Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2604.22945 · Annals of Physics

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

D. Ojeda-Guillén, R. D. Mota, M. Salazar-Ramírez

quant-phmath-phSubmitted Apr 24, 2026• Announced Apr 28, 2026• 1 citation
arXivHTMLPDF

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 $μ$ 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 $t \approx π$. The standard Kerr medium dynamics are exactly recovered in the limit $μ\rightarrow 0$.

Journal reference
Annals of Physics 494 (2026) 170644
DOI
10.1016/j.aop.2026.170644
Comments
18 pages, 3 figures

Related papers

Phase-space dynamics and multicomponent wave-packet structures in the Dunkl anharmonic oscillator

D. Ojeda-Guillén, R. D. Mota

We study the phase-space dynamics and multicomponent wave-packet structures of the Dunkl anharmonic oscillator (Kerr medium). Starting from the time-evolved state, we derive analytical expressions for the Dunkl-Husimi $Q$-function and a normalized angular Husimi profile. The Dunkl parameter modifies the coherent-state excitation weights and the parity-dependent phase accumulation, producing angular anisotropy and asymmetric phase-space separation. For the half-integer deformation $μ=0.5$, the dynamics generate asymmetric multicomponent Husimi patterns with unequal peak heights and an early reconstruction at $t=π/λ$. For the deformation $μ=1.0$, the Husimi distribution exhibits broadening while retaining its multimodal structure and unitary evolution. At the fundamental nonlinear revival time, the wave packet is reconstructed for the selected parameter ratio. The standard Kerr dynamics are recovered in the limit $μ\rightarrow 0$.

math-phCross-listedarXiv:2611.00280•Nov 3, 2026

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

D. Ojeda Guillén, R. D. Mota, J. C. Vega

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.

quant-pharXiv:2602.16743•Physica Scripta•Feb 19, 2026•1 citation

su(1,1) Symmetry and Exact Solutions of the Dunkl-Klein-Gordon Equation in Higher Dimensions

M. Salazar-Ramírez, B. C. Lütfüoğlu, Jesús Alfredo Martínez-Nuño

We investigate the $d$-dimensional Dunkl--Klein--Gordon equation for a scalar particle within an algebraic framework. By employing Schrödinger factorization, we construct the generators of the $\mathfrak{su}(1,1)$ algebra and establish the associated symmetry of the radial sector. The corresponding energy spectra are derived from unitary irreducible representations, and the Sturmian radial bases are obtained analytically. We analyze the $d$-dimensional Dunkl--Klein--Gordon oscillator and the bound-state sector of the $d$-dimensional Dunkl--Klein--Gordon equation with a Dunkl--Coulomb-like potential. Furthermore, $\mathrm{SU}(1,1)$ coherent states are constructed and their time evolution is analyzed, revealing a characteristic radial oscillation behavior. The results show that the Dunkl deformation modifies the spatial and spectral structure of the system and, in the oscillator sector, introduces explicit parity-dependent energy shifts while preserving the underlying radial algebraic dynamics.

quant-pharXiv:2606.27589•Physica Scripta•Jun 29, 2026•0 citations

Deformed quantum mechanics with extended Dunkl derivative

Yusmantoro Yusmantoro

In this research, we investigate the extended Dunkl quantum mechanics with two parameters. We apply this formalism to study general quantum phenomena. Using cartesian coordinate, we calculate the solution to problem of a spinless particle confined to a box in three dimensions. Then, we derive the creation and annihilation operators to analyze harmonic oscillator in three dimensions via algebraic approach. Subsequently, utilizing the same operators and the generators of [Formula: see text] Lie algebra, we examine the Dirac harmonic oscillator in [Formula: see text] dimension. Finally, we construct the hydrodynamical formulation of the extended Dunkl quantum mechanics by deriving continuity and quantum Hamilton–Jacobi equations.

Modern Physics Letters A•Aug 18, 2025•0 citations

Generalized Dunkl Quantum Systems with Energy-Dependent Interactions: Exact Solvability and Thermodynamic Properties

B. Hamil, R. Boudjouraf, A. Benchikha +1 more

This work investigates a class of exactly solvable quantum systems described by a generalized Dunkl--Schrödinger equation with energy-dependent interactions. A two-parameter extension of the Dunkl differential operator is introduced, incorporating both the conventional reflection contribution and an additional derivative-reflection coupling. The resulting formalism leads to a modified Heisenberg algebra, from which the continuity equation, conserved probability current, and generalized normalization condition are derived for energy-dependent potentials. As an application, an energy-dependent harmonic oscillator is studied in detail. Exact analytical expressions for the eigenfunctions and energy spectrum are obtained in terms of associated Laguerre polynomials. The combined effects of the Wigner parameter, the deformation parameter, and the energy-dependence parameter are shown to produce parity splitting, nonlinear energy spectra, and nonuniform level spacing while preserving exact solvability. The thermodynamic properties of the model are then examined through the canonical partition function using the Euler--Maclaurin summation formula. Analytical expressions for the partition function, internal energy, entropy, and specific heat are derived and analyzed numerically. The results demonstrate that the generalized Dunkl deformation and energy-dependent interaction provide effective control over both the spectral and thermal characteristics of the system, extending the class of exactly solvable quantum models with reflection symmetry.

quant-pharXiv:2610.04322•Oct 6, 2026•0 citations

Quantum Effects in the Interaction of an Anharmonic Oscillator with Binomial and Squeezed Coherent Fields

Amitabh Joshi, Ravinder Rupchand Puri

We show that a single mode of an electromagnetic field in the binomial state or in the squeezed coherent state, propagating in a non-absorbing nonlinear medium modelled as an anharmonic oscillator, gives rise to interesting quantum effects such as collapses and revivals in the energy exchange between the field and the medium. Analytic expressions are obtained for the energy exchange variable. We have exploited the characteristics of the binomial state which provides a means of systematic study of the collapses and revivals with variation in the field statistics from Poissonian to sub-Poissonian. Effects of variation in initial squeezing in the field are also included.

Journal of Modern Optics•Mar 1, 1991•1 citation
We study the Dunkl anharmonic oscillator (Kerr medium) Hamiltonian from an algebraic approach of the SU(1,1)SU(1,1)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)su(1,1)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 μμμ 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≈πt \approx πt≈π. The standard Kerr medium dynamics are exactly recovered in the limit μ→0μ\rightarrow 0μ→0.
We study the phase-space dynamics and multicomponent wave-packet structures of the Dunkl anharmonic oscillator (Kerr medium). Starting from the time-evolved state, we derive analytical expressions for the Dunkl-Husimi QQQ-function and a normalized angular Husimi profile. The Dunkl parameter modifies the coherent-state excitation weights and the parity-dependent phase accumulation, producing angular anisotropy and asymmetric phase-space separation. For the half-integer deformation μ=0.5μ=0.5μ=0.5, the dynamics generate asymmetric multicomponent Husimi patterns with unequal peak heights and an early reconstruction at t=π/λt=π/λt=π/λ. For the deformation μ=1.0μ=1.0μ=1.0, the Husimi distribution exhibits broadening while retaining its multimodal structure and unitary evolution. At the fundamental nonlinear revival time, the wave packet is reconstructed for the selected parameter ratio. The standard Kerr dynamics are recovered in the limit μ→0μ\rightarrow 0μ→0.
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)su(1,1)su(1,1) Lie algebra. The spectral problem is solved exactly using two algebraic methods: the su(1,1)su(1,1)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 QQQ parameter and the second-order correlation function g(2)(0)g^{(2)}(0)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 μμμ, 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.
We investigate the ddd-dimensional Dunkl--Klein--Gordon equation for a scalar particle within an algebraic framework. By employing Schrödinger factorization, we construct the generators of the su(1,1)\mathfrak{su}(1,1)su(1,1) algebra and establish the associated symmetry of the radial sector. The corresponding energy spectra are derived from unitary irreducible representations, and the Sturmian radial bases are obtained analytically. We analyze the ddd-dimensional Dunkl--Klein--Gordon oscillator and the bound-state sector of the ddd-dimensional Dunkl--Klein--Gordon equation with a Dunkl--Coulomb-like potential. Furthermore, SU(1,1)\mathrm{SU}(1,1)SU(1,1) coherent states are constructed and their time evolution is analyzed, revealing a characteristic radial oscillation behavior. The results show that the Dunkl deformation modifies the spatial and spectral structure of the system and, in the oscillator sector, introduces explicit parity-dependent energy shifts while preserving the underlying radial algebraic dynamics.
In this research, we investigate the extended Dunkl quantum mechanics with two parameters. We apply this formalism to study general quantum phenomena. Using cartesian coordinate, we calculate the solution to problem of a spinless particle confined to a box in three dimensions. Then, we derive the creation and annihilation operators to analyze harmonic oscillator in three dimensions via algebraic approach. Subsequently, utilizing the same operators and the generators of [Formula: see text] Lie algebra, we examine the Dirac harmonic oscillator in [Formula: see text] dimension. Finally, we construct the hydrodynamical formulation of the extended Dunkl quantum mechanics by deriving continuity and quantum Hamilton–Jacobi equations.
This work investigates a class of exactly solvable quantum systems described by a generalized Dunkl--Schrödinger equation with energy-dependent interactions. A two-parameter extension of the Dunkl differential operator is introduced, incorporating both the conventional reflection contribution and an additional derivative-reflection coupling. The resulting formalism leads to a modified Heisenberg algebra, from which the continuity equation, conserved probability current, and generalized normalization condition are derived for energy-dependent potentials. As an application, an energy-dependent harmonic oscillator is studied in detail. Exact analytical expressions for the eigenfunctions and energy spectrum are obtained in terms of associated Laguerre polynomials. The combined effects of the Wigner parameter, the deformation parameter, and the energy-dependence parameter are shown to produce parity splitting, nonlinear energy spectra, and nonuniform level spacing while preserving exact solvability. The thermodynamic properties of the model are then examined through the canonical partition function using the Euler--Maclaurin summation formula. Analytical expressions for the partition function, internal energy, entropy, and specific heat are derived and analyzed numerically. The results demonstrate that the generalized Dunkl deformation and energy-dependent interaction provide effective control over both the spectral and thermal characteristics of the system, extending the class of exactly solvable quantum models with reflection symmetry.
We show that a single mode of an electromagnetic field in the binomial state or in the squeezed coherent state, propagating in a non-absorbing nonlinear medium modelled as an anharmonic oscillator, gives rise to interesting quantum effects such as collapses and revivals in the energy exchange between the field and the medium. Analytic expressions are obtained for the energy exchange variable. We have exploited the characteristics of the binomial state which provides a means of systematic study of the collapses and revivals with variation in the field statistics from Poissonian to sub-Poissonian. Effects of variation in initial squeezing in the field are also included.