Quantum Dynamics and Collapse-and-Revival Phenomena in the Dunkl Anharmonic Oscillator

TL;DR

This paper exactly solves the Dunkl anharmonic oscillator Hamiltonian using algebraic methods, explores its quantum dynamics including collapse and revival phenomena, and shows how Dunkl deformation affects these phenomena.

Contribution

It introduces an algebraic solution for the Dunkl anharmonic oscillator and analyzes how Dunkl deformation influences quantum collapse, revival, and squeezing effects.

Findings

Exact energy spectrum depends on parity and Dunkl parameter μ.

Dunkl deformation modulates fractional revivals and enables perfect state reconstructions.

Dunkl parameter μ induces interference-based squeezing near t ≈ π.

Abstract

We study the Dunkl anharmonic oscillator (Kerr medium) Hamiltonian from an algebraic approach of the $SU(1,1)$ group. In order to obtain the exact energy spectrum of this problem, we write its Hamiltonian in terms of the Dunkl creation and annihilation operators, which close the $su(1,1)$ Lie algebra. This allows us to exactly solve this Hamiltonian and obtain its parity-dependent energy spectrum. Then, we investigate the quantum dynamics of the system, particularly the collapse and revival phenomena, by using an initial state given by a superposition of even and odd Dunkl coherent states. We compute the field quadrature and the survival probability, showing that the Dunkl parameter $\mu$ modulates the fractional revivals and produces perfect state reconstructions at half-periods for specific deformation values. We analyze the quadrature variance to show that the Dunkl deformation generates interference-induced squeezed states around $t \approx \pi$. The standard Kerr medium dynamics are exactly recovered in the limit $\mu \rightarrow 0$.