($80^\circ$). The black arrow denotes $\epsilon_{dd}=1.46$ at 180 mG where we focus on. (i)-(iii), exemplary single images of samples at $\theta=80^\circ$. (iv)-(vi), exemplary single images of samples at $\theta=70^\circ$. d. Formation dynamics of crystalline order at $\epsilon_{dd}=1.46$, $\theta=70^\circ,80^\circ$. Insets are simulated 2D profiles at $t_h=5,10,15$ ms. Errorbars correspond to the standard error of the mean.
• Figure 3: Probing phase coherence and phase rigidity of 2D dipolar stripes.a. Integrated density distribution after 8.5 ms TOF interference, gray lines are individual measurements, yellow and red lines correspond to the average. Clear density modulation of TOF images is preserved at $\theta=70^\circ$ after averaging more than 50 trials, showing the global coherence. Meanwhile, interference patterns with random distribution are observed at $\theta=80^\circ$. b,c Averaged 2D density distribution after 8.5 ms and 16 ms TOF (averaged more than 50 trials). Upper panel for $\theta=70^\circ$ and lower panel for $\theta=80^\circ$. d. Upper panel: the gray dashed line corresponds to the 1D in situ density distribution integrated along the $y$ direction with three stripes. The blue solid line is the fitting profile to extract the displacement $\Delta x$ and the imbalance $\eta$. Lower panel: $\Delta x$vs.$\eta$ at $70^\circ$ (yellow) and $80^\circ$ (red). Samples at $70^\circ$ show linear $\Delta x-\eta$ correlations, signaling an effective supersolid low-energy Goldstone mode. The gray dashed line is the linear fit of the experimental data at $\theta=70^\circ$. The gray solid line is the $\Delta x-\eta$ linear correlation obtained from teGPE simulations SI. Errorbars represent fitting errors.
• Figure 4: Superfluid hydrodynamics of a 2D dipolar supersolid. Yellow (red) circles correspond to the aspect ratio of averaged in situ images at different $t_h$ with $\theta=70^\circ (80^\circ)$, fitting errors are smaller than the symbol size. The periodic oscillations of the AR indicate a collective quadrupole mode excited in the system. Solid curves are damped sinusoidal fittings starting from $t_h=10$ ms. The green dashed line denotes the aspect ratio of the trap $\omega_y/\omega_x$.