Direct experimental test of commutation relation via weak value

TL;DR

This paper proposes and experimentally demonstrates a novel method to directly test the canonical commutation relation in quantum mechanics using weak measurements and neutron interferometry.

Contribution

It introduces a new scheme employing weak values to directly verify the commutation relation between two dichotomic observables.

Findings

Successful experimental verification of the commutation relation

Use of weak measurement to access the imaginary part of weak values

Implementation with neutron interferometry and spin-energy degrees of freedom

Abstract

The canonical commutation relation is the hallmark of quantum theory and Heisenberg's uncertainty relation is a direct consequence of it. But despite its fundamental role in quantum theory, surprisingly, its genuine direct experimental test has hitherto not been performed. In this article, we present a novel scheme to directly test the canonical commutation relation between two dichotomic observables, by exploiting the notion of weak measurement. The imaginary part of a suitably formulated weak value enables this direct test. The measurement of the weak value of a path-qubit observable in a neutron interferometer experiment is used to verify the approach. The experiment is realized using a newly developed technique in our neutron interferometric setup where the neutron's spin/energy degree of freedom serves as ancilla.