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

• URL: http://arxiv.org/abs/2602.16743v1
• Date: Wed, 18 Feb 2026 02:46:00 GMT
• Title: A Generalization of the Parametric Amplifier with Dunkl Derivative: Spectral and Statistical Properties
• Authors: D. Ojeda Guillén, R. D. Mota, J. C. Vega,
• Abstract summary: We study the parametric amplifier Hamiltonian within the framework of the Dunkl formalism.<n>We show that our results reduce to the standard parametric amplifier case in the limit of vanishing Dunkl deformation parameter.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $μ$, 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.

Related papers

• Transmutation based Quantum Simulation for Non-unitary Dynamics [35.35971148847751]
  We present a quantum algorithm for simulating dissipative diffusion dynamics generated by positive semidefinite operators of the form $A=Ldagger L$.<n>Our main tool is the Kannai transform, which represents the diffusion semigroup $e-TA$ as a Gaussian-weighted superposition of unitary wave propagators.
  arXiv  Detail & Related papers  (2026-01-07T05:47:22Z)
• SU(1,1) coherent states for the Dunkl- Klein-Gordon equation in its canonical form [0.0]
  We construct Perelomov coherent states for the Dunkl-Klein-Gordon equation in its canonical form.<n>Our analysis is restricted to the even-parity sector and to the regime where the curvature constant $R$ is much smaller than the system's kinetic energy.
  arXiv  Detail & Related papers  (2025-07-15T03:27:19Z)
• A Statistical Analysis for Supervised Deep Learning with Exponential Families for Intrinsically Low-dimensional Data [32.98264375121064]
  We consider supervised deep learning when the given explanatory variable is distributed according to an exponential family.<n>Under the assumption of an upper-bounded density of the explanatory variables, we characterize the rate of convergence as $tildemathcalOleft( dfrac2lfloorbetarfloor(beta + d)2beta + dn-frac22beta + dn-frac22beta + dn-frac22beta + dn-
  arXiv  Detail & Related papers  (2024-12-13T01:15:17Z)
• A Statistical Analysis of Deep Federated Learning for Intrinsically Low-dimensional Data [31.52603443208588]
  This paper delves into the generalization properties of deep federated regression within a two-stage sampling model.<n>Our findings reveal that the intrinsic dimension, characterized by the entropic dimension, plays a pivotal role in determining the convergence rates for deep learners.
  arXiv  Detail & Related papers  (2024-10-28T01:36:25Z)
• Nonasymptotic Analysis of Stochastic Gradient Descent with the Richardson-Romberg Extrapolation [22.652143194356864]
  We address the problem of solving strongly convex and smooth problems using gradient descent (SGD) with a constant step size.<n>We provide an expansion of the mean-squared error of the resulting estimator with respect to the number of iterations $n$.<n>Our analysis relies on the properties of the SGDs viewed as a time-homogeneous Markov chain.
  arXiv  Detail & Related papers  (2024-10-07T15:02:48Z)
• Quantum coherence measures for generalized Gaussian wave packets under a Lorentz boost [0.10713888959520207]
  We consider a single particle, spin-momentum entangled state and measure the effect of relativistic boost on quantum coherence. The coherence of the wave function as measured by the boosted observer is studied as a function of the momentum and the boost parameter. A more prominent loss of coherence due to relativistic boost is observed for a single particle electron than that of a neutron.
  arXiv  Detail & Related papers  (2024-07-09T07:02:45Z)
• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• Berry phase and the Mandel parameter of the non-degenerate parametric amplifier [0.0]
  We write the Hamiltonian of the non-degenerate parametric amplifier problem. By assuming that our Hamiltonian is an explicit function of time we calculate its Mandel Berry phase.
  arXiv  Detail & Related papers  (2023-12-22T23:20:11Z)
• The Tempered Hilbert Simplex Distance and Its Application To Non-linear Embeddings of TEMs [36.135201624191026]
  We introduce three different parameterizations of finite discrete TEMs via Legendre functions of the negative tempered entropy function. Similar to the Hilbert geometry, the tempered Hilbert distance is characterized as a $t$-symmetrization of the oriented tempered Funk distance.
  arXiv  Detail & Related papers  (2023-11-22T15:24:29Z)
• On the $\mathcal{P}\mathcal{T}$-symmetric parametric amplifier [0.0]
  General time-dependent PT-symmetric parametric oscillators for unbroken parity and time reversal regimes are studied theoretically. We have demostrated the time variation of the Wigner distribution for the system consisting of two spatially separated ground state of the TD-parametric amplifier.
  arXiv  Detail & Related papers  (2023-05-20T19:45:22Z)
• Theory of free fermions under random projective measurements [43.04146484262759]
  We develop an analytical approach to the study of one-dimensional free fermions subject to random projective measurements of local site occupation numbers. We derive a non-linear sigma model (NLSM) as an effective field theory of the problem.
  arXiv  Detail & Related papers  (2023-04-06T15:19:33Z)
• Entanglement of inhomogeneous free fermions on hyperplane lattices [0.0]
  We introduce an ingeneous model of free fermions on a $(D-1)$-dimensional lattice with $D(D-1)/2$ continuous parameters. We compute the entanglement entropy numerically for $D=2,3,4$, for a wide range of parameters.
  arXiv  Detail & Related papers  (2022-06-13T22:40:39Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.