Laguerre-Gaussian modes: entangled state representation and generalized Wigner transform in quantum optics

TL;DR

This paper introduces a new entangled state representation for Laguerre-Gaussian modes, linking them to eigenstates of quantum operators, and derives their Wigner representation as a generalized transform of Hermite Gaussian modes.

Contribution

It presents a novel entangled state framework for LG modes and derives their Wigner function using Weyl ordering invariance, connecting classical and quantum optical representations.

Findings

LG modes are eigenstates of orbital angular momentum and photon number operators.

The Wigner function of LG modes is obtained as a generalized transform of Hermite Gaussian modes.

LG modes can be generated by a 50:50 beam splitter with a specific phase difference.

Abstract

By introducing a new entangled state representation, we show that the Laguerre-Gaussian (LG) mode is just the wave function of the common eigenvector of the orbital angular momentum and the total photon number operators of 2-d oscillator, which can be generated by 50:50 beam splitter with the phase difference phi=Pi/2{\phi} between the reflected and transmitted fields. Based on this and using the Weyl ordering invariance under similar transforms, the Wigner representation of LG is directly obtained, which can be considered as the generalized Wigner transform of Hermite Gaussian modes.