Electromagnetic field expectations as measures of photon localization

TL;DR

This paper introduces a physically meaningful measure of photon localization based on electromagnetic field expectations, demonstrating that photons can be localized arbitrarily closely without a lower bound on localization scale.

Contribution

It proposes a new measure of photon localization using electromagnetic field expectations and shows photons can be localized arbitrarily tightly according to this measure.

Findings

Field expectations match previously proposed complex amplitude measures

Gaussian localization observed in one test state with well-resolved momentum

Almost exponential and scale-invariant localization in a symmetric momentum distribution

Abstract

The questions of whether a photon can be localized in an arbitrarily small volume and what is the allowable strength of that localization (the decrease with distance of the functional form) are questions of current interest. We propose a measure of localization for the single photon that is the expectation values of the electromagnetic field strength components in a coherent wavepacket state of mean photon number unity. As such, we deal with real quantities that have a physical meaning rather than complex amplitudes. It is seen that the real parts of complex amplitudes proposed previously as measures of localization are equal to our field expectations. With this measure, we examine two test states. The first has a well-resolved momentum. The field expectations show Gaussian (quadratic exponential) localization in all directions, although the localization length scale is much larger than the mean wavelength. For the other test state, with a spherically symmetric momentum distribution, we find almost exponential localization in all directions at t = 0. The profiles are scale invariant, so choosing the momentum width very large would make the localization length arbitrarily small. We conclude that there is no lower bound on the localization length scale of a photon as determined by this measure.