🔬 mesoscale physics

Probing Cooper pair momentum by quasiparticle steering with planar Josephson junctions

This paper demonstrates that ballistic planar Josephson junctions enable a kinematic probe of Cooper pair momentum by steering Andreev bound states into an adjacent normal region at a phase-controlled angle that significantly exceeds conventional momentum scales, offering a distinct alternative to energy-shift-based detection methods.

Original authors: Isidora Araya Day, Anton R. Akhmerov, Antonio R. L. Manesco

Published 2026-07-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Isidora Araya Day, Anton R. Akhmerov, Antonio R. L. Manesco

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a superconductor as a bustling dance floor where electrons pair up to form "Cooper pairs," gliding in perfect unison. Usually, these pairs move so slowly compared to the frantic speed of the individual dancers (the Fermi momentum) that trying to measure their collective drift is like trying to spot a snail's crawl in a hurricane. For years, scientists have tried to catch this drift by listening for a change in the pitch of the music (a Doppler shift in energy), but the signal is often too faint to hear clearly.

This paper proposes a different way to see the dance: instead of listening to the pitch, let's watch the direction the dancers jump off the floor.

The Main Discovery: A Phase-Controlled Jet
The authors, using detailed computer simulations, show that if you build a specific kind of "dance floor" called a planar Josephson junction—a short bridge between two superconductors—you can make the electrons shoot out into a neighboring normal region at a very specific, controllable angle.

Think of the junction as a narrow hallway with a special, shifting wall (the phase difference). Inside this hallway, the electron pairs bounce back and forth, picking up momentum with every bounce. When they finally reach the end of the hall and exit into the open space, they don't just walk straight out. Instead, they are ejected like a water hose that has been twisted. The angle of this "electron jet" isn't random; it is controlled by the phase difference (the twist) applied to the superconductors.

Why This Angle is a Big Deal
Here is the clever part. In the past, scientists expected the exit angle to be tiny, scaling with the ratio of the superconducting gap to the chemical potential ( $\Delta/\mu$ ). It's like expecting a gentle nudge.

However, this paper demonstrates that the angle is actually much larger, scaling with the square root of that ratio ( $\sqrt{\Delta/\mu}$ ).

The Analogy: Imagine the old method was like a single tap on a shoulder, barely moving a person. This new method is like a person running down a hallway, bouncing off walls that are moving in sync with them, gaining speed and direction with every bounce until they launch out of the door at a sharp angle. Because of this "repeated bouncing" effect, the angle is significantly bigger than anyone thought possible within the standard rules of the game.

What the Paper Rules Out
The authors are careful to clarify what this is not.

It is not just a simple Doppler shift where the energy changes but the direction stays mostly the same.
It is not caused by magnetic fields bending the path. The paper calculates that the magnetic field generated by the current itself would bend the path by less than $10^{-4}$ radians over a 1-micron distance—a tiny, negligible wiggle compared to the massive angle caused by the phase control.
It is not a result of the electrons hitting a wall and bouncing back (normal reflection). The setup relies on the electrons passing through cleanly into the normal region.

How Sure Are We?
The paper does not claim to have measured this in a real lab experiment yet. Instead, the authors have built a rigorous mathematical model and run high-fidelity simulations to prove the effect exists.

They used a "tight-binding" model (a digital grid of atoms) and a "continuum" model (smooth math) and found they agreed perfectly.
They show that even when the superconducting gap is small compared to the chemical potential (a condition called the Andreev approximation, where $\Delta/\mu \ll 1$ ), this large angle effect still holds up.
The paper suggests that this effect is "sizeable" enough to be seen with existing tools.

How to Catch It in the Real World
The authors propose two ways to actually see this electron jet in a real device, though they haven't done it yet:

Quantum Point Contacts: Imagine placing a series of tiny, adjustable gates (like turnstiles) in the path of the exiting electrons. By measuring how much current flows through each gate as you change the phase, you could map out the angle of the jet.
Scanning Gate Microscopy: This is like using a super-sensitive, invisible finger to scan the area and "see" where the electron current is flowing, creating a direct image of the jet.

The Numbers
The paper notes that for real-world materials like InAs–Al devices, the ratio $\Delta/\mu$ is typically between $10^{-3}$ and $10^{-2}$ . This results in an ejection angle of just a few degrees. While that sounds small, it is huge in the world of quantum physics and is large enough to be detected by the methods mentioned above.

In short, the paper suggests that by tuning the phase of a superconductor, we can steer quasiparticles out of a junction at a sharp, predictable angle, offering a new, kinematic way to measure the momentum of the superconducting condensate that is much easier to spot than the old energy-shift methods.

Technical Summary: Probing Cooper Pair Momentum by Quasiparticle Steering with Planar Josephson Junctions

Problem Statement
The momentum of a Cooper pair in a superconductor, defined by the phase gradient of the order parameter ( $\hbar q = \hbar \nabla \phi$ ), is typically small compared to the Fermi momentum ( $\hbar k_F$ ). Consequently, measuring this momentum directly is challenging. Existing probes, such as tunneling spectroscopy and nonlocal conductance measurements, detect the condensate momentum indirectly through the Doppler energy shift of Andreev bound states (ABS). However, because the Cooper pair wavevector is limited by the inverse coherence length ( $\xi^{-1} = \Delta/\hbar v_F$ ), the ratio $\xi^{-1}/k_F \sim \Delta/\mu$ is a small parameter. This makes the Doppler shift a subtle effect that is difficult to observe kinematically through the direction of quasiparticle motion, particularly within the standard Andreev approximation regime ( $\Delta \ll \mu$ ).

Methodology
The authors investigate the kinematic detection of condensate momentum by studying the scattering of propagating Andreev bound states at the interface between a short, ballistic planar Josephson junction (SNS) and a semi-infinite normal region.

Theoretical Framework: The system is modeled using a Bogoliubov-de Gennes (BdG) Hamiltonian with a phase difference $\phi$ applied across the junction. The authors employ a perturbative expansion beyond the standard Andreev approximation to derive the transverse momentum of the ABS.
Scaling Analysis: By rescaling variables, the authors demonstrate that the transverse wavevector of the ABS scales as $\sqrt{\Delta/a}$ (where $a = \hbar^2/2m$ ), leading to an angular scale $\theta \sim \sqrt{\Delta/\mu}$ . This is parametrically larger than the conventional Cooper pair momentum scale $\Delta/\mu$ .
Scattering Calculation: The study solves the continuum scattering problem at the SNS-normal interface. The authors match the propagating Andreev modes (and evanescent modes) in the junction to pure electron modes in the normal region. They calculate the transmitted electron field and determine the ejection angle from the ratio of transverse to longitudinal electron currents ( $j_y/j_x$ ).
Numerical Verification: The analytical continuum results are cross-verified using tight-binding simulations via the Kwant package, ensuring the findings hold even when finite-bandwidth effects are considered.

Key Contributions and Results

Phase-Controlled Ejection Angle: The central finding is that subgap quasiparticles ejected from a planar Josephson junction into a normal region emerge at a finite, phase-controlled angle $\Theta$ .
Enhanced Scaling: The ejection angle scales as $\Theta \sim \sqrt{\Delta/\mu}$ . This scaling is significantly larger than the $\Delta/\mu$ scale associated with the Doppler shift of single reflection events. The enhancement arises because the Andreev bound states acquire momentum through repeated Andreev reflections from the phase-biased condensate, rather than a single scattering event.
Kinematic Probe: Unlike previous methods that rely on energy shifts, this work establishes a "kinematic probe" where the signal manifests as a momentum-space deflection (directional change) of the emitted quasiparticles.
Numerical Confirmation: The tight-binding simulations confirm that the continuum prediction remains valid even at $\Delta/\mu = 0.1$ , demonstrating the robustness of the effect.
Experimental Feasibility: The paper proposes two experimental detection schemes:
- Quantum Point Contacts (QPCs): Measuring nonlocal conductance across a series of QPCs to map the angular distribution of the current.
- Scanning Gate Microscopy (SGM): Directly imaging the ejected quasiparticle current density.
  The authors note that for epitaxial InAs-Al devices, the expected ejection angle is on the order of a few degrees, which is within the resolution of these techniques. They also suggest that depletion of the normal region could further amplify the angle by reducing the Fermi momentum.

Significance
The paper claims to establish a new mechanism for detecting Cooper pair momentum that is distinct from and more sensitive than existing Doppler-shift-based probes. By leveraging the unique momentum structure of Andreev bands in ballistic planar junctions, the authors demonstrate that the condensate momentum can be transferred to quasiparticles in a way that results in a macroscopically observable deflection angle. This effect persists even in the limit where $\Delta \ll \mu$ , making it accessible to current experimental capabilities in electron optics. The work provides a theoretical foundation for using directional quasiparticle ejection as a tool to probe the dynamics of superconducting condensates.

Technical Summary: Probing Cooper Pair Momentum by Quasiparticle Steering with Planar Josephson Junctions

More like this