Locally-acting mirror Hamiltonians

TL;DR

This paper develops a position-space quantisation of the electromagnetic field to model local interactions with optical elements, enabling the construction of locally-acting Hamiltonians for semi-transparent mirrors.

Contribution

It introduces a position-space quantisation method that accounts for both positive and negative frequency solutions, allowing for locally-acting Hamiltonians in quantum optics.

Findings

Constructed annihilation operators for localised field excitations.

Developed locally-acting interaction Hamiltonians for semi-transparent mirrors.

Facilitated modelling of local optical interactions in quantum field theory.

Abstract

Photons, i.e. the basic energy quanta of monochromatic waves, are highly non-localised and occupy all available space in one dimension. This non-local property can complicate the modelling of the quantised electromagnetic field in the presence of optical elements that are local objects. Therefore, in this paper, we take an alternative approach and quantise the electromagnetic field in position space. Taking into account the negative- {\em and} the positive-frequency solutions of Maxwell's equations, we construct annihilation operators for highly-localised field excitations with bosonic commutator relations. These provide natural building blocks of wave packets of light and enable us to construct locally-acting interaction Hamiltonians for two-sided semi-transparent mirrors.