Merging and oscillations of dipolar Bose-Einstein condensate droplets

This study investigates the dynamics of dipolar $^{164}$Dy Bose-Einstein condensate droplets in a double-well potential, revealing how spontaneous symmetry breaking, atom number, and interaction strength govern oscillatory behaviors and merger events following barrier removal.

The Gist

Imagine a bowl of ultra-cold, magical soup made of atoms. This isn't just any soup; it's a Bose-Einstein Condensate (BEC), a state of matter where thousands of atoms lose their individual identities and act as a single, giant "super-atom." Now, imagine these atoms have tiny magnets attached to them (they are dipolar). Because they are magnets, they don't just bump into each other; they pull and push on each other from a distance, like invisible rubber bands.

This paper is a computer simulation of what happens when you take two separate pools of this magnetic soup, remove the wall between them, and watch them dance, crash, and merge.

Here is the story of the experiment, broken down into simple concepts:

1. The Setup: Two Pools and a Wall

The scientists set up a "double-well" trap. Think of this as a landscape with two deep valleys separated by a hill. They put their magnetic atoms into these valleys.

• The Atoms: They used Dysprosium (Dy), an element with a very strong magnetic personality.
• The Magic: Because the atoms are magnetic, they want to clump together to save energy (attraction), but they also push apart sideways (repulsion). This tug-of-war creates stable, self-contained "droplets" of atoms, like tiny, floating islands of soup.

2. The Ground Rules: The Phase Diagram

Before removing the wall, the researchers asked: "How many atoms do we need to make these islands?"
They found that the number of atoms changes the shape of the landscape:

• Too few atoms: They just spread out like a thin mist. No islands form.
• Just enough atoms: A single, dense island forms in one valley. Interestingly, if the atoms are too "magnetic" (strong dipolar interaction), the system breaks symmetry. Even though the two valleys are identical, the atoms decide to put all the heavy island in just one valley and leave the other empty. It's like a coin toss where the coin decides to land on heads every single time just because it felt like it.
• More atoms: The islands get crowded. To save space, they split up. You might get two islands, then three, then four, arranging themselves like a tiny crystal grid.

3. The Big Event: Knocking Down the Wall

The main experiment begins when the scientists suddenly remove the hill (the barrier) between the two valleys. The atoms are now free to roam the whole bowl. What happens next depends on how many atoms they started with and how strong their magnetic pull is.

Scenario A: The "Ghost" Droplet (Low Atom Count)

If there aren't many atoms, they don't form solid islands. Instead, the whole cloud just sloshes back and forth in the bowl, like water in a bathtub. It's a gentle, rhythmic wobble.

Scenario B: The "Revolving Door" (Edge of Formation)

If the atom count is right on the edge of forming a droplet, something weird happens. The atoms rush to the center, form a dense ball, but then immediately explode back out into a cloud. They try to become a droplet, fail, and try again. It's like a balloon that keeps inflating and deflating rapidly because the air pressure is just barely enough to hold it together.

Scenario C: The "Crash and Merge" (Medium Atom Count)

This is the most dramatic part. You have two distinct islands (droplets) on either side of the center. When the wall is removed, they rush toward each other.

• The Barrier: Even though they are attracted, there is a "force field" (caused by the repulsive tails of their magnetic fields) that keeps them apart.
• The Crash: If they have enough speed (energy) from the initial rush, they smash through this force field and merge into one giant, super-dense island. It's like two cars crashing and fusing into one massive vehicle.
• The Aftermath: Once merged, this giant blob vibrates and breathes (expands and contracts) as it settles down.

Scenario D: The "Elastic Dance" (High Atom Count)

If there are too many atoms, the repulsive force between the islands is too strong. They rush toward each other, get close, but then bounce off like rubber balls.

• They don't merge. Instead, they start oscillating. They move back and forth, passing each other but never touching.
• The more atoms you have, the stronger the repulsion, and the faster they vibrate. It's like two dancers who are magnetically attracted but wearing stiff, bouncy suits; they can't hug, so they just bounce around each other in a rhythmic pattern.

4. Why Does the Dance Stop? (Damping)

In a perfect world, these bouncing droplets would dance forever. But in reality, the dance slows down and stops. Why?

• The "Breathing" Effect: Every time the droplets get close to each other, they squish and deform. This squishing turns their movement energy into internal heat (vibrations inside the droplet).
• Leaking Atoms: Sometimes, during a close call, a few atoms get knocked off the main island and float away.
• Result: The droplets lose their momentum and eventually stop moving, settling into a calm, merged state or a stationary pattern.

The Big Picture

This paper is essentially a study of how magnetic atoms behave when forced to interact.

• It shows that nature is picky: sometimes atoms prefer to be in one spot, sometimes they split up, and sometimes they refuse to merge.
• It reveals that symmetry can break spontaneously: Even in a perfectly balanced setup, the atoms might choose to be unbalanced.
• It highlights the delicate balance between attraction (wanting to merge) and repulsion (wanting to stay apart).

The Takeaway:
Imagine a group of people in a room who are both best friends and bitter rivals. If you remove the wall between two groups, some will hug and merge into one big group, some will bounce off each other and dance around the room, and some will just stand there. This paper maps out exactly who does what, depending on how "friendly" (magnetic) they are and how many people are in the room. This helps scientists understand the weird, quantum behavior of matter that could one day lead to new technologies like super-conductors or quantum computers.

Technical Summary

1. Problem Statement

The paper investigates the non-equilibrium dynamics of dipolar Bose-Einstein condensate (BEC) droplets composed of $^{164}\text{Dy}$ atoms. Specifically, the study focuses on systems confined in a double-well potential where the interwell barrier is suddenly removed. The primary objectives are to:

• Map the ground-state phase diagrams of these droplets as a function of atom number ($N$) and dipolar interaction strength ($\epsilon_{dd}$).
• Analyze the time evolution following barrier removal, specifically distinguishing between droplet merging (coalescence) and oscillatory dynamics (where droplets remain distinct).
• Understand the mechanisms driving symmetry breaking, the role of the repulsive tails of the dipolar interaction, and the damping of oscillations due to internal modes.

2. Methodology

The authors employ a theoretical approach based on the extended Gross-Pitaevskii equation (eGPE) in three dimensions.

• Hamiltonian: The system is described by the Hamiltonian:
  $$H_{eGP} = -\frac{\hbar^2}{2m}\nabla^2 + V_{ext} + g|\Psi|^2 + \gamma|\Psi|^3 + V_{dd}$$
  • Contact Interaction ($g|\Psi|^2$): Tuned via Feshbach resonance.
  • Lee-Huang-Yang (LHY) Correction ($\gamma|\Psi|^3$): Crucial for stabilizing the droplets against collapse by accounting for quantum fluctuations.
  • Dipolar Interaction ($V_{dd}$): Long-range, anisotropic interaction ($1/r^3$) dependent on the angle between the polarization axis and the relative position vector.
• Potential Geometry:
  • Transverse ($y, z$): Harmonic confinement ($\omega_y = 60$ Hz, $\omega_z = 120$ Hz).
  • Longitudinal ($x$): A "Mexican-hat" double-well potential ($V_x = \alpha x^4 - \beta x^2$). The barrier is removed by setting $\beta = 0$, converting the potential to a single well.
• Simulation Protocol:
  1. Ground State: Obtained via imaginary-time evolution of the eGPE for a specific $N$ and $\epsilon_{dd}$.
  2. Dynamics: Real-time evolution is simulated using the Crank-Nicolson scheme after the barrier is removed.
  3. Parameters: The study varies the atom number ($N$), dipolar strength ($\epsilon_{dd} \in [1.4, 1.5]$), and interwell distance ($2d = 3, 5, 7$ $\mu$m).

3. Key Contributions and Results

A. Ground-State Phase Diagrams

The authors constructed phase diagrams showing the lowest-energy configurations as a function of $N$:

• Spontaneous Symmetry Breaking (SSB): For strong dipolar interactions ($\epsilon_{dd} = 1.5$), the system exhibits SSB. In certain atom number ranges (Region II), the ground state is asymmetric, with a fully developed droplet in one well and a dilute cloud in the other, despite the symmetric potential. This optimizes interaction energy.
• Configuration Transitions: As $N$ increases, the system transitions through distinct phases:
  • Region I: Delocalized gas (no droplets).
  • Region II: Asymmetric single droplet (SSB).
  • Region III: Symmetric two-droplet state.
  • Region IV/V: Asymmetric (2+1) and Symmetric (2x2) multi-droplet states.
• Dependence on $\epsilon_{dd}$: As $\epsilon_{dd}$ decreases (e.g., to 1.4), the SSB regions vanish, and transitions become smoother. The peak density of droplets saturates at $\sim 2250$ atoms/$\mu$m$^3$ for $\epsilon_{dd}=1.5$.

B. Dynamical Regimes: Merging vs. Oscillations

Upon removing the barrier, the system evolves into one of two regimes depending on the initial excess energy and the inter-droplet potential barrier:

1. Droplet Merging:

   • Occurs when the initial excess kinetic energy is sufficient to overcome the repulsive barrier formed by the dipolar tails.
   • Signature: A sharp drop in interaction energy and a redistribution of kinetic energy.
   • Outcome: Two or more droplets coalesce into a single, larger droplet (sometimes transiently forming a "giant" droplet before settling).
   • Threshold: The merging threshold increases with larger initial well separation ($2d$). For $2d=3\mu$m, merging occurs for $N \lesssim 8800$; for $2d=5\mu$m, the threshold rises to $\sim 1.65 \times 10^4$.

2. Oscillatory Dynamics:

   • Occurs when the inter-droplet barrier prevents merging. Droplets oscillate around their equilibrium positions.
   • Driving Forces: External potential and the repulsive tails of the in-plane dipolar interaction.
   • Periodicity: The oscillation period ($T$) decreases as $N$ increases (due to stronger repulsion). For $N=10^4$, $T \approx 9.8$ ms; for $N=2\times 10^4$, $T \approx 8.2$ ms.
   • Damping: Oscillations are damped over time. The primary mechanism is the transfer of center-of-mass kinetic energy into internal breathing modes during close droplet encounters.
   • Symmetry Effects:
     • Symmetric (2 vs 2): Highly regular, long-lived oscillations (lifetimes up to seconds for large $N$).
     • Asymmetric (2 vs 1): Complex, multi-frequency motion with strong coupling between $x$ and $y$ directions due to long-range dipolar forces.

C. Influence of Geometry and Parameters

• Interwell Distance ($2d$): Increasing $2d$ from 3 $\mu$m to 5 $\mu$m increases the initial excess energy. This leads to larger oscillation amplitudes but significantly reduced lifetimes (by a factor of $\sim 6$) due to stronger deformation of the droplets as they penetrate deeper into the potential barrier.
• Interaction Strength ($\epsilon_{dd}$): Lowering $\epsilon_{dd}$ to 1.45 or 1.4 reduces the stability of the droplets, leading to irregular dynamics, frequent merging/splitting cycles, and a loss of clear periodicity.

4. Significance

• Control of Quantum Phases: The study demonstrates how to control the transition between distinct quantum phases (merging vs. oscillating) by tuning atom number and trap geometry.
• Symmetry Breaking: It provides a clear theoretical example of spontaneous symmetry breaking in dipolar systems driven by interaction energy minimization, a phenomenon relevant to supersolid formation.
• Damping Mechanisms: The identification of breathing modes as the primary source of damping in dipolar droplet oscillations offers insight into the stability and lifetime of such quantum fluids.
• Experimental Relevance: The predicted phenomena (SSB, controlled merging, long-lived oscillations) are accessible in current experiments with highly magnetic atoms like Dysprosium ($^{164}\text{Dy}$), providing a roadmap for investigating non-equilibrium dynamics in dipolar quantum fluids.

In conclusion, the paper establishes a comprehensive framework for understanding the interplay between interaction strength, geometry, and initial conditions in dipolar BECs, revealing rich dynamical behaviors ranging from symmetry breaking to complex multi-droplet oscillations.