Generalized Susskind-Glogower coherent states

TL;DR

This paper introduces generalized Susskind-Glogower coherent states using extended Bessel functions, explores their mathematical properties including identity resolution, and demonstrates their connection to Lie algebra representations and optical properties.

Contribution

It develops new families of coherent states based on extended Bessel functions, providing their quantization maps and Lie algebra representations, advancing the mathematical framework of coherent states.

Findings

New Susskind-Glogower-I and II coherent states are introduced.

These states realize representations of $rak{su}(1,1)$ and $rak{su}(2)$ Lie algebras.

Optical properties of the new states are analyzed and compared.

Abstract

Susskind-Glogower coherent states, whose Fock expansion coefficients include Bessel functions, have recently attracted considerable attention for their optical properties. Nevertheless, identity resolution is still an open question, which is an essential mathematical property that defines an overcomplete basis in the Fock space and allows a coherent state quantization map. In this regard, the modified Susskind-Glogower coherent states have been introduced as an alternative family of states that resolve the identity resolution. In the present manuscript, the quantization map related to the modified Susskind-Glogower coherent states is exploited, which naturally leads to a particular representation of the $\mathfrak{su}(1,1)$ Lie algebra in its discrete series. The latter provides evidence about further generalizations of coherent states, built from the Susskind-Glogower ones by extending the indexes of the Bessel functions of the first kind and, alternatively, by employing the modified Bessel functions of the second kind. In this form, the new families of Susskind-Glogower-I and Susskind-Glogower-II coherent states are introduced. The corresponding quantization maps are constructed so that they lead to general representations of elements of the $\mathfrak{su}(1,1)$ and $\mathfrak{su}(2)$ Lie algebras as generators of the SU$(1,1)$ and SU$(2)$ unitary irreducible representations respectively. For completeness, the optical properties related to the new families of coherent states are explored and compared with respect to some well-known optical states.