# Generalized Grassmann variables for quantum kit (k-level) systems and Barut-Girardello coherent states for su(r+1) algebras

## TL;DR

This paper develops a framework using generalized Grassmann variables to construct and analyze Barut-Girardello coherent states for $su(r+1)$ algebras, relevant for multi-level quantum systems.

## Contribution

It introduces a generalized Weyl-Heisenberg algebra ${
mf A}(r)$ and constructs explicit Barut-Girardello coherent states for $su(r+1)$, expanding the mathematical tools for quantum multi-level systems.

## Key findings

- Explicit construction of $su(r+1)$ Barut-Girardello coherent states.
- Introduction of generalized Grassmann variables for Fock--Bargmann space.
- Discussion of over--completion properties of these states.

## Abstract

This paper concerns the construction of $su(r+1)$ Barut--Girardello coherent states in term of generalized Grassmann variables. We first introduce a generalized Weyl-Heisenberg algebra ${\cal A}(r)$ ($r \geq 1$) generated by $r$ pairs of creation and annihilation operators. This algebra provides a useful framework to describe qubit and qukit ($k$-level) systems. It includes the usual Weyl-Heisenberg and $su(2)$ algebras. We investigate the corresponding Fock representation space. The generalized Grassmann variables are introduced as variables spanning the Fock--Bargmann space associated with the algebra ${\cal A}(r)$. The Barut--Girardello coherent states for $su(r+1)$ algebras are explicitly derived and their over--completion properties are discussed.