Supercurrent from the imaginary part of the Andreev levels in non-Hermitian Josephson junctions

This paper investigates a non-Hermitian Josephson junction to identify and propose an experimental protocol for detecting a novel supercurrent contribution arising from the phase derivative of Andreev level broadening, demonstrating that this fingerprint of non-Hermiticity can be observed even in the absence of exceptional points.

The Gist

The Big Picture: A Leaky Boat in a Storm

Imagine a superconducting wire (a "Josephson junction") as a boat trying to sail across a river. The "current" is the water flowing through the boat, carrying energy without resistance. Usually, physicists think of this boat as a perfect, sealed vessel where the water level (energy) is perfectly stable.

However, in the real world, nothing is perfectly sealed. This boat has a hole in the bottom. Water leaks in and out. In physics, this "leakiness" is called being Non-Hermitian. It means the system is "open" to its environment, losing energy (dissipation) just like a real boat losing water.

For a long time, scientists thought that to understand the boat's movement, they only needed to look at the average water level (the real part of the energy). They ignored the fact that the water level was fluctuating or "blurring" because of the leak (the imaginary part).

This paper argues that the leak itself is actually steering the boat.

The Main Discovery: The "Leakage Steering" Effect

The researchers discovered a new way the boat moves. They found that the supercurrent (the flow of electricity) isn't just determined by the height of the energy levels, but also by how fast the width of those levels is changing as you twist a knob (the phase).

• The Analogy: Imagine you are driving a car. Usually, you steer based on the road ahead (the energy levels). But this paper says: "Wait! If your tires are deflating at a specific rate depending on how you turn the wheel, that deflation itself creates a force that pushes the car sideways."
• The Physics: In this quantum boat, the "deflation" is the imaginary part of the energy. The paper shows that if this "leakiness" changes as you change the phase, it creates a brand new type of electric current that no one had successfully measured before.

The Special Zones: "The Ghostly Flatlands"

The paper focuses on specific settings where this effect is easiest to see. They call these Global Zero-Energy States (G-ZES).

• The Analogy: Imagine a roller coaster. Usually, the track goes up and down. But in these special "G-ZES" zones, the track flattens out completely at the bottom. The cars (electrons) aren't moving up or down in energy; they are stuck at zero.
• The Twist: Even though the cars aren't moving up or down, the width of the track (how much they are wobbling or leaking) is changing wildly as you move along the flat section.
• Why it matters: In these flat zones, the "normal" current (from the height of the track) disappears. The only thing pushing the car forward is the "leakage steering" effect. This makes it the perfect place to spot the new phenomenon.

The "Magic Symmetry" Break

The paper also explains why this happens using a concept called Symmetry.

• The Analogy: Think of a perfectly balanced seesaw. If you sit on one side, the other goes up. This is "Time-Reversal Symmetry." If you run the movie backward, the physics looks the same.
• The Break: In these quantum systems, there is a special "shifted" version of the seesaw. When the system is in a "broken" phase, the seesaw is no longer balanced. The "leakiness" (the imaginary part) starts to depend on which way you are facing (the phase).
• The Result: This breaking of symmetry is what allows the "leakage steering" current to exist. The authors found a way to tune the system so that this symmetry is always broken, making the effect visible everywhere, not just in tiny, fragile spots.

How to Catch It in the Lab

The authors propose a "detective game" to prove this exists.

1. Measure the Boat's Speed: Measure the total electric current flowing through the junction (the CPR).
2. Measure the Water Level: Use a special microscope (Andreev spectroscopy) to measure the energy levels of the electrons.
3. Do the Math: Calculate what the current should be if you only looked at the water levels (ignoring the leaks).
4. The Smoking Gun: If the measured current is different from the calculated current, the difference is the "Leakage Steering" current!

They suggest looking for this in a specific setup: a tiny quantum dot connected to superconductors and a magnetic metal, with a magnetic field tilted at a specific angle. This setup creates the "flat track" (G-ZES) where the effect is strongest and easiest to spot.

Why This Is a Big Deal

Previously, scientists thought you had to find a very specific, fragile point called an Exceptional Point (where two energy levels merge) to see non-Hermitian effects. It was like trying to find a needle in a haystack.

This paper says: "You don't need the needle."

You can find this effect in a much broader, more stable range of conditions. It opens the door to seeing "ghostly" quantum effects in everyday electronic devices, proving that the "leakiness" of the quantum world isn't just noise—it's a fundamental part of how electricity flows.

Summary in One Sentence

The paper reveals that in open quantum systems, the way energy "leaks" out can actually push electric current, and by tuning a specific type of quantum junction, we can finally see and measure this hidden force.

Technical Summary

1. Problem Statement

The paper addresses a fundamental gap in the understanding of electronic transport in open quantum systems described by non-Hermitian (NH) Hamiltonians. Specifically, it investigates Josephson junctions (JJs) coupled to normal-metal reservoirs (ferromagnetic leads) where the interaction with the environment leads to complex energy eigenvalues (Andreev quasi-bound states, or quasi-ABS).

• The Core Issue: In standard Hermitian systems, the supercurrent ($J$) is derived from the phase derivative of the real part of the energy levels ($J \propto \partial_\phi \sum \epsilon_A$). In NH systems, eigenvalues are complex ($z_j = \epsilon_j - i\lambda_j$), where the imaginary part ($\lambda_j$) represents energy broadening (dissipation).
• The Open Question: How does the imaginary part of these levels contribute to the supercurrent? Previous studies suggested that the phase dispersion of the imaginary part ($\partial_\phi \lambda_j$) generates a novel current contribution ($J_{Im}$). However, this contribution was often overlooked or predicted to cause unphysical "spikes" in the current-phase relation (CPR) near Exceptional Points (EPs)—spectral points where eigenvalues and eigenvectors coalesce. The paper aims to resolve the physical nature of $J_{Im}$, identify regimes where it is dominant and smooth, and propose an experimental protocol for its detection.

2. Methodology

The authors employ a combination of analytical derivations and numerical simulations based on the Green's Function (GF) formalism.

• System Model: A superconductor–quantum dot–superconductor (SQDFS) Josephson junction coupled to a ferromagnetic (F) metal reservoir. The system is subjected to an external magnetic field ($\vec{B}$) applied to the quantum dot, which is non-collinear with the ferromagnet's magnetization ($\vec{M}_F$).
• Effective Hamiltonian: By tracing out the degrees of freedom of the leads, the authors derive an effective frequency-independent NH Hamiltonian ($\check{H}_{eff}$) for the quantum dot. The eigenvalues of this Hamiltonian are the complex Andreev levels.
• Current Formula: They utilize a specific NH current formula derived in their previous work (Ref. [54]):
  $$J_{ABS}(\phi) = J_{Re} + J_{Im} = -\frac{e}{\pi} \sum_j \left[ (\partial_\phi \epsilon_j) \text{Im}(L_j) - (\partial_\phi \lambda_j) \text{Re}(L_j) \right]$$
  where $J_{Re}$ arises from the dispersion of the real energy part, and $J_{Im}$ arises from the dispersion of the imaginary broadening part.
• Symmetry Analysis: The authors analyze the symmetries of the shifted Hamiltonian ($\check{H}' = \check{H}_{eff} + i\Gamma_N \mathbb{1}$) to understand the origin of the phase dependence in $\lambda_j$. They identify a Time-Reversal-like Symmetry (TRS) that emerges in the resonant regime ($\epsilon_d = 0$).
• Regime Exploration: They systematically vary system parameters (junction asymmetry $\gamma$, spin-dependent dissipation imbalance $\gamma_N$, magnetic field strength $B$, and angle $\theta$) to map out spectral configurations, specifically looking for Global Zero-Energy States (G-ZES) where zero-energy states exist across the entire phase range.

3. Key Contributions

1. Resolution of the $J_{Im}$ Contribution: The paper clarifies that the imaginary part of the levels contributes to the supercurrent via its phase dispersion. It demonstrates that this term is essential for maintaining the continuity of the CPR and preventing unphysical divergences (spikes) predicted by formulas considering only the real part of the energy near EPs.
2. Identification of Optimal Detection Regimes (G-ZES): The authors identify specific parameter regimes, termed Global Zero-Energy States (G-ZES), where the system hosts zero-energy states for all phase biases. In these regimes:
   • The imaginary part of the levels is phase-dispersive everywhere.
   • The $J_{Im}$ contribution is smooth and sizable (up to ~30% of the total current).
   • Crucially, these regimes can be achieved without the presence of Exceptional Points (EPs) in the spectrum, broadening the scope of detectable NH effects.
3. Symmetry-Current Connection: The work establishes a deep link between the phase dispersion of the imaginary part and the breaking of a Time-Reversal-like Symmetry (TRS) in the shifted Hamiltonian.
   • In the "unbroken" TRS phase, $\lambda_j$ is constant, and $J_{Im} = 0$.
   • In the "broken" TRS phase (which occurs between EPs or in G-ZES regimes), $\lambda_j$ becomes phase-dependent, generating $J_{Im}$.
4. Experimental Protocol: The authors propose a concrete detection method:
   • Measure the total CPR ($J_{tot}$) directly.
   • Measure the Andreev spectrum via differential conductance ($dI/dV$) spectroscopy to extract the real energy levels ($\epsilon_j$) and their broadening ($\lambda_j$).
   • Reconstruct the expected current ($J_{Re}$) from the spectroscopy data.
   • A discrepancy between the measured $J_{tot}$ and the reconstructed $J_{Re}$ provides a direct fingerprint of the non-Hermitian $J_{Im}$ contribution.

4. Key Results

• Spectral Engineering: By tuning the junction asymmetry ($\gamma$) and the spin-dissipation imbalance ($\gamma_N$), the authors can control the position of EPs. They show that for highly asymmetric junctions, the EPs can be pushed out of the physical phase interval $[-\pi, \pi]$, resulting in a spectrum with no EPs but with globally extended zero-energy states (G-ZES).
• Smoothness of $J_{Im}$: In the G-ZES regime (specifically the "1 G-ZES" region), the $J_{Im}$ component is smooth across the entire phase range, unlike the singular behavior near EPs. This makes it experimentally accessible.
• Quantitative Impact: In the optimal 1 G-ZES regime (low dissipation, $\Gamma_N \approx 0.5\Gamma$), the $J_{Im}$ contribution can reach ~30% of the maximum supercurrent. The total current at $\phi = \pi/2$ is found to be ~50% larger than the current calculated from the real part alone.
• Off-Resonance Effects: In the off-resonant regime ($\epsilon_d \neq 0$), the TRS is explicitly broken, causing all levels to have phase-dispersive imaginary parts. While this enhances the visibility of $J_{Im}$, it complicates the extraction of $J_{Im}$ from $J_{Re}$ because all levels contribute to both terms. However, the effect is generally larger in this regime.

5. Significance

• Beyond Exceptional Points: Historically, experimental searches for NH physics in condensed matter have focused heavily on detecting EPs. This paper demonstrates that pure non-Hermitian effects (specifically the supercurrent from level broadening) can be robustly detected even in the absence of EPs, significantly expanding the experimental landscape.
• New Transport Fingerprint: The identification of $J_{Im}$ as a distinct, measurable component of the supercurrent provides a new "fingerprint" for non-Hermiticity in open superconducting devices.
• Symmetry-Based Design: By linking the phenomenon to the breaking of a TRS-like symmetry, the paper offers a theoretical framework for engineering NH effects in solid-state systems, potentially guiding the design of devices for topological quantum computing or spintronics.
• Experimental Feasibility: The proposed protocol relies on standard transport measurements (CPR and $dI/dV$ spectroscopy) available in modern nanofabrication labs, making the observation of these effects highly feasible with current technology.

In summary, the paper provides a comprehensive theoretical framework and a practical roadmap for detecting and quantifying the supercurrent contribution arising from the imaginary part of Andreev levels, moving the field of non-Hermitian physics from abstract spectral singularities to measurable transport phenomena in Josephson junctions.