Perfect quantum protractors

TL;DR

This paper introduces the concept of perfect quantum protractors, states that generate three orthogonal bases under rotations, and demonstrates their utility in quantum metrology through theoretical analysis and an experimental demonstration with rubidium-87 atoms.

Contribution

It defines perfect quantum protractors, analyzes their existence for different angular momentum systems, and demonstrates their application in optimal rotation angle estimation.

Findings

Perfect quantum protractors maximize angular momentum uncertainty measures.

They do not exist for certain spin values but do for others, with numerical evidence for some cases.

Experimentally demonstrated with rubidium-87 atoms for spin-1 system.

Abstract

In this paper we introduce and investigate the concept of a perfect quantum protractor, a pure quantum state $|\psi\rangle\in\mathcal{H}$ that generates three different orthogonal bases of $\mathcal{H}$ under rotations around each of the three perpendicular axes. Such states can be understood as pure states of maximal uncertainty with regards to the three components of the angular momentum operator, as we prove that they maximise various entropic and variance-based measures of such uncertainty. We argue that perfect quantum protractors can only exist for systems with a well-defined total angular momentum $j$, and we prove that they do not exist for $j\in\{1/2,2,5/2\}$, but they do exist for $j\in\{1,3/2,3\}$ (with numerical evidence for their existence when $j=7/2$). We also explain that perfect quantum protractors form an optimal resource for a metrological task of estimating the angle of rotation around (or the strength of magnetic field along) one of the three perpendicular axes, when the axis is not $\textit{a priori}$ known. Finally, we demonstrate this metrological utility by performing an experiment with warm atomic vapours of rubidium-87, where we prepare a perfect quantum protractor for a spin-1 system, let it precess around $x$, $y$ or $z$ axis, and then employ it to optimally estimate the rotation angle.