Superfluid Fraction of a 2D Bose-Einstein Condensate in a Triangular Lattice

TL;DR

This study reports the first coherent measurement of the two-dimensional superfluid fraction tensor in a Bose-Einstein condensate subjected to a triangular optical lattice. By combining a density-profile analysis, Leggett bounds, and a dynamic compressibility/sound-velocity approach, the authors extract $f_s$ and demonstrate its isotropy due to the lattice's threefold symmetry; results are in good agreement with Gross-Pitaevskii simulations. The work establishes two complementary, robust methods for probing superfluidity in 2D modulated systems and highlights the density-profile bound as a practical upper bound, with potential extensions to dipolar supersolids and strongly interacting gases. Overall, the methods provide a versatile framework for characterizing superfluid response in complex lattice geometries with implications for 2D quantum fluids and beyond.

Abstract

We experimentally investigate the superfluid properties of a two-dimensional, weakly interacting Bose-Einstein condensate in the zero-temperature regime, when it is subjected to a triangular optical lattice potential. We implement an original method, which involves solving the hydrodynamic continuity equation to extract the superfluid fraction tensor from the measured in situ density distribution of the fluid at rest. In parallel, we apply an independent dynamical approach that combines compressibility and sound velocity measurements to determine the superfluid fraction. Both methods yield consistent results in good agreement with simulations of the Gross-Pitaevskii equation as well as with the Leggett bounds determined from the measured density profiles.

Figures (9)

• Figure 1: Sketch of the experiment and density distribution. A triangular lattice potential is projected onto a 2D Bose-Einstein condensate using a spatial light modulator and repulsive light. High-resolution imaging is used to determine the atomic density distribution. The zoomed picture is an absorption image, averaged 40 times, of the atomic cloud subjected to a lattice potential with a depth $V_0\simeq4.7\,\mu_0$, where $\mu_0/k_B = 45(2)\,$nK is the chemical potential of the cloud in the absence of the lattice potential. The color bar encodes the surface density. The length of the scale bar is $10\um$. Integrated density profiles along the $x$ and $y$ directions are also shown.
• Figure 2: Leggett bounds. (a) Measured upper bound $f_s^{+} (\boldsymbol{e}_x)$ (blue circles) and lower bound $f_s^{-} (\boldsymbol{e}_y)$ (green diamonds). Error bars correspond to the error propagation of the 1-$\sigma$ uncertainties in the calibration of the imaging system response. The black solid line is the predicted superfluid fraction assuming a pure BEC described by the GPE. The colored regions define the excluded regions for the superfluid fraction according to Leggett bounds measurements. The darker-colored regions correspond to an exclusion beyond experimental errors. The white region shows the corresponding allowed region. The lighter-colored regions correspond to the intermediate region defined by the extension of experimental errors. (b) Experimentally determined upper $f_s^{+}(\boldsymbol{e}_{1})$ (solid blue) and lower $f_s^{-}(\boldsymbol{e}_{1})$ (solid green) Leggett bounds as a function of $\theta$, the angle between unit vector $\boldsymbol{e}_{1}$ and $\boldsymbol{e}_x$ for $V_0\simeq 4.0\,\mu_0$. The solid black line gives the value of $f_s$ already shown in (a) for this lattice depth.
• Figure 3: Superfluid fraction measurement. Measured superfluid fraction as a function of the normalized lattice amplitude using the method based on the density profile (violet diamonds) and the dynamic approach (red circles). Error bars for the density profile methods are determined as in Fig. \ref{['fig:Leggett_bounds']}. Error bars for the dynamic approach are deduced from the sound velocity and compressibility measurements. The solid line (same as in Fig. \ref{['fig:Leggett_bounds']}) is the predicted $f_s$. The shaded regions represent the excluded areas for the superfluid fraction according to the Leggett bounds measurements reported in Fig. \ref{['fig:Leggett_bounds']}.
• Figure 4: Compressibility and speed of sound measurements. (a) Compressibility determined by measuring the displacement $\delta_y$ of the cloud CoM as a function of the applied static force $M a_y$ to the cloud. The solid line is a fit to the data, giving a slope of $\delta_y / a_y = 34.0 +- 0.7\times 10^{-6}\, \s ^2$. From this slope the compressibility of the cloud is found using Eq. \ref{['eq:kappa_meas']}. (b) Speed of sound determined from the oscillation of the CoM, $\delta_y$, after an abrupt release of a static force. The oscillation data is fitted to an exponentially damped sine yielding an oscillation frequency $\nu_y = 22.4 +- 0.2Hz$. In both figures, $V_0\simeq 2.4 \,\mu_0$. Injecting the measured slope and oscillation frequency in Eq. \ref{['eq:fs_slope+frequ']}, we find $f_s^{(y, y)} = 0.82 +- 0.02$.
• Figure 5: Compressibility measurement. Normalized compressibility as a function of the normalized lattice amplitude. The solid line represents the corresponding prediction given by the GPE. Error bars correspond to the 1-$\sigma$ statistical uncertainty obtained from typically 40 repetitions of the experiment.