Quantum thermodynamic uncertainty relation and macroscopic superconducting coherence

TL;DR

This paper explores how macroscopic superconducting coherence influences thermodynamic uncertainty relations in hybrid normal-superconducting devices, establishing a new bound and linking coherence to fluctuations.

Contribution

It derives a hybrid quantum TUR for N-S junctions, showing deviations relate to superconducting pair amplitude and coherence effects.

Findings

Deviations from normal TUR are governed by superconducting coherence.

Introducing dephasing suppresses coherence and restores the TUR bound.

The derived TUR is never violated and relates to the normal quantum bound.

Abstract

Stability and efficiency are mutually exclusive in a thermodynamic process, e.g. in a thermal machine. Any effort to reduce the fluctuations of a certain output quantity is necessarily accompanied by an increase of entropy production, therefore lowering its efficiency. This interplay is beautifully captured by the so called Thermodynamic Uncertainty Relations (TURs) which set a lower bound on the relative uncertainty of a current for a given rate of entropy production. Their status in hybrid normal-superconducting (N-S) devices has remained unsettled. We show that, in the subgap regime, departures from the normal quantum TUR are governed by {\it macroscopic} superconducting coherence quantified by the pair amplitude, and that introducing a dephasing probe suppresses this coherence and restores the bound. We further derive a hybrid quantum TUR that is general for two-terminal N-S junctions in the Andreev regime: the inequality is never violated, is saturated only at vanishing current, and is related to the normal quantum bound under the replacement (e to 2e). For N-S quantum dot and Cooper-pair-splitter systems we compute current and noise and show that deviations from the normal bound track the pair amplitude on the central region. The results establish a direct link between superconducting macroscopic coherence and nonequilibrium fluctuations and supply a general bound for the Andreev regime.