Quantum probes for quantum wells

TL;DR

This paper investigates optimal quantum measurement strategies for estimating the width of an infinite potential well, revealing that entanglement enhances measurement precision and that time evolution can serve as a resource under certain conditions.

Contribution

It introduces a quantum estimation framework for potential well characterization, demonstrating the benefits of entanglement and dynamical evolution in improving measurement accuracy.

Findings

Position measurement on delocalized particles yields a width-independent QSNR.

Time evolution increases QSNR quadratically with time, aiding precision.

Entangled probes exhibit super-additivity, improving estimation accuracy.

Abstract

We seek for the optimal strategy to infer the width $a$ of an infinite potential wells by performing measurements on the particle(s) contained in the well. In particular, we address quantum estimation theory as the proper framework to formulate the problem and find the optimal quantum measurement, as well as to evaluate the ultimate bounds to precision. Our results show that in a static framework the best strategy is to measure position on a delocalized particle, corresponding to a width-independent quantum signal-to-noise ratio (QSNR), which increases with delocalisation. Upon considering time-evolution inside the well, we find that QSNR increases as $t^2$. On the other hand, it decreases with $a$ and thus time-evolution is a metrological resource only when the width is not too large compared to the available time evolution. Finally, we consider entangled probes placed into the well and observe super-additivity of the QSNR: it is the sum of the single-particle QSNRs, plus a positive definite term, which depends on their preparation and may increase with the number of entangled particles. Overall, entanglement represents a resource for the precise characterization of potential wells.