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

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.

17 pages

References in corpus (31)

• Displaced and Squeezed Number States
• The Dunkl oscillator in the plane II : representations of the symmetry algebra
• The Dunkl oscillator in the plane I : superintegrability, separated wavefunctions and overlap coefficients
• Coupled harmonic oscillators and their quantum entanglement
• The Dunkl oscillator in three dimensions
• Exact Solutions of the 2D Dunkl--Klein--Gordon Equation: The Coulomb Potential and the Klein--Gordon Oscillator
• $SU(1,1)$ solution for the Dunkl oscillator in two dimensions and its coherent states
• Exact Solution of the Relativistic Dunkl Oscillator in $(2+1)$ Dimensions
• Dunkl-Klein-Gordon equation in three-dimensions: The Klein-Gordon oscillator and Coulomb Potential
• Thermal Properties of Relativistic Dunkl Oscillators
• Algebraic approach for the one-dimensional Dirac-Dunkl oscillator
• $SU(1,1)$ solution for the Dunkl-Coulomb problem in two dimensions and its coherent states
• On the Path Integral Formulation of Wigner-Dunkl Quantum Mechanics
• Relativistic Solutions of Generalized-Dunkl Harmonic and Anharmonic Oscillators
• Dunkl-Pauli Equation in the Presence of a Magnetic Field
• The condensation of ideal Bose gas in a gravitational field in the framework of Dunkl-statistic
• Rationally-extended Dunkl oscillator on the line
• The SU(1,1) Perelomov number coherent states and the non-degenerate parametric amplifier
• Dunkl-Schrödinger equation with time-dependent harmonic oscillator potential
• Generalized Rabi models: diagonalization in the spin subspace and differential operators of Dunkl type
• Time-dependent Mandel Q parameter analysis for a hexagonal boron nitride single photon source
• Dunkl-Schrodinger Equation in Higher Dimension
• Time-Dependent Dunkl-Schrödinger Equation with an Angular-Dependent Potential
• The Condensation of Ideal Dunkl-Bose Gas in Power-Law Traps
• Matrix diagonalization and exact solution of the k-photon Jaynes-Cummings model
• Dunkl-Klein-Gordon Equation in Higher Dimensions
• Bounding the Wigner Deformation Parameter in Harmonically Trapped Bose Gases
• Berry phase and the Mandel parameter of the non-degenerate parametric amplifier
• Algebraic approach and Berry phase of a Hamiltonian with a general $SU(1,1)$ symmetry
• $Sp(4,R)$ algebraic approach of the most general Hamiltonian of a two-level system in two-dimensional geometry
• $SU(1,1)\times SU(2)$ approach and the Mandel parameter to the Hamiltonian of two oscillators with weak coupling