Countable basis for free electromagnetic fields

TL;DR

This paper introduces a countable, Hilbert space-compatible basis for free electromagnetic fields, enabling more rigorous and practical analysis of light-matter interactions with quantized energy states.

Contribution

It develops a novel countable basis of polychromatic electromagnetic fields as eigenstates of commuting operators, addressing the divergence issues of monochromatic expansions.

Findings

Basis vectors are in the Hilbert space, ensuring mathematical rigor.

The basis is countable and isomorphic to , facilitating practical computations.

Three types of basis (regular, incoming, outgoing) are defined with smoothness properties.

Abstract

Polychromatic electromagnetic fields are typically expanded as integrals over monochromatic fields, such as plane waves, multipolar fields, or Bessel beams. However, monochromatic fields do not belong to the Hilbert space of free Maxwell fields, since their norms diverge. Moreover, the continuous frequency integrals involved in such expansions complicate the treatment of light--matter interactions via the scattering operator. Here, we identify and study a polychromatic basis for free Maxwell fields whose basis vectors belong to the Hilbert space. These vectors are defined as simultaneous eigenstates of four commuting operators with integer eigenvalues. As a consequence, the basis set is countable, and the Hilbert space is separable and isomorphic to $\ell^2$, the Hilbert space of square-summable sequences. Each basis vector represents a polychromatic single-photon wave with quantized energy and a wavelet--like temporal dependence. Three versions of this basis are defined: Regular, incoming, and outgoing. The fields of the regular basis are smooth in both space and time. The incoming and outgoing fields are likewise smooth, except at the spatial origin. These results support and motivate the use of countable bases for both the theoretical description and the practical computation of light--matter interactions.