Algebraic properties of Rogers-Szego functions: I. Applications in quantum optics

TL;DR

This paper explores algebraic properties of Rogers-Szego functions within q-special functions, revealing their role in quantum optics, especially in describing coherent and phase states, and analyzing uncertainty relations.

Contribution

It introduces Rogers-Szego functions in a quantum optics context and connects them to eigenfunctions of coherent and phase states, advancing understanding of q-deformed states.

Findings

Eigenfunctions of coherent and phase states are expressed as expansions of Rogers-Szego functions.

Uncertainty relations for cosine, sine, and number operators are analyzed, supporting features of q-deformed states.

The algebraic framework clarifies the role of Rogers-Szego functions in quantum optical systems.

Abstract

By means of a well-established algebraic framework, Rogers-Szego functions associated with a circular geometry in the complex plane are introduced in the context of q-special functions, and their properties are discussed in details. The eigenfunctions related to the coherent and phase states emerge from this formalism as infinite expansions of Rogers-Szego functions, the coefficients being determined through proper eigenvalue equations in each situation. Furthermore, a complementary study on the Robertson-Schrodinger and symmetrical uncertainty relations for the cosine, sine and nondeformed number operators is also conducted, corroborating, in this way, certain features of q-deformed coherent states.