Generalized Delta Functions and Their Use in Quantum Optics

TL;DR

This paper introduces a generalized delta function with complex arguments, revealing its distinct properties and applying it to derive a simplified form of the P-function for Schrödinger cat states, enhancing understanding in quantum optics.

Contribution

It presents a novel generalization of the delta function for complex arguments and demonstrates its application in deriving quantum optical functions.

Findings

Generalized delta functions behave like poles in the complex plane.

Derived a simple form of the P-function for Schrödinger cat states.

Insights into the diagonal P-function's ability to describe off-diagonal density matrix elements.

Abstract

The Dirac delta function is widely used in many areas of physics and mathematics. Here we consider the generalization of a Dirac delta function to allow the use of complex arguments. We show that the properties of a generalized delta function are very different from those of a Dirac delta function and that they behave more like a pole in the complex plane. We use the generalized delta function to derive the Glauber-Sudarshan P-function for a Schr\"odinger cat state in a surprisingly simple form. Aside from their potential applications in classical electromagnetism and quantum optics, these results provide insight into the ability of the diagonal P-function to describe density operators with off-diagonal elements.