Quantum vortex channels as Josephson junctions

Problem
In quantum gases, weak links—localized constrictions or barriers between superfluid reservoirs—are essential for studying quantum transport and Josephson dynamics. Traditionally, these links are realized using externally imposed optical potentials (e.g., laser barriers) or, in superfluids, microapertures. While effective, these methods require external structuring. The paper addresses the possibility of creating "self-induced" weak links that arise naturally from internal interactions within the system, specifically within rotating binary Bose-Einstein condensates (BECs). The central challenge is to demonstrate that quantized vortices can act as reconfigurable, interaction-controlled weak links without external optical engineering, and to characterize the transport regimes (hydrodynamic vs. tunneling) and circuit properties of these vortex-mediated channels.

Methodology
The authors investigate a binary dipolar-nondipolar mixture (specifically $^{162}\text{Dy}$ and a nondipolar component) confined in a three-dimensional infinite tube geometry with harmonic transverse confinement. The system is modeled using coupled dipolar Gross-Pitaevskii equations (GPEs) at zero temperature.

1. System Setup: The system is tuned into an immiscible regime where the two components phase-separate. A quantized vortex is imprinted in one component (either the dipolar or nondipolar one), creating a hollow core that acts as a channel for the other component to flow through.
2. Simulation Protocol: To induce a steady supercurrent, a linear phase gradient is imprinted on the component filling the vortex core. The coupled GPEs are evolved in imaginary time to relax the system into a stationary DC state carrying a quantized current.
3. Analysis: The authors calculate the Current-Phase Relation (CPR) by varying the total phase difference across the system. They analyze the effective potential barrier created by quantum pressure within the vortex channel, defined by the radial confinement width.
4. Modeling: The numerical results are compared against circuit models. For short junctions, a Deaver-Pierce model (a kinetic inductance in series with an ideal Josephson junction) is used. For long junctions, where the vortex core structure is modified by long-range dipolar interactions, an extended circuit model incorporating a second junction and an additional inductance is developed.

Key Contributions

• Self-Induced Weak Links: The paper demonstrates that rotation-induced vortices in binary condensates naturally form hollow channels that act as weak links for the non-vortex component, eliminating the need for external optical barriers.
• Tunable Barrier Mechanism: The authors identify a novel barrier mechanism where the "height" of the barrier is determined by the quantum pressure associated with the vortex channel width. This width, and consequently the barrier height, is tunable via interspecies scattering length ($a_{12}$) and dipolar interactions.
• Regime Crossover: The study maps the crossover from a hydrodynamic transport regime (nearly linear CPR) to a Josephson tunneling regime (sinusoidal CPR) as the interspecies interaction strength increases, effectively narrowing the vortex channel.
• Double-Junction Formation: In the long-junction configuration (vortex in the nondipolar component), long-range dipolar interactions reshape the vortex core, creating a local density maximum that splits the channel into two coupled Josephson junctions in series.
• Circuit Modeling: The authors successfully quantitatively capture the DC CPRs for both single and double-junction configurations using extended circuit models, validating the Deaver-Pierce framework for atomtronic circuits.

Results

• Short Vortex Junction: When the vortex resides in the dipolar component, magnetostriction compresses the domain, creating a short channel. As $a_{12}$ increases, the channel width ($w$) decreases relative to the healing length ($\xi_2$).
  • At low $a_{12}$ ($125 a_0$), the barrier is low ($V_b < \mu$), and the CPR is nearly linear, indicating hydrodynamic flow.
  • At high $a_{12}$ ($170 a_0$), the barrier exceeds the chemical potential, the density is strongly depleted, and the CPR becomes sinusoidal, characteristic of Josephson tunneling.
  • The circuit model (inductor + Josephson junction) accurately reproduces these transitions without fitting the inductance.
• Long Vortex Junction: When the vortex resides in the nondipolar component, the channel is elongated.
  • At high $a_{12}$ ($220 a_0$), the system behaves as a single tunneling junction with a sinusoidal CPR.
  • At lower $a_{12}$ ($130-150 a_0$), dipolar interactions induce a local density maximum in the center of the vortex core. This creates a "double junction" structure: two barriers separated by a small central reservoir.
  • The CPRs in this regime exhibit features distinct from single junctions, such as stable solutions beyond $\pi$ and zero-current states at $2\pi$. An extended circuit model (two junctions in series with an additional inductance) is required to quantitatively match the numerical data.

Significance
The paper establishes vortices as versatile, reconfigurable Josephson elements in superfluids. By demonstrating that weak links can be self-induced and tuned via interaction parameters (scattering length and dipolar strength), the work provides a new platform for exploring Josephson dynamics in atomtronic circuits. The ability to transition between hydrodynamic and tunneling regimes, and to engineer complex structures like double junctions purely through internal interactions, offers a pathway to creating atomtronic circuits without the need for complex external optical structuring. The quantitative agreement with circuit models suggests these vortex-mediated links are suitable for engineering multijunction atomtronic circuits in binary dipolar condensates.