Multi-slit interferometry and commuting functions of position and momentum

TL;DR

This paper analyzes a modified double-pinhole experiment showing how it prepares quantum states that are approximate joint eigenstates of certain commuting functions of position and momentum, enabling near-independent measurement of these observables.

Contribution

It provides a theoretical framework linking multi-slit interferometry with commuting functions of position and momentum, and explicitly constructs states localized in both spaces.

Findings

The experiment can prepare states close to joint eigenstates of position and momentum functions.

It is possible to measure momentum distribution with minimal impact on position.

Changing experimental settings reveals mutual disturbance between position and momentum measurements.

Abstract

In a recent, modified double-pinhole diffraction experiment the existence of an interference pattern was established indirectly along with a near-perfect imaging of the double pinhole. Our theoretical analysis shows that the experiment constitutes a preparation of a quantum state that is, to a good approximation, a joint eigenstate of commuting functions of position and momentum. Gaining information about the momentum distribution by means of the particular experimental setup is thus possible with negligible impact on the position distribution. Furthermore, we construct explicitly a class of states simultaneously localised on periodic sets in position and momentum space, which are therefore eigenstates of the observables being measured jointly (to a good approximation) in multi-slit interferometry. Finally, we show that with an appropriate change of settings the experiment demonstrates the mutual disturbance of position and momentum measurements.