A Generalization of the Parametric Amplifier with Dunkl Derivative: Spectral and Statistical Properties

TL;DR

This paper extends the parametric amplifier model using Dunkl calculus, providing exact spectral solutions and analyzing the impact of Dunkl deformation on quantum statistical properties like photon bunching.

Contribution

It introduces Dunkl operators into the parametric amplifier framework, derives exact spectra, and explores how Dunkl deformation influences quantum statistical behaviors.

Findings

Exact energy spectrum obtained via algebraic methods.

Mandel parameter remains unaffected by Dunkl deformation.

Correlation function depends explicitly on the Dunkl parameter.

Abstract

We study the parametric amplifier Hamiltonian within the framework of the Dunkl formalism. We introduce the Dunkl creation and annihilation operators and show that their quadratic combinations generate an $su(1,1)$ Lie algebra. The spectral problem is solved exactly using two algebraic methods: the $su(1,1)$ tilting transformation and the generalized Bogoliubov transformation. The exact energy spectrum and the corresponding eigenfunctions are obtained in terms of the Dunkl number coherent states. Furthermore, we compute the Mandel $Q$ parameter and the second-order correlation function $g^{(2)}(0)$ to analyze the statistical properties of the Dunkl squeezed states. We show that, for the squeezed vacuum, the Mandel parameter remains independent of the Dunkl deformation, whereas the correlation function exhibits an explicit dependence on the Dunkl parameter $\mu$, which modifies the photon bunching effects. Finally, we show that our results reduce to the standard parametric amplifier case in the limit of vanishing Dunkl deformation parameter.