Fresnel-transform's quantum correspondence and quantum optical ABCD Law

TL;DR

This paper establishes a quantum optical analogue of the classical Fresnel transform through the Fresnel operator, deriving the quantum ABCD law and illustrating its application via the damping oscillator's evolution.

Contribution

It introduces the Fresnel operator as a quantum counterpart to the classical Fresnel transform and derives the quantum ABCD law using operator methods and coherent states.

Findings

Fresnel operator's multiplication rule leads to the quantum ABCD law.

The quantum ABCD law is demonstrated through the damping oscillator's evolution.

Explicit connection between ray-transfer matrices and quantum operators is established.

Abstract

Corresponding to Fresnel transform there exists a unitary operator in quantum optics theory, which could be named Fresnel operator (FO). We show that the multiplication rule of FO naturally leads to the quantum optical ABCD law. The canonical operator methods as mapping of ray-transfer ABCD matrix is explicitly shown by FO's normally ordered expansion through the coherent state representation and the technique of integration within an ordered product of operators. We show that time evolution of the damping oscillator embodies the quantum optical ABCD law.