Sheffer Polynomials and the s-ordering of Exponential Boson Operators

TL;DR

This paper derives explicit s-ordered forms of boson operator products using Sheffer polynomials, providing a unified framework that interpolates between different orderings in quantum optics.

Contribution

It introduces a novel application of Sheffer polynomial sequences to explicitly compute s-orderings of boson operators, extending the family to interpolate between known orderings.

Findings

Explicit s-ordered expressions derived using Sheffer polynomials.

Extension of the polynomial family to interpolate between normal and anti-normal orderings.

Provides a unified mathematical framework for different operator orderings in quantum optics.

Abstract

The s-ordered form of any product of single-mode boson creation and annihilation operators, containing only a single annihilator, is computed explicitly. The s-ordering concept originated in quantum optics, but subsumes normal, symmetric (Weyl), and anti-normal ordering for any two operators satisfying a canonical commutation relation. Because the s-ordering map can be viewed as producing a function of a complex variable, its inverse is a quantization map that takes such "classical" functions to quantum operators. The explicit s-ordered expressions are derived with the aid of a parametric family of Sheffer polynomial sequences (or equivalently a parametric exponential Riordan array of polynomial coefficients), called the Hsu-Shiue family. To yield orderings interpolating between normal and anti-normal, this family must be extended.