Revisiting the adiabatic limit in ballistic multiterminal Josephson junctions

Motivated by recent experiments on multiterminal Josephson junctions, this paper proposes a model for the intermediate bias regime that treats the central normal metal as a continuum with nonequilibrium electronic populations, predicting characteristic voltage scales that govern mesoscopic critical current oscillations and bridge interpretations across quartet, topological, and Floquet physics.

The Gist

The Big Picture: A Superconducting Traffic Jam

Imagine a city made of superconductors (materials where electricity flows with zero resistance). In this city, there are special "traffic lights" called Josephson Junctions. Usually, these junctions have two roads meeting. But scientists are now building junctions with three, four, or even more roads meeting at a central square (a 2D metal sheet).

When electricity flows through these multi-road intersections, something magical happens: electrons don't just travel alone; they team up in groups.

• Pairs: Usually, electrons travel in pairs (Cooper pairs).
• Quartets: In these special multi-road junctions, four electrons can link up to form a "quartet" and travel together.

The paper investigates what happens when you turn up the voltage (speed up the traffic) in these large, complex junctions.

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The Problem: The "Adiabatic" vs. "Floquet" Dilemma

To understand the paper, we need to look at two ways scientists usually describe this traffic:

1. The Slow Motion Camera (Adiabatic Limit): Imagine the traffic lights change so slowly that the cars (electrons) never get confused. They adjust perfectly to every change. This works great for very small, slow systems.
2. The Strobe Light (Floquet Theory): Imagine the traffic lights are flashing rapidly. The cars are constantly jumping between different states, creating a complex, vibrating pattern. This works for small, quantum dots.

The Conflict:
The researchers looked at large-scale devices (big central squares with many lanes). They found that the old "Slow Motion" model was too simple, and the "Strobe Light" model was too complicated.

• In a huge city with thousands of lanes, the quantum "magic" of electrons jumping between lanes (called Landau-Zener tunneling) becomes diluted. It's like trying to hear a whisper in a stadium full of people; the signal gets lost in the noise.
• However, the electrons are still in a state of non-equilibrium. They are being pushed by a voltage, so they aren't sitting still. They are "excited."

The Solution: The "Adiabatic Approximation with a Twist"

The authors propose a new way to model this. Think of it like this:

• The Stage (The Metal): They treat the central metal sheet as a continuous, smooth ocean (a continuum) rather than a grid of individual stepping stones.
• The Actors (The Electrons): They assume the electrons are "excited" and moving because of the voltage (non-equilibrium), but they move in a way that is still smooth and predictable (adiabatic).
• The Analogy: Imagine a dance floor.
  • Old Model 1: Everyone moves in perfect, slow synchronization.
  • Old Model 2: Everyone is jumping randomly to a strobe light.
  • New Model: The music is loud and fast (voltage), so the crowd is energetic and moving in a specific, non-equilibrium pattern. But the dancers themselves are moving smoothly across the floor without tripping over each other (no quantum tunneling chaos).

The Key Discovery: The "Dilution" Effect

The paper makes a fascinating mathematical observation about collisions.

Imagine two groups of dancers (energy levels) moving toward each other on the dance floor. In a small room (1D), they might crash into each other and swap partners perfectly (quantum tunneling).

But in a huge stadium (2D with many lanes/channels):

• There are so many lanes that only a tiny fraction of the dancers actually collide and swap partners.
• The paper calculates that the chance of this "quantum swap" happening is 1 divided by the number of lanes.
• The Metaphor: If you have 1,000 lanes, the chance of a specific quantum collision is 1 in 1,000. It's so rare that for all practical purposes, you can ignore the "quantum swapping" and just treat the dancers as flowing smoothly.

The Result: The "Inversion" Switch

The most exciting part of the paper is how this new model explains a real-world experiment (done by Harvard and Penn State).

They observed a phenomenon called "Inversion."

• Non-Inversion: Usually, the supercurrent (the flow of electricity) is strongest when there is no magnetic field.
• Inversion: Sometimes, as you increase the voltage, the current becomes stronger when there is a magnetic field, and weaker when there isn't. It's like the traffic flow suddenly prefers the red light over the green light.

How the paper explains it:
The researchers found that the "voltage" changes the energy of the electrons (the electrochemical potential). This shift interacts with the magnetic field in a way that creates a checkerboard pattern of interference.

• At low voltages, the waves cancel out at certain points.
• As you turn up the voltage, the waves shift. Suddenly, the point where they used to cancel out becomes a point where they boost each other.
• This causes the current to flip-flop between "stronger with no field" and "stronger with a field" as you adjust the voltage.

Why This Matters

This paper provides a rulebook for engineers building these complex quantum devices.

1. Simplification: It tells us we don't need to do impossible calculations for every single electron in a large device. We can use a simpler model that accounts for the "voltage excitement" without getting bogged down in quantum chaos.
2. Prediction: It predicts exactly when these "inversion" switches will happen, which is crucial for designing future quantum computers and sensors.
3. Unification: It bridges the gap between the "slow" world of large devices and the "fast" world of quantum theory, showing that they are actually two sides of the same coin.

Summary in One Sentence

The paper discovers that in large, multi-road superconducting junctions, the "quantum magic" of electrons jumping lanes becomes so rare that we can ignore it, allowing us to use a simpler model that perfectly explains how voltage and magnetic fields can flip the direction of supercurrent flow.

Technical Summary

1. Problem Statement

Multiterminal Josephson junctions (MJJs) have emerged as a platform for exploring complex quantum phenomena, including Cooper pair quartets, topological states, and Floquet physics. However, a theoretical gap exists in understanding the behavior of large-scale, ballistic 2D-metal-based MJJs under finite bias voltage.

• The Conflict: Standard theoretical approaches often rely on either:
  1. Strict Adiabatic Limit ($V \to 0$): Assumes phase variables evolve slowly, neglecting non-equilibrium electronic populations.
  2. Full Floquet Theory: Necessary for finite voltage but computationally intractable for large 2D systems with continuous spectra and many transverse channels.
• The Specific Challenge: Recent experiments (e.g., by the Harvard and Penn State groups) on 2D MJJs show critical current oscillations and "inversion" phenomena (where the critical current at half-flux quantum exceeds that at zero flux) that depend on bias voltage. Existing models fail to simultaneously explain the magnetic field dependence and the voltage dependence in the intermediate regime where the electrochemical potential becomes comparable to the 1D energy level spacing.

2. Methodology

The authors propose a hybrid theoretical framework that bridges the gap between the adiabatic limit and full non-equilibrium Floquet theory.

• Physical Argument (Dimensionality Analysis):

  • The authors analyze the collision of Floquet–Kulik quartet superlevels in a voltage-biased junction.
  • In a 1D system, all levels are quantum mechanically correlated.
  • In a 2D ballistic metal with $M$ transverse channels, the authors demonstrate that only $M$ pairs of colliding levels are quantum mechanically correlated (due to the 1D nature of individual Andreev tubes), while $M^2 - M$ pairs are uncorrelated.
  • Key Insight: As the device size increases ($M \to \infty$), the relative weight of quantum mechanical Landau-Zener tunneling (which causes level repulsion) scales as $1/M$ and vanishes. This justifies neglecting Landau-Zener tunneling in large-scale 2D devices.

• The "Enhanced Adiabatic" Model:

  • Phase Dynamics: Treated adiabatically (evaluated at $V=0^+$), effectively neglecting Landau-Zener transitions.
  • Electronic Populations: Retained as non-equilibrium. The central 2D metal is treated as a continuum with a shifted electrochemical potential $\mu_N$.
  • Coupling: The model introduces a coupling between the supercurrent and the non-equilibrium electronic populations via the electroflux effect. The shift in electrochemical potential $\delta\mu_N$ depends on both the bias voltage $V$ and the magnetic flux $\Phi$.

• Calculations:

  • The authors use a tight-binding Hamiltonian for the 2D metal and BCS Hamiltonians for the superconducting leads.
  • They employ Keldysh Green's function techniques to calculate current-phase relations.
  • They derive analytical expressions for both short-junction (discrete Andreev bound states) and long-junction (continuous spectrum) limits, focusing on quartet and split-quartet processes.

3. Key Contributions

1. Justification of the Adiabatic Approximation in 2D: The paper provides a rigorous argument that for large-scale 2D MJJs, quantum coherence between Floquet levels is "diluted" by the number of transverse channels. This validates using an adiabatic phase treatment even at finite bias, provided non-equilibrium populations are included.
2. The Electroflux Effect: The authors identify and model a mechanism where the magnetic flux $\Phi$ modulates the electrochemical potential $\mu_N$ of the central normal metal. This arises from the interference of phase-sensitive Andreev reflections between grounded leads.
3. Unified Framework for Quartet Signals: The model successfully unifies the description of:
   • Quartet Critical Currents: Arising from the interference of odd-in-phase (short-junction) and even-in-phase (long-junction/continuum) components.
   • Voltage Dependence: How the critical current evolves as the bias voltage increases.
   • Magnetic Flux Dependence: How the critical current oscillates with flux.

4. Results

• Critical Current Oscillations: The model predicts that the critical current $I_{q,c}$ exhibits mesoscopic oscillations as a function of the electrochemical potential shift $\delta\mu_N$ (which is proportional to bias voltage $V$).
• Noninversion vs. Inversion:
  • Noninversion: $I_{q,c}(\Phi=0) > I_{q,c}(\Phi=\Phi_0/2)$. This is the standard expectation (constructive interference at zero flux).
  • Inversion: $I_{q,c}(\Phi=0) < I_{q,c}(\Phi=\Phi_0/2)$. This is a non-standard behavior observed in experiments.
• The Crossover Mechanism: The paper demonstrates that the electroflux effect ($s_0$) shifts the $\Phi=0$ and $\Phi=\Phi_0/2$ curves in opposite directions as a function of $\delta\mu_N$.
  • As the bias voltage increases, $\delta\mu_N$ changes, causing the system to cross over between noninversion and inversion regimes.
  • This explains the experimental observation of voltage-tunable inversion in 4-terminal MJJs.
• Numerical Validation: Numerical simulations of the interference between odd-in-phase (split quartets) and even-in-phase (continuum quartets) terms reproduce the "checkerboard" patterns and the specific crossover points observed in recent experiments (e.g., Huang et al., Nat. Comm. 2022).

5. Significance

• Interpretation of Experiments: The model provides a direct theoretical explanation for recent experimental data on 2D MJJs, specifically the voltage-dependent inversion of the quartet critical current, which previous zero-dimensional or strict adiabatic models could not fully capture.
• Scalability: By establishing that Landau-Zener tunneling is negligible in large 2D systems, the paper simplifies the theoretical landscape for future MJJ research, allowing researchers to focus on non-equilibrium population effects rather than complex multi-level Floquet dynamics.
• New Physics: It highlights the role of the electroflux effect (flux-dependent electrochemical potential) as a critical parameter in mesoscopic superconducting devices, linking dissipative quasiparticle transport with non-dissipative supercurrents.
• Topological and Floquet Context: The work clarifies the regime where Floquet theory applies (discrete levels) versus where continuum approximations hold, offering a roadmap for interpreting experiments in the "intermediate" regime relevant to current device fabrication.

In summary, the paper resolves a long-standing discrepancy between theory and experiment in large-scale MJJs by proposing that non-equilibrium electronic populations, rather than quantum level repulsion, dominate the voltage-dependent behavior of quartet currents in 2D ballistic systems.