Optimal current-based sensing of phonon temperature using a finite reservoir

TL;DR

This paper investigates how finite reservoirs influence phonon temperature sensing in nanoscale systems, proposing optimal current-based thermometry strategies and analyzing their precision limits.

Contribution

It introduces a thermodynamic framework for finite reservoirs in nanoscale thermometry, comparing three current-based measurement strategies and optimizing their precision.

Findings

Monitoring quanta exchanged yields optimal precision in the long time limit.

Fisher information captures the finite reservoir's impact on sensitivity.

Tuning gate voltage enhances measurement precision within each strategy.

Abstract

In realistic nanoscale transport set-ups, electron-phonon coupling leads to the exchange of heat between phonon baths and electronic reservoirs with finite heat capacities. Such exchange affects the finite reservoir's temperature. However, this sensitivity of the finite reservoir temperature to the exchange of heat with the finite reservoir has remained unexplored for thermometry. Here, we fill this gap by combining current metrology techniques with a thermodynamic framework encompassing finite reservoirs. We focus on an experimentally realizable set-up with a quantum dot coupled to a finite reservoir and consider two distinct current-based strategies in the long time limit, namely monitoring quanta exchanged between the quantum dot and finite reservoir and the measurement of the total current flowing from the quantum dot into an infinite reservoir. A third strategy involves measurements of the quantum dot occupation. For a large but finite reservoir, we show that the Fisher information for all three strategies captures the finite reservoir's contribution to sensitivity through common factors. We also demonstrate that monitoring quanta exchanged between the system and finite reservoir in the long time limit achieves optimal precision. Finally, we provide an optimization analysis that explores how maximal precision can be achieved within each of the current-based strategies by tuning the gate voltage.