Quantum theory of structured monochromatic light

TL;DR

This paper develops a quantum framework for structured monochromatic light, especially orbital angular momentum states, revealing their non-orthogonality but valid eigenstate properties, aiding quantum information applications.

Contribution

It provides a first-principles quantization of beam-like solutions, clarifying the quantum degrees of freedom and operator properties of structured light.

Findings

Photon states are not orthogonal despite being eigenstates of number and Hamiltonian.

Beam-photon operators do not satisfy canonical commutation relations.

The explicit operator representation aids quantum information processing.

Abstract

Applications that envisage utilizing the orbital angular momentum (OAM) at the single photon level assume that the OAM degrees of freedom that the photons inherit from the classical wave solutions are orthogonal. To test this critical assumption, we quantize the beam-like solutions of the vector Helmholtz equation from first principles to delineate its elementary quantum mechanical degrees of freedom. We show that although the beam-photon operators do not in general satisfy the canonical commutation relations, implying that the photon states they create are not orthogonal, the states are nevertheless bona fide eigenstates of the number and Hamiltonian operators. The explicit representation for the photon operators presented in this work forms a natural basis to study light-matter interactions and quantum information processing at the single photon level.