Polynomial Heisenberg algebras, multiphoton coherent states and geometric phases

TL;DR

This paper constructs polynomial Heisenberg algebras via the harmonic oscillator, develops multiphoton coherent states, and analyzes their quantum properties, uncertainty relations, Wigner functions, and geometric phases, revealing their intrinsic quantum and cyclic nature.

Contribution

It introduces a method to realize polynomial Heisenberg algebras and constructs multiphoton coherent states with detailed analysis of their quantum features and geometric phases.

Findings

States are intrinsically quantum and cyclic.

States exhibit specific uncertainty relations and Wigner distributions.

Geometric phases are explicitly calculated.

Abstract

In this paper we will realize the polynomial Heisenberg algebras through the harmonic oscillator. We are going to construct then the Barut-Girardello coherent states, which coincide with the so-called multiphoton coherent states, and we will analyze the corresponding Heisenberg uncertainty relation and Wigner distribution function for some particular cases. We will show that these states are intrinsically quantum and cyclic, with a period being a fraction of the oscillator period. The associated geometric phases will be as well evaluated.