Detecting crossed Andreev reflection in a quantum Hall interferometer with a superconducting beam splitter

This paper proposes using time-domain electron interferometry in a Hong-Ou-Mandel geometry with a superconducting beam splitter to detect and characterize local and crossed Andreev reflection processes by analyzing distinct changes in current cross-correlation signals compared to normal-conducting setups.

The Gist

Imagine you are watching a very fancy, high-speed dance competition. The dancers are electrons (tiny particles of electricity), and the stage is a special material called graphene that is being squeezed by a powerful magnet.

In this world, the electrons don't just wander around randomly; they are forced to march in a single file line along the very edges of the stage, like cars on a one-way highway. This is called the Quantum Hall Effect.

The Setup: The "Splitter"

The scientists in this paper built a special intersection on this highway. Usually, in these experiments, they use a tiny gate (a "beam splitter") that randomly sends an electron either left or right.

But in this new experiment, they replaced that normal gate with a thin strip of superconductor (a material that conducts electricity with zero resistance). This is the "Superconducting Beam Splitter."

The Dance: The Hong-Ou-Mandel (HOM) Experiment

To test how this new gate works, they sent two identical electrons onto the stage at almost the exact same time.

• The Goal: They wanted to see what happens when two identical dancers arrive at the split at the same time.
• The Old Rule (Normal Gate): In the normal world, because electrons are "antisocial" (a rule called the Pauli Exclusion Principle), they hate being in the same place. If two identical electrons hit a normal splitter at the same time, they will always split up: one goes left, and one goes right. They never both go left or both go right. This creates a "dip" in the data, like a valley in a mountain range.

The Twist: The Superconductor's Magic Trick

Here is where the superconductor changes the rules. When an electron hits a superconductor, something magical happens called Andreev Reflection.

Think of an electron as a person wearing a red shirt. When they hit the superconductor, the superconductor doesn't just send them away; it grabs a "hole" (which acts like a person wearing a blue shirt) from the crowd and swaps it with the electron.

• Local Andreev Reflection: The electron hits the superconductor and bounces back as a "hole" (red shirt becomes blue shirt).
• Crossed Andreev Reflection: This is the star of the show. An electron comes from the left side, hits the superconductor, and instead of bouncing back, it turns into a "hole" that travels to the right side. It's like a dancer jumping from the left stage, changing costumes mid-air, and landing on the right stage.

The Result: The Dance Floor Flips

The scientists wanted to see if they could spot this costume change (the "Crossed Andreev Reflection") just by watching the traffic at the end of the highway.

1. Without the Superconductor: The two electrons split up (one left, one right). The data shows a "valley" (a dip).
2. With the Superconductor: Because of the magic costume swap, the electrons sometimes end up doing the opposite of what they usually do. Instead of splitting up, they might both end up on the same side, or their behavior flips completely.

The Big Discovery:
When the scientists looked at the data, the "valley" they expected to see flipped upside down and became a "peak."

It's as if the dance competition judges suddenly decided that the rule "dancers must split up" was now "dancers must stay together." This flip in the signal is the "smoking gun" that proves the electrons are interacting with the superconductor in this special, magical way.

Why Does This Matter?

This isn't just about watching electrons dance.

• Detecting the Invisible: This method gives scientists a new, very clear way to "see" these tricky quantum processes that are usually very hard to measure.
• Future Computers: The paper mentions that these interactions are the first step toward creating topological quantum computers. These are super-powerful computers that don't crash easily because they use these special "dancers" (particles) that are protected by the laws of physics.

In Summary:
The paper shows that by replacing a normal traffic light with a superconducting one, the behavior of electron traffic flips completely. By watching this flip, scientists can prove that a special quantum magic trick (Crossed Andreev Reflection) is happening, paving the way for the next generation of super-computers.

Technical Summary

1. Problem Statement

The paper addresses the challenge of experimentally detecting and characterizing Andreev reflection processes (specifically local and crossed Andreev reflection) at the interface between a superconductor (SC) and a Quantum Hall (QH) edge state.

• Context: While SC/QH interfaces are crucial for topological quantum computing (e.g., hosting Majorana or parafermion bound states), current experimental characterization relies primarily on measuring "negative downstream resistance" in four-terminal geometries.
• Gap: There is a lack of methods to distinguish between normal electron transmission/reflection and Andreev processes (where an electron converts to a hole) in a time-domain setting. Specifically, the two-particle interference signatures of these processes in a Hong-Ou-Mandel (HOM) geometry have not been fully explored.
• Goal: To propose and theoretically demonstrate a method to identify crossed Andreev reflection (CAR) by analyzing the charge cross-correlations in a time-domain electron interferometer where the beam splitter is a superconductor.

2. Methodology

The authors employ a combination of scattering theory and numerical tight-binding simulations on a graphene honeycomb lattice.

• System Geometry:

  • A Quantum Hall (QH) bar (graphene) is subjected to a perpendicular magnetic field.
  • Two chiral edge states (from source contacts 1 and 2) propagate toward a central scattering region acting as a beam splitter.
  • Beam Splitters: The study compares two cases:
    1. Normal Insulator: A gapped region acting as a standard electronic beam splitter (reference).
    2. Superconductor: A thin superconducting wire crossing the QH bar, acting as a superconducting beam splitter.
  • Configuration: The study focuses on a "p-S-n" junction configuration, where the chemical potentials of the left and right QH regions have opposite signs ($\mu_L = -\mu_R$). This alignment favors the coupling of electron-like states on one side to hole-like states on the other, enhancing crossed Andreev processes.

• Theoretical Framework:

  • Input States: Identical single-particle wavepackets (modeled as Gaussian or Lorentzian "levitons") are injected from the two source contacts with a time delay $\tau$.
  • Formalism: The authors use the Bogoliubov-de Gennes (BdG) formalism to describe the scattering matrix ($S$), which relates incoming electron/hole operators to outgoing ones. This allows for the inclusion of particle-hole mixing (Andreev processes).
  • Observable: The primary observable is the charge covariance (cross-correlation) between the drain contacts (3 and 4): $\langle \hat{Q}_3 \hat{Q}_4 \rangle(\tau)$.
  • Simulation: Numerical calculations are performed using the Kwant package to compute the scattering matrix for specific parameters (magnetic field, superconducting gap $\Delta_S$, and region length $L_{SC}$).

3. Key Contributions

• Extension of HOM Interferometry to Superconductors: The paper extends the standard electronic HOM interferometer concept to include a superconducting beam splitter, moving beyond single-particle coherence studies to the two-particle sector.
• Identification of a New Signature: The authors identify a qualitative inversion of the HOM signal (a change from a peak to a dip, or vice versa) as a robust signature of crossed Andreev reflection.
• Analytical and Numerical Validation: They provide an analytical derivation of the charge correlator for both normal and superconducting cases and validate it against numerical tight-binding simulations of a graphene QH bar.
• Parameter Optimization: The study identifies the optimal working regime where normal and Andreev processes coexist with comparable weights (specifically when the superconducting region length $L_{SC}$ is comparable to the coherence length $\xi_0$).

4. Key Results

• Normal Insulator Case (Benchmark):

  • For a normal beam splitter, the system reproduces the standard HOM effect: fermionic anti-bunching.
  • The charge covariance $\langle \hat{Q}_3 \hat{Q}_4 \rangle$ shows a positive peak at zero time delay ($\tau \approx 0$). This indicates that two indistinguishable electrons tend to exit the same port, reducing the probability of them exiting different ports (anti-bunching).
  • The peak height is slightly reduced due to energy-dependent scattering phases but remains positive.

• Superconducting Case (Main Finding):

  • When the beam splitter is a superconductor with a "p-S-n" configuration, the HOM signal undergoes a dramatic change.
  • Signal Inversion: As the superconducting gap $\Delta_S$ increases, the HOM peak inverts to a dip (becomes negative) at $\tau \approx 0$.
  • Physical Mechanism: This inversion is driven by crossed Andreev reflection (CAR). In CAR, an electron from one side is converted into a hole on the other side. The unitarity of the extended BdG scattering matrix (Eq. 19 in the paper) dictates that if CAR amplitudes are non-zero, the interference term changes sign.
  • Regime Dependence:
    • For small $\Delta_S$, the signal is negative (dip).
    • As $\Delta_S$ increases further, the signal oscillates (inverting back to a peak and then dip) before eventually vanishing as transmission is exponentially suppressed by the gap.
    • The most distinct signature (a clear downward dip) occurs when $L_{SC} \approx 2\xi_0$, where normal and Andreev processes compete.

• Robustness: The authors argue that this sign inversion is a direct consequence of the non-zero Andreev amplitudes and is distinct from effects caused by Coulomb interactions or thermal broadening, which typically reduce visibility but do not invert the sign.

5. Significance

• Direct Detection of CAR: The paper proposes a direct, time-domain method to detect crossed Andreev reflection, which is notoriously difficult to isolate from local Andreev reflection and normal transmission in DC transport measurements.
• Characterization Tool: The inversion of the HOM peak serves as a "smoking gun" for the presence of superconducting correlations and specific Andreev processes in QH systems.
• Pathway to Topological Quantum Computing: Since crossed Andreev reflection is a fundamental mechanism for generating non-local entangled states and potentially non-Abelian anyons (like Majorana modes) in hybrid SC/QH systems, this interferometric technique offers a vital diagnostic tool for future topological quantum computing architectures.
• Experimental Feasibility: The proposed setup uses parameters (graphene, magnetic fields, superconducting gaps) that are consistent with recent experimental capabilities, suggesting this signature could be observed in current or near-future experiments.

In summary, the paper demonstrates that time-domain electron interferometry in a superconducting QH setup transforms the standard fermionic anti-bunching signature into a sign-inverted HOM dip, providing a powerful and unambiguous method to detect and characterize crossed Andreev reflection.