Josephson vortices and persistent current in a double-ring supersolid system

This paper theoretically investigates ultra-cold dipolar atoms in radially coupled concentric annular traps, revealing how rotation and barrier strength induce particle imbalances, density modulations, and distinct vortex configurations—including unique Josephson vortices at ring junctions—that can be experimentally identified through characteristic interference patterns.

The Gist

Imagine a microscopic world where atoms behave not just like individual particles, but like a single, giant, super-cooled wave. This is a Bose-Einstein Condensate (BEC), a state of matter where atoms lose their individual identities and move in perfect unison, like a synchronized dance troupe.

This paper explores what happens when you trap these "super-atoms" in a very specific shape: two concentric rings, like a target with a bullseye and an outer ring, or a donut sitting inside a larger donut.

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

1. The Setup: Two Rings and a Wall

The scientists created a trap using lasers and magnetic fields to hold the atoms in these two rings.

• The Barrier: Between the inner ring and the outer ring, there is an invisible "wall" (a potential barrier).
• The Twist: These atoms are dipolar, meaning they act like tiny magnets. They repel each other sideways but attract each other along their poles. This magnetic personality makes them behave differently than normal atoms.

2. The "Super-Solid" Mystery

Usually, these atoms act like a superfluid (a liquid with zero friction that flows forever without stopping). But under certain conditions, they can become a supersolid.

• The Analogy: Imagine a crowd of people running in a circle.
  • In a Superfluid, everyone runs smoothly and evenly spaced, like a perfectly smooth river.
  • In a Supersolid, the crowd suddenly clumps together into distinct groups (like islands in a river) while still flowing without friction. It's a solid structure that flows like a liquid.

What the paper found:
In their double-ring setup, the atoms naturally prefer to crowd into the outer ring. As the magnetic "personality" of the atoms gets stronger, the outer ring spontaneously turns into this clumpy "supersolid" state, even without anyone spinning the system. The inner ring, however, usually stays smooth and fluid-like.

3. Spinning the Dance Floor

The researchers then started rotating the entire trap, like spinning a record player. This is where things get interesting.

The "Josephson Vortex" (The Bridge Breaker)

When the rings are separated by a strong wall, the atoms in the outer ring start to flow, but the inner ring stays still.

• The Metaphor: Imagine the outer ring is a highway with cars speeding around, and the inner ring is a parking lot with no cars. The "Josephson Vortex" is like a traffic jam or a break in the flow that happens exactly at the gate (the barrier) between the highway and the parking lot.
• The paper calls this a JV1. It's a defect that sits right on the wall between the two rings.

The "Central Vortex" (The Eye of the Storm)

If the wall between the rings is weak (or the atoms can spill over), the whole system can spin together.

• The Metaphor: Imagine a whirlpool forming right in the center of the target. The entire system (both rings) spins around this empty hole in the middle.
• The paper calls this a CV (Central Vortex).

The Unique Discovery: The "Inner Ring" Transformation

This is the paper's biggest surprise. Usually, the inner ring is just a passive bystander. But if the rotation is fast enough and the wall between the rings is weak, the inner ring suddenly wakes up!

• The Metaphor: The inner ring, which was previously a smooth, empty parking lot, suddenly develops its own "islands" of atoms (clumps), just like the outer ring did earlier.
• The New Vortex (JV2): Because these new clumps form in the inner ring, new "traffic jams" (vortices) appear between these clumps. The paper calls these JV2s.
• Why it's special: This is a unique behavior found only in this specific double-ring system. The rotation itself forces the inner ring to become a supersolid, creating a new type of vortex that has never been seen in this configuration before.

4. How Do We See This? (The Interference Pattern)

You can't see these tiny atoms with a regular microscope. So, how do the scientists know what's happening?

• The Analogy: Imagine taking a photo of two overlapping ripples in a pond. Where the ripples meet, they create a complex pattern of light and dark stripes.
• The Experiment: The scientists suddenly turn off the trap (the "walls" disappear). The atoms expand outward like an exploding balloon. As the inner and outer rings expand and crash into each other, they create an interference pattern.
• The Result:
  • If there is a Central Vortex, the pattern looks like concentric circles (like ripples from a stone dropped in water).
  • If there is a Josephson Vortex (JV1), the pattern looks like a spiral (like a galaxy).
  • If there are JV2s (the new discovery), the pattern shows clumpy, wavy structures in the center.

Summary

The paper describes a theoretical experiment where scientists spin a double-ring trap of magnetic atoms. They discovered that:

1. The outer ring naturally wants to become a "clumpy" supersolid.
2. Spinning the system creates different types of "defects" (vortices) depending on how strong the wall between the rings is.
3. Most importantly, spinning fast enough can force the inner ring to become a supersolid too, creating a brand-new type of vortex (JV2) that sits between the clumps in the inner ring.
4. These invisible quantum states can be "seen" by letting the atoms expand and looking at the unique ripple patterns they leave behind.

The paper confirms that these states are real and can be observed in current state-of-the-art experiments, offering a new way to study how matter behaves when it is both solid and fluid at the same time.

Technical Summary

Technical Summary: Josephson Vortices and Persistent Current in a Double-Ring Supersolid System

Problem Statement
This work investigates the theoretical properties of ultra-cold dipolar atoms confined in radially coupled, concentric annular traps (a double-ring geometry). The study focuses on the interplay between the superfluid (SF) and supersolid (SS) phase transitions, persistent currents (PCs), and the nucleation of topological defects (vortices) under rotation. Specifically, the authors address how the long-range dipolar interaction influences the distribution of particles between the inner and outer rings, how spontaneous density modulation characteristic of the SS state evolves in this multi-ring geometry, and how rotation induces distinct vortex configurations. The system is motivated by the unique physics of dipolar Bose-Einstein condensates (BECs), where the SS state arises from a periodic density modulation while maintaining phase coherence, and the potential for Josephson tunneling between rings separated by an azimuthally symmetric barrier.

Methodology
The authors model the system using the extended Gross-Pitaevskii equation (eGPE) for dysprosium-164 atoms ($^{164}$Dy), incorporating contact interactions, long-range dipole-dipole interactions (DDI), and the Lee-Huang-Yang (LHY) correction to account for quantum fluctuations. The trapping potential consists of a toroidal well with a Gaussian barrier centered at radius $r_0$, creating an inner and an outer ring. The barrier strength ($V_B$) and the relative dipolar interaction strength ($\epsilon_{dd} = a_{dd}/a$) are key control parameters.

The study employs two primary numerical approaches:

1. Imaginary-time evolution: Used to determine the ground-state wavefunctions and identify static density profiles and phase transitions.
2. Real-time evolution: Used to explore dynamical behavior and the system's response to rotation.

To analyze rotational properties, the authors calculate the "yrast line" (the lowest energy state for a given angular momentum $L$) by minimizing an energy functional $E(L) = E + C(L - L_0)^2$. They also compute the angular momentum $L$ as a function of rotation frequency $\Omega$ and the superfluid fraction $f_s$. Finally, to propose experimental detection, the authors simulate the expansion of the condensate after switching off the trap to analyze interference patterns.

Key Contributions and Results

• Population Imbalance and Supersolidity: In the non-rotating ground state, the long-range repulsive nature of dipolar interactions in the radial plane causes a population imbalance, with the outer ring consistently hosting a higher particle number than the inner ring. As $\epsilon_{dd}$ increases, the outer ring preferentially undergoes spontaneous density modulation, transitioning into a supersolid state while the inner ring remains largely superfluid. The barrier strength has a weak effect on the superfluid fraction but determines the density overlap between rings.

• Vortex Nucleation in Separated Rings: When the rings are separated by a strong barrier ($V_B \approx 10^5 a_{dd}^2$), rotation induces the formation of Josephson vortices (JV1) located at the junction barrier between the rings. These vortices allow for persistent currents where the angular momentum is carried primarily by the outer ring. At higher rotation frequencies, a central vortex (CV) can coexist with JV1s.

• Vortex Nucleation in Connected Rings: When the barrier is weaker ($V_B \approx 10^4 a_{dd}^2$), allowing density overlap, the system supports central vortices (CV) where the entire system carries angular momentum. Crucially, at high angular momentum, rotation can induce density modulation in the inner ring, transforming it into a supersolid state even if the non-rotating ground state was superfluid.

• Discovery of JV2 (Unique to Double-Ring Dipolar Systems): The authors identify a unique class of vortices, termed JV2, which form at the junctions between localized density sites induced by rotation in the inner ring. Unlike JV1s (which form at the inter-ring barrier) or CVs (at the center), JV2s arise specifically from the interplay between the double-ring geometry and rotation-induced supersolidity. These vortices are distinct to the double-ring dipolar system.

• Phase Diagrams: The study delineates phase diagrams mapping the existence of CVs, JV1s, and JV2s as functions of barrier strength ($V_B$), interaction strength ($\epsilon_{dd}$), and rotation frequency ($\Omega$). A critical barrier strength ($V_{B,c}$) separates regimes dominated by JV1s from those dominated by CVs.

• Detection via Interferometry: The paper proposes that distinct interference patterns emerge upon trap release, allowing for the experimental differentiation of these vortex configurations.

  • CVs produce concentric circular density patterns.
  • JV1s result in a single spiral density pattern due to the phase winding in the outer ring only.
  • JV2s are identified by the persistence of modulated density structures in the inner region, with vortices located between these sites.

Significance and Claims
The paper claims to provide a theoretical framework for understanding the complex interplay between Josephson tunneling and persistent currents in dipolar supersolids within a double-ring geometry. The primary significance lies in the prediction of rotation-induced supersolidity in the inner ring and the consequent formation of JV2 vortices, a phenomenon the authors state is unique to this specific system. The work suggests that these distinct topological defects can be experimentally distinguished using current state-of-the-art interferometric techniques. The authors modestly frame their work as a theoretical investigation that identifies specific vortex configurations and phase behaviors, offering a roadmap for their observation in future experiments with dipolar gases (specifically dysprosium or erbium) in double-ring traps. They do not claim to have experimentally realized these states but rather to have provided the theoretical conditions and detection protocols necessary for their observation.