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:2602.16743 · Physica Scripta

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

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

quant-phmath-phSubmitted Feb 18, 2026• Announced Feb 19, 2026• 1 citation
arXivHTMLPDF

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.

Journal reference
Phys. Scr. 101 225206 (2026)
DOI
10.1088/1402-4896/ae70dc
Comments
17 pages

Related papers

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

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

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 $μ\rightarrow 0$.

quant-pharXiv:2604.22945•Annals of Physics•Apr 28, 2026•1 citation

Berry phase and the Mandel parameter of the non-degenerate parametric amplifier

J. C. Vega, E. Choreño, D. Ojeda-Guillén +1 more

We study the non-degenerate parametric amplifier problem from an algebraic approach of the SU(1,1) group. We write the Hamiltonian of this problem in terms of the boson generators of the SU(1,1) group and the difference operator. We apply the tilting transformation to our results to exactly solve this Hamiltonian and obtain its energy spectrum and eigenfunctions. Then, by assuming that our Hamiltonian is an explicit function of time, we calculate its Berry phase. Finally we obtain the Mandel Q -parameter of the photon numbers n a and n b .

arXiv:2312.15114•Journal of the Optical Society of America B•Mar 15, 2024•4 citations

Quadrature-dependent Bogoliubov transformations and multiphoton squeezed states

Silvio De Siena, Antonio Di Lisi, Fabrizio Illuminati

We introduce a linear, canonical transformation of the fundamental single-mode field operators a and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ that generalizes the linear Bogoliubov transformation familiar in the construction of the harmonic oscillator squeezed states. This generalization is obtained by adding a nonlinear function of any of the fundamental quadrature operators ${X}_{1}$ and ${X}_{2}$ to the linear transformation, thus making the original Bogoliubov transformation quadrature dependent. Remarkably, the conditions of canonicity do not impose any constraint on the form of the nonlinear function, and lead to a set of nontrivial algebraic relations between the c-number coefficients of the transformation. We examine in detail the structure and the properties of the quantum states defined as eigenvectors of the transformed annihilation operator b. These eigenvectors define a class of multiphoton squeezed states. The structure of the uncertainty products and of the quasiprobability distributions in phase space shows that besides coherence properties, these states exhibit a squeezing and a deformation (cooling) of the phase-space trajectories, both of which strongly depend on the form of the nonlinear function. The presence of the extra nonlinear term in the phase of the wave functions has also relevant consequences on photon statistics and correlation properties. The nonquadratic structure of the associated Hamiltonians suggests that these states be generated in connection with multiphoton processes in media with higher nonlinearities. We give a detailed description of the quadratic nonlinear transformation, which defines four-photon squeezed states. In particular, the behaviors of the second-order correlation function ${g}^{(2)}(0)$ and of the fourth-order correlation function ${g}^{(4)}(0)$ are studied. The former exhibits super-Poissonian statistics, while the latter indicates photon bunching in the four-photon emissions.

arXiv:quant-ph/0105070•Physical Review A•Nov 9, 2001•21 citations

Coherent and squeezed states of quantum Heisenberg algebras

Starting from deformed quantum Heisenberg Lie algebras some realizations are given in terms of the usual creation and annihilation operators of the standard harmonic oscillator. Then the associated algebra eigenstates are computed and give rise to new classes of deformed coherent and squeezed states. They are parametrized by deformed algebra parameters and suitable redefinitions of them as paragrassmann numbers. Some properties of these deformed states also are analyzed.

•Jan 1, 2008•0 citations

Coherent and squeezed states of quantum Heisenberg algebras

Nibaldo Alvarez-Moraga

Starting from deformed quantum Heisenberg Lie algebras some realizations are given in terms of the usual creation and annihilation operators of the standard harmonic oscillator. Then the associated algebra eigenstates are computed and give rise to new classes of deformed coherent and squeezed states. They are parametrized by deformed algebra parameters and suitable redefinitions of them as paragrassmann numbers. Some properties of these deformed states are also analysed.

arXiv:math-ph/0503055•Journal of Physics A Mathematical and General•Mar 3, 2005•4 citations

The SU(1, 1) Perelomov number coherent states and the non-degenerate parametric amplifier

D. Ojeda-Guillén, R. D. Mota, V. D. Granados

We construct the Perelomov number coherent states for an arbitrary su(1, 1) group operation and study some of their properties. We introduce three operators which act on Perelomov number coherent states and close the su(1, 1) Lie algebra. By using the tilting transformation we apply our results to obtain the energy spectrum and eigenfunctions of the non-degenerate parametric amplifier. We show that these eigenfunctions are the Perelomov number coherent states of the two-dimensional harmonic oscillator.

arXiv:1311.6867•Journal of Mathematical Physics•Apr 1, 2014•17 citations
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 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 non-degenerate parametric amplifier problem from an algebraic approach of the SU(1,1) group. We write the Hamiltonian of this problem in terms of the boson generators of the SU(1,1) group and the difference operator. We apply the tilting transformation to our results to exactly solve this Hamiltonian and obtain its energy spectrum and eigenfunctions. Then, by assuming that our Hamiltonian is an explicit function of time, we calculate its Berry phase. Finally we obtain the Mandel Q -parameter of the photon numbers n a and n b .
We introduce a linear, canonical transformation of the fundamental single-mode field operators a and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ that generalizes the linear Bogoliubov transformation familiar in the construction of the harmonic oscillator squeezed states. This generalization is obtained by adding a nonlinear function of any of the fundamental quadrature operators X1{X}_{1}X1​ and X2{X}_{2}X2​ to the linear transformation, thus making the original Bogoliubov transformation quadrature dependent. Remarkably, the conditions of canonicity do not impose any constraint on the form of the nonlinear function, and lead to a set of nontrivial algebraic relations between the c-number coefficients of the transformation. We examine in detail the structure and the properties of the quantum states defined as eigenvectors of the transformed annihilation operator b. These eigenvectors define a class of multiphoton squeezed states. The structure of the uncertainty products and of the quasiprobability distributions in phase space shows that besides coherence properties, these states exhibit a squeezing and a deformation (cooling) of the phase-space trajectories, both of which strongly depend on the form of the nonlinear function. The presence of the extra nonlinear term in the phase of the wave functions has also relevant consequences on photon statistics and correlation properties. The nonquadratic structure of the associated Hamiltonians suggests that these states be generated in connection with multiphoton processes in media with higher nonlinearities. We give a detailed description of the quadratic nonlinear transformation, which defines four-photon squeezed states. In particular, the behaviors of the second-order correlation function g(2)(0){g}^{(2)}(0)g(2)(0) and of the fourth-order correlation function g(4)(0){g}^{(4)}(0)g(4)(0) are studied. The former exhibits super-Poissonian statistics, while the latter indicates photon bunching in the four-photon emissions.
We construct the Perelomov number coherent states for an arbitrary su(1, 1) group operation and study some of their properties. We introduce three operators which act on Perelomov number coherent states and close the su(1, 1) Lie algebra. By using the tilting transformation we apply our results to obtain the energy spectrum and eigenfunctions of the non-degenerate parametric amplifier. We show that these eigenfunctions are the Perelomov number coherent states of the two-dimensional harmonic oscillator.