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

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 $μ$, 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.

💡 Research Summary

In this work the authors extend the well‑known degenerate parametric amplifier to a Dunkl‑deformed setting, thereby introducing a parity‑dependent deformation parameter μ into the quantum‑optical model. The construction begins with the Dunkl derivative Dμ = d/dx + μ x⁻¹(1 − R), where R is the Z₂ reflection operator. Using Dμ they define Dunkl‑deformed annihilation and creation operators aμ and aμ†. These operators obey modified commutation relations that involve R, yet the anticommutation {R, aμ}=0 guarantees that the bilinear combinations Kμ⁺ = (aμ†)²/2, Kμ⁻ = aμ²/2 and Kμ⁰ = (aμ†aμ + aμaμ†)/4 satisfy the exact su(1,1) Lie algebra. Consequently the Dunkl‑deformed system retains the algebraic backbone that underlies many quantum‑optical Hamiltonians.

The Dunkl number operator Nμ = aμ†aμ has eigenvalues