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

This paper experimentally determines the superfluid fraction of a two-dimensional Bose-Einstein condensate in a triangular optical lattice using two consistent methods—hydrodynamic analysis of in situ density profiles and dynamical measurements of compressibility and sound velocity—which align with Gross-Pitaevskii simulations and Leggett bounds.

The Gist

Imagine a crowd of people in a large, empty room. If you ask them to walk together in a specific direction, they might stumble over each other, bump into walls, or get distracted. This is like a normal fluid, where friction and chaos slow things down.

Now, imagine that same crowd, but they are all holding hands, moving in perfect, silent unison. If you push the room, they all glide together without any internal friction. This is a superfluid, a state of matter where atoms act like a single, giant quantum wave.

In this paper, scientists at the Laboratoire Kastler Brossel in Paris decided to test how well this "perfect dance" works when the floor isn't flat. Instead of a smooth room, they built a triangular grid of invisible hills and valleys (an optical lattice) for the atoms to stand on. Think of it like placing a trampoline with a honeycomb pattern under the dancers.

The Big Question

When you put a superfluid on a bumpy, patterned floor, does it still glide perfectly? Or does the pattern of the floor make it stumble? The scientists wanted to measure exactly how much of the fluid remains "super" (frictionless) versus how much gets "stuck" in the bumps.

The Experiment: A Quantum Dance Floor

1. The Dancers: They used a cloud of Rubidium atoms cooled down to almost absolute zero (colder than outer space). At this temperature, the atoms become a Bose-Einstein Condensate (BEC), acting as one giant super-atom.
2. The Grid: They projected a laser pattern onto the atoms to create a triangular grid of light. The atoms couldn't go through the bright spots (the hills), so they settled into the dark spots (the valleys).
3. The Test: They wanted to see how the fluid responded if they tried to move the whole grid.

Two Ways to Measure the "Super"

The team used two clever, different methods to figure out the "superfluid fraction" (the percentage of atoms that are still gliding perfectly).

Method 1: The Snapshot (The Static Map)
Imagine taking a high-resolution photo of the crowd. Even if they aren't moving, the way they are packed together in the valleys tells a story.

• The scientists took a picture of the atoms sitting still in the grid.
• They used a mathematical trick (solving a "continuity equation") to ask: "If we tried to slide this entire pattern, how much of this crowd would actually move with it, and how much would be left behind in the valleys?"
• It's like looking at a frozen wave in the ocean and calculating how much water would actually flow if the wind started blowing.

Method 2: The Push (The Dynamic Test)
This method was more like a physical experiment.

• They gently pushed the cloud of atoms with a magnetic force.
• They measured two things:
  1. How squishy the cloud was (Compressibility): How much did the cloud shrink or expand when pushed?
  2. How fast a ripple traveled (Sound speed): They gave the cloud a little tap and timed how fast a wave of atoms rippled through it.
• By combining these two measurements, they could calculate the superfluid fraction. It's like testing how fast a wave moves through a crowd to see if they are holding hands (superfluid) or just standing loosely (normal fluid).

The Results

Both methods gave the same answer, which is a great sign that the experiment worked.

• The Finding: As the "hills" of the light grid got higher (stronger), the superfluid fraction went down. The atoms got more "stuck" in the valleys.
• The Agreement: Their real-world measurements matched perfectly with computer simulations (using the Gross-Pitaevskii equation) and theoretical limits (Leggett bounds) that predict what should happen.

Why This Matters

This paper is a success story for measurement. Before this, it was very hard to measure how "super" a fluid is when it's trapped in a 2D pattern (like a honeycomb). The scientists proved they could do it accurately using two different tools.

They didn't invent a new machine or cure a disease; instead, they built a better ruler. They showed that even when you force a superfluid to dance on a bumpy floor, you can still precisely measure how much of it keeps its magical, frictionless nature. This helps scientists understand the rules of quantum mechanics in complex environments, which is a stepping stone for understanding more exotic states of matter, like "supersolids" (materials that are both solid and superfluid at the same time).

Technical Summary

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

Problem Statement
The superfluid fraction ($f_s$) is a fundamental quantity characterizing superfluid states, traditionally defined by the system's response to a moving external potential. While $f_s$ is well-understood in homogeneous systems (where $f_s=1$ at zero temperature) and in one-dimensionally modulated systems, determining the superfluid fraction in two-dimensional (2D) modulated systems remains challenging. In 2D, superfluidity is characterized by a tensor rather than a scalar. Existing methods for extracting $f_s$, such as measuring sound velocity ratios or calculating Leggett bounds from density profiles, are often restricted to 1D modulations or separable density functions. With the recent realization of 2D supersolids, there is a need for robust, general methods to measure the superfluid tensor in arbitrary 2D lattice potentials.

Methodology
The authors experimentally investigate a weakly interacting, quasi-2D Bose-Einstein condensate (BEC) of $^{87}$Rb atoms confined in a square box and subjected to a triangular optical lattice potential. The lattice is generated using a Digital Micromirror Device (DMD) with blue-detuned repulsive light, creating a potential with three-fold symmetry. The system is maintained at temperatures below 20 nK, approximating the zero-temperature limit.

To determine the superfluid fraction tensor, the authors employ two independent methods:

1. Hydrostatic Continuity Equation Approach:

   • This method relies on solving the hydrodynamic continuity equation for a fluid at rest in a moving frame.
   • The authors measure the in situ atomic density distribution using high-resolution absorption imaging.
   • To account for the finite resolution of the imaging system, they model the measured density as a low-pass filtered version of the true density, calibrating attenuation factors for the first two Fourier components of the lattice modulation.
   • They reconstruct the actual density profile $\rho_{eq}(\mathbf{r})$ and solve the continuity equation $\nabla \cdot [\rho_{eq}(\mathbf{r})(\mathbf{v}(\mathbf{r}) - \mathbf{v}_0)] = 0$ to find the phase profile $S(\mathbf{r})$.
   • The superfluid fraction tensor components are then extracted directly from the phase gradient using the definition $f^{(i,j)}_s = \delta_{ij} - \lim_{v_0 \to 0} \frac{\hbar}{NMv_{0,i}} \int d^2r \, \rho_{eq}(\mathbf{r}) [\partial_j S(\mathbf{r})]$.

2. Dynamic Transport Approach:

   • This method combines independent measurements of the cloud's compressibility ($\kappa$) and the speed of sound ($c$).
   • Compressibility: Measured by applying a static magnetic force gradient along the $y$-axis and observing the displacement of the cloud's center of mass (CoM).
   • Speed of Sound: Determined by abruptly switching off the static force and measuring the frequency of the resulting CoM oscillations in the box potential.
   • The superfluid fraction is calculated using the relation $f^{(i,i)}_s = \kappa M c_i^2$.

Key Results

• Consistency of Methods: Both the hydrostatic (density profile) and dynamic (transport) approaches yield consistent results for the superfluid fraction as a function of the normalized lattice amplitude ($V_0/\mu_0$).
• Isotropy: Despite the complex triangular lattice potential, the measured superfluid fraction tensor is found to be nearly isotropic, with off-diagonal components and anisotropy ratios typically below 5%. This confirms theoretical expectations for a triangular lattice with three-fold symmetry.
• Agreement with Theory: The experimental data aligns well with numerical simulations of the Gross-Pitaevskii Equation (GPE), which accurately describes the weakly interacting BEC.
• Leggett Bounds: The measured superfluid fractions lie within the tightest Leggett bounds derived from the reconstructed density profiles. The bounds are shown to be orientation-dependent, with the tightest constraints found along specific lattice directions.
• Deviations at High Amplitude: At large lattice depths, small deviations between the dynamic approach and the GPE prediction are observed. The authors attribute this to the damping of sound waves, potentially arising from finite-temperature effects, though a detailed study of this regime is beyond the scope of the work.

Significance and Claims
The paper demonstrates the successful implementation of two distinct methods for determining the superfluid fraction tensor in a 2D modulated weakly-interacting Bose gas. The authors claim that:

1. The hydrostatic method, based on solving the continuity equation from the density profile, provides a direct and exact determination for mean-field systems.
2. The dynamic method, combining compressibility and sound velocity, offers an independent transport-based verification.
3. The agreement between these methods and GPE simulations validates the experimental approach for studying 2D superfluidity.
4. The methodology is not restricted to scalar superfluid fractions or specific lattice geometries; it can be extended to arbitrary lattice potentials and other systems where a mean-field description is valid, such as dipolar supersolids.
5. The density-profile-based method generally provides an upper bound that is tighter than the standard Leggett bounds, suggesting potential utility for characterizing strongly interacting systems (e.g., Fermi gases) where mean-field approximations may not hold, provided the density profile can be accurately measured.

The work addresses the open problem of measuring the superfluid tensor in 2D modulated systems, providing a quantitative framework relevant to the study of 2D supersolids and other spatially modulated quantum fluids.