The good, the bad and the ugly coherent states through polynomial Heisenberg algebras

TL;DR

This paper constructs and analyzes three distinct types of coherent states—good, bad, and ugly—derived from polynomial Heisenberg algebras related to the harmonic oscillator, and explores their physical properties.

Contribution

It introduces a novel classification of coherent states based on polynomial Heisenberg algebras and provides explicit constructions and property analyses.

Findings

Explicit solutions to Painleve IV equation are obtained.

Three new sets of eigenstates are constructed and characterized.

Physical properties of the coherent states are analyzed.

Abstract

Second degree polynomial Heisenberg algebras are realized through the harmonic oscillator Hamiltonian, together with two deformed ladder operators chosen as the third powers of the standard annihilation and creation operators. The corresponding solutions to the Painleve IV equation are easily found. Moreover, three different sets of eigenstates of the deformed annihilation operator are constructed, called the good, the bad and the ugly coherent states. Some physical properties of such states will be as well studied.