Coherent states on a circle: the Higgs-like approach

TL;DR

This paper applies a Higgs-like approach to analyze the quantum behavior of a harmonic oscillator constrained on a circle, deriving its spectrum and constructing coherent states with nonclassical properties.

Contribution

It introduces a novel method to quantize and analyze a harmonic oscillator on a circular constraint using shape invariance and f-deformed algebra, leading to new insights into its quantum states.

Findings

Coherent states exhibit squeezing and sub-Poissonian statistics.

Spectrum derived using shape-invariant Hamiltonian form.

Nonclassical features persist even with small curvature.

Abstract

In this paper, the Higgs-like approach is used to analyze the quantum dynamics of a harmonic oscillator constrained on a circle. We obtain the Hamiltonian of this system as a function of the Cartesian coordinate of the tangent line through the gnomonic projection and then quantize it in the standard way. We then recast the Hamiltonian in a shape-invariant form and derive the spectrum energy of the confined harmonic oscillator on the circle. With help of the f-deformed oscillator algebra, we construct the coherent states on the circle and investigate their quantum statistical properties. We find that such states show nonclassical features like squeezing and sub-Poissonian statistics even in small curvatures of the circle.