In-situ Observation of Magnetostriction Crossover in a Strongly Dipolar Two-Dimensional Bose Gas

This paper reports the in-situ observation of a magnetostriction crossover from an anisotropic superfluid to an isotropic normal phase in a quasi-two-dimensional dysprosium gas, supported by a new mean-field framework that enables precise thermometry and reveals the distinct spatial characteristics of the coherent core versus thermal wings.

The Gist

The Big Picture: A Crowd of Magnetic Dancers

Imagine a ballroom filled with thousands of tiny, invisible dancers. These aren't just any dancers; they are atoms of a rare earth metal called Erbium, and they are super cold—so cold that they move almost like a single, synchronized entity.

What makes them special is that every single dancer has a tiny magnet inside them (like a microscopic compass needle). In physics, we call these "dipolar" atoms. Because they are magnets, they don't just bump into each other like billiard balls (which is how normal atoms interact); they also pull and push on each other from a distance, just like how two magnets on a fridge can snap together or push apart even when they aren't touching.

The scientists in this paper wanted to watch how this crowd of magnetic dancers behaves when they change the rules of the dance floor. Specifically, they wanted to see what happens when the dancers start acting like a superfluid (a frictionless, magical liquid) versus when they are just a normal gas (a hot, chaotic crowd).

The Problem: The "Magnetic Stretch" (Magnetostriction)

In a normal gas, if you squeeze the ballroom, the crowd squishes evenly. But because these dancers are magnets, they have a weird trick up their sleeve called magnetostriction.

Think of it like this: If you have a crowd of people holding hands in a circle, and you tell them to lean to the left, the whole circle stretches out sideways. In the world of these magnetic atoms, if you tilt their magnetic "compass needles" in a certain direction, the entire cloud of atoms physically deforms. It stretches in one direction and shrinks in another.

The Challenge:
For a long time, scientists couldn't easily measure the temperature of these magnetic clouds. Why? Because the "stretching" (magnetostriction) messed up the usual ways we measure heat. It was like trying to weigh a balloon while someone was constantly blowing air into it and changing its shape. You couldn't tell if the balloon was heavy because it was full of air (hot) or just because it was stretched out.

The Discovery: Two Different Worlds in One Image

The team managed to take a "snapshot" (an in-situ image) of these atoms and discovered something fascinating: The crowd behaves differently depending on whether they are dancing in a chaotic way or a synchronized way.

1. The Normal Phase (The Chaotic Crowd):
   When the atoms are hot and moving randomly (the "normal" phase), they act like a loose crowd of people. Even if you tilt their magnetic needles, they don't stretch. They stay round and happy.

   • Analogy: Imagine a crowd of people running around a field. If you tell them to lean left, they just stumble a bit, but the overall shape of the crowd doesn't change. They are too busy bumping into each other to coordinate a stretch.
   • Why this matters: Because they don't stretch, scientists can finally use standard tools to measure their temperature accurately, even when the magnetic fields are tilted.

2. The Superfluid Phase (The Synchronized Dance):
   When the atoms get cold enough to become a superfluid, they lock into a perfect rhythm. Suddenly, the magnetic stretching kicks in hard.

   • Analogy: Now imagine the crowd is a perfectly synchronized marching band. If the conductor tells them to lean left, they all lean together in perfect unison, and the whole formation stretches out dramatically.
   • The Result: The center of the cloud (the superfluid part) gets very long and skinny, while the outer edges (the normal part) stay round.

The "Crossover": A Smooth Transition

The coolest part of the paper is that they didn't just see two separate states; they saw the transition between them in a single picture.

Imagine a photo of a crowd where the people in the very center are holding hands and stretching out into a long line, but the people on the very edge are still running around in a circle.

• The scientists found a "sweet spot" where the cloud smoothly changes from a round, normal cloud to a stretched, superfluid cloud.
• They call this the Magnetostriction Crossover. It's like watching a rubber band slowly snap from being loose to being pulled tight.

Why This Matters: The "Thermometer" for Magnetic Atoms

Before this paper, measuring the temperature of these magnetic atoms was like trying to read a thermometer while someone was shaking it violently.

The scientists developed a new mathematical recipe (a theory) that accounts for this stretching.

• The Breakthrough: They realized that the "loose" outer edges of the cloud (the thermal wings) don't stretch. This means you can look at the edges, measure the temperature there, and know the temperature of the whole system.
• The Payoff: Now, scientists have a reliable "thermometer" for these exotic magnetic gases. This allows them to study even stranger phenomena, like supersolids (materials that are both a solid crystal and a frictionless fluid at the same time) or quantum vortices (tiny tornadoes in the quantum world).

Summary in a Nutshell

• The Setup: Scientists studied a cloud of magnetic atoms (Erbium) that can stretch like taffy when tilted.
• The Surprise: The "hot" part of the cloud doesn't stretch, but the "cold, superfluid" part stretches wildly.
• The Solution: They created a new way to measure temperature by looking at the non-stretching edges of the cloud.
• The Future: This opens the door to understanding the most exotic states of matter, helping us build better quantum computers and sensors in the future.

It's essentially a story about how scientists finally figured out how to take a clear photo of a wiggly, magnetic crowd, allowing them to finally understand the rules of the dance.

Technical Summary

1. Problem Statement

• Context: Magnetostriction (anisotropic spatial deformation) is a hallmark of strongly dipolar quantum gases due to long-range, anisotropic dipole-dipole interactions (DDI).
• Challenge: While local probing (in-situ imaging) and the Local Density Approximation (LDA) have revolutionized the study of contact-interacting gases (e.g., extracting equations of state and critical points), they face significant hurdles in dipolar systems.
• Specific Gap: It was unclear how to perform thermometry or extract the Equation of State (EOS) in dipolar gases when the dipole orientation is tilted. Strong DDI induces magnetostriction, distorting the cloud shape. Standard LDA assumes the cloud shape follows the trap potential, but strong anisotropy breaks this assumption. Furthermore, it was unknown whether the thermal (normal) phase exhibits magnetostriction similar to the superfluid phase, complicating the separation of thermal and quantum effects.

2. Methodology

• Experimental System: The authors utilized ultracold $^{166}$Er (Erbium) atoms confined in a quasi-two-dimensional (quasi-2D) harmonic trap.
  • Trap Frequencies: $(\omega_x, \omega_y, \omega_z) \approx 2\pi \times (20.0, 21.3, 983)$ Hz.
  • Interaction Strength: Relative dipolar strength $\epsilon_{dd} = a_{dd}/a_s \approx 0.98$, where $a_{dd} = 65.5 a_0$ and $a_s = 67 a_0$.
  • Control: The dipole orientation angle $\theta$ (relative to the $z$-axis) was tuned from $0^\circ$ (perpendicular to the 2D plane) to $90^\circ$ (in the plane).
• Regimes Studied:
  • Superfluid Phase: $T/T_c \approx 0.35$ (approx. 43 nK).
  • Normal (Thermal) Phase: $T/T_c \approx 1.1$ (approx. 70 nK).
• Theoretical Framework:
  • Developed a Quasi-2D Hartree-Fock-Mean-Field (HFMF) theory.
  • Decomposed the effective interaction into local ($U_{eff}^l$) and non-local ($U_{eff}^{nl}$) momentum-dependent terms.
  • Included the von-Weizsäcker quantum correction ($f^{vW}$) in the phase-space density expansion to account for gradient effects beyond the Thomas-Fermi approximation.
  • Used Fast Fourier Transforms (FFT) to solve the self-consistent equations for density profiles.

3. Key Contributions

1. Discovery of Phase-Dependent Magnetostriction: The paper establishes a fundamental dichotomy:
   • The superfluid core exhibits strong magnetostriction (anisotropic deformation) when dipoles are tilted.
   • The normal (thermal) phase remains magnetostriction-free (isotropic) even under strong DDI and large tilt angles.
2. Robust Thermometry Framework: The authors developed a method to determine temperature ($T$) and chemical potential ($\mu$) across all dipole orientations using a single fit to in-situ images. This is achieved by fitting the normal-phase wings with the HFMF theory, which accounts for non-local interactions.
3. Validation of LDA in the Dilute Limit: They demonstrated that the low-density "wings" of the strongly dipolar gas obey a local-density Equation of State, validating the use of LDA for thermodynamic extraction in this regime despite non-local interactions.
4. Observation of the Crossover: They captured the smooth transition from an isotropic thermal periphery to an anisotropic coherent core in a single in-situ image, linking the onset of magnetostriction to the buildup of phase coherence.

4. Key Results

• Aspect Ratio Behavior (Fig. 1):
  • Superfluid: As $\theta$ increases from $0^\circ$ to $90^\circ$, the aspect ratio of the superfluid core changes significantly (deviating from the trap aspect ratio), consistent with 2D dipolar Gross-Pitaevskii Equation (GPE) predictions.
  • Normal Gas: The aspect ratio of the thermal cloud remains constant and matches the trap aspect ratio, regardless of $\theta$.
• Theoretical Explanation:
  • In the normal phase, the anisotropic Fock term (non-local exchange) integrates out in real space at the Thomas-Fermi level, and the Hartree term (local density) is small due to the derivative form of the interaction for slowly varying densities.
  • Quantum corrections (von-Weizsäcker term) are parametrically suppressed by $(\lambda_T/l_r)^4 (a_{dd}/\lambda_T)^2 \ll 1$, preventing significant anisotropy in the thermal cloud.
  • In contrast, the superfluid possesses global coherence; the order parameter directly samples the anisotropy of the interaction in momentum space, leading to pronounced deformation.
• Thermometry Accuracy (Fig. 2):
  • The HFMF model accurately fits density profiles for various angles ($\theta = 0^\circ, 90^\circ$) using a single $(N, T)$ pair.
  • Neglecting the non-local term leads to systematic errors in the extracted chemical potential, particularly at high tilt angles.
• Equation of State (Fig. 3):
  • While the homogeneous EOS breaks scale invariance due to non-local interactions, the trapped system's low-density wings converge to the homogeneous EOS, confirming the validity of LDA-based extraction in the dilute regime.
• Magnetostriction Crossover (Fig. 4):
  • Near the superfluid transition ($T \approx 43$ nK, $\theta = 90^\circ$), the aspect ratio of density contours transitions smoothly from the trap value (isotropic) to the Thomas-Fermi value (anisotropic) as density increases.
  • The crossover center density ($n_0 \approx 29.5 \, \mu m^{-2}$) aligns closely with the estimated Berezinskii-Kosterlitz-Thouless (BKT) critical density ($n_c \approx 29.0 \, \mu m^{-2}$), suggesting that magnetostriction emerges as phase coherence builds up.

5. Significance

• Methodological Breakthrough: This work provides the first reliable tool for in-situ thermometry in strongly dipolar 2D gases with arbitrary dipole orientations. Previous methods failed when magnetostriction distorted the cloud shape.
• Fundamental Physics: It clarifies the distinct roles of coherence and thermal fluctuations in dipolar systems. The absence of magnetostriction in the normal phase simplifies the study of thermal properties, while the crossover offers a new observable for tracking the onset of superfluidity.
• Future Applications: The developed framework enables precise characterization of exotic phases in dipolar systems, including:
  • Supersolids and crystalline order.
  • Quantized vortices.
  • 2D quantum droplets.
  • Studies of BKT physics and beyond-mean-field fluctuations in 2D.

In summary, the paper resolves a long-standing challenge in dipolar gas physics by demonstrating that thermal clouds remain isotropic, thereby enabling accurate thermometry and revealing a smooth crossover to anisotropic superfluidity that serves as a direct probe of coherence in 2D dipolar systems.