🔬 mesoscale physics

A thermodynamic uncertainty relation for (hybrid) N--S coherent conductors

This paper establishes a hybrid quantum thermodynamic uncertainty relation for normal-superconducting conductors by demonstrating that, despite complex interference between quasiparticle and Andreev processes, the nonequilibrium excess noise admits a positive representation that preserves a universal dissipation-precision constraint across arbitrary superconducting gaps.

Original authors: Sergi Vidal, Rosa Lopez

Published 2026-06-23

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CC BY 4.0

Original authors: Sergi Vidal, Rosa Lopez

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 busy highway where cars (electrons) are trying to get from one city (a normal metal) to another (a superconductor). In a normal highway, traffic flows smoothly, but sometimes cars get stuck, merge, or crash, creating "noise" or fluctuations in the flow. Scientists have long known a fundamental rule about this: the more energy you waste (dissipation) to keep traffic moving, the more precise and steady that flow can be. This is called the Thermodynamic Uncertainty Relation (TUR). It's like a cosmic speed limit: you can't have a perfectly smooth, high-speed flow without paying a high energy cost.

However, things get weird when you introduce a superconductor. In this special material, electrons don't just drive alone; they pair up into "dance couples" called Cooper pairs. When a single electron hits the superconductor, it doesn't just pass through or bounce back normally. Instead, it performs a magic trick called Andreev reflection: the electron turns into a "hole" (a missing electron) and bounces back, while simultaneously dropping off a whole new pair of electrons into the superconductor's dance floor.

This creates a chaotic mix of two types of traffic:

The Solo Drivers (Quasiparticles): Electrons that just pass through normally.
The Dance Couples (Andreev Reflection): The magic trick where one electron becomes two.

The Problem:
For a long time, scientists tried to apply the standard "traffic rule" (the TUR) to this hybrid highway. But it kept failing. Why? Because the two types of traffic interfere with each other. Sometimes the solo drivers and the dance couples cancel each other out, and sometimes they amplify the chaos. It was like trying to measure the noise of a band where the drummer and the guitarist were playing different songs at the same time; the math got messy, and the old rules didn't add up.

The Breakthrough:
The authors of this paper, Sergi Vidal and Rosa López, decided to break the traffic noise down into three distinct parts to see what was really happening:

The Andreev Noise: The noise caused purely by the dance couples.
The Quasiparticle Noise: The noise caused by the solo drivers.
The Interference Noise: The messy part where the two groups clash.

They discovered something surprising: While the "Interference Noise" could actually be negative (which sounds impossible in normal physics, like having "negative traffic"), when you add it all up, the total noise is always positive. It's like a financial ledger where one account shows a loss, but when you combine it with the profits from the other accounts, the total balance is still positive.

The New Rule (HQTUR):
By proving that the total "shot noise" (the random bumps in the traffic) is always positive, even with the messy interference, they derived a new, universal rule for these hybrid highways. They call it the Hybrid Quantum Thermodynamic Uncertainty Relation (HQTUR).

Think of it this way:

Old Rule: "You can't have a smooth flow without paying a price." (True for normal roads).
The Failure: "This rule breaks on superconductor roads because the traffic is too weird."
The New Rule: "Even on these weird superconductor roads, where electrons are dancing and turning into holes, there is still a strict, unbreakable limit. If you want a precise, steady flow of electricity, you must pay a specific amount of energy cost. No matter how you tune the system, you cannot cheat this trade-off."

Why It Matters:
This paper doesn't just fix a math problem; it gives us a new "law of the universe" for a specific type of advanced electronic device. It tells engineers and physicists that no matter how they design these hybrid superconductor circuits, they cannot escape the fundamental cost of precision. The paper proves that even in the quantum world, where particles can do magic tricks, nature still demands a fair exchange: Precision costs Energy.

In short, the authors found a way to look past the confusing interference of electrons and holes to reveal a hidden, unbreakable balance sheet that governs how energy and precision interact in these futuristic devices.

Problem Statement
Thermodynamic uncertainty relations (TURs) establish fundamental bounds between current fluctuations and entropy production in nonequilibrium systems. While the classical TUR and its quantum generalization (QTUR) for normal metallic conductors are well-established, their application to hybrid Normal–Superconductor (N–S) junctions remains incomplete. In N–S systems, transport is governed by the coexistence of two distinct channels: Andreev reflection (where an incident electron is retroreflected as a hole, transferring a Cooper pair) and quasiparticle transmission. Previous studies have reported deviations from the classical TUR in these systems, and while a "hybrid" quantum TUR (HQTUR) was derived for specific limits (e.g., purely Andreev regimes or low-voltage perturbative schemes), a rigorous, non-perturbative derivation valid for arbitrary bias, temperature, and finite superconducting gaps has been lacking. The central challenge is that the interference between Andreev and quasiparticle amplitudes introduces noise terms that are not positive-definite, complicating the establishment of a universal dissipation-precision bound.

Methodology
The authors employ the Anantram–Datta scattering formalism to model a coherent scatterer (such as a quantum dot) coupled to a normal reservoir and a grounded superconductor. Working within the Nambu (electron–hole) basis, they derive exact expressions for the charge current, energy current, and zero-frequency charge noise.

The core of the methodology involves a rigorous decomposition of the total charge noise ( $S$ ) into three physically distinct components:

Andreev noise ( $S_A$ ): Associated with electron-hole conversion processes.
Quasiparticle noise ( $S_Q$ ): Associated with the transmission of electrons and holes as quasiparticles.
Interference noise ( $S_{Cross}$ ): Arising from the quantum interference between Andreev and quasiparticle amplitudes.

Crucially, the authors demonstrate that while $S_{Cross}$ can attain negative values (preventing a simple additive bound at the component level), the total noise can be reorganized into thermal ( $S_{th}$ ) and shot ( $S_{sh}$ ) contributions. They prove that the total shot noise is non-negative ( $S_{sh} \geq 0$ ) through an exact algebraic identity, despite the negativity of the interference term. This positivity is established by expressing the shot noise as a sum of squared moduli and non-negative transmission factors.

Key Contributions and Results

Noise Decomposition: The paper provides a detailed decomposition of the zero-frequency noise in hybrid N–S conductors, explicitly identifying the roles of Andreev, quasiparticle, and interference terms. Numerical results illustrate that $S_{Cross}$ is negative and significant far from equilibrium, particularly in the crossover region where both transport channels are active.
Proof of Positivity: A key technical contribution is the proof that the total nonequilibrium shot noise remains non-negative. This is achieved by showing that the negative interference term is exactly compensated by the structure of the thermal and shot noise partitioning, ensuring $S_{sh} \geq 0$ .
Derivation of the Hybrid Quantum TUR (HQTUR): Leveraging the positivity of the shot noise and the convexity of the function $\phi(x) = x \text{arsinh}(2ex)$ , the authors derive a rigorous thermodynamic uncertainty relation for hybrid N–S systems:
$\sigma \geq \frac{k_B |I|}{e} \text{arsinh}\left(\frac{2e|I|}{S}\right)$
where $\sigma$ is the entropy production rate, $I$ is the charge current, and $S$ is the total zero-frequency noise.
Validation: The authors numerically test the derived HQTUR against the classical TUR and the standard QTUR across a wide range of bias voltages and superconducting gaps. The results show that while the classical TUR and the QTUR are violated in regimes where Andreev and quasiparticle channels interfere strongly, the HQTUR holds universally for the entire parameter space explored.

Significance
The paper claims to establish the HQTUR as the natural and tight dissipation-precision constraint for hybrid normal–superconducting quantum conductors. By extending the pure-Andreev quantum TUR and the normal-conductor QTUR to the full transport regime where quasiparticle processes and Cooper-pair correlations coexist, the work clarifies how superconducting coherence reshapes current fluctuations. The result demonstrates that despite the complex interplay of transport channels and the presence of negative interference terms in the noise, a universal thermodynamic bound persists, preserving the fundamental trade-off between thermodynamic cost and the precision of nonequilibrium currents in hybrid quantum devices.

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