Dunkl Squeezed States & SU(1,1) Parity Effects

• Dunkl squeezed states are quantum squeezed states constructed via Dunkl‐deformed ladder operators, featuring a parity-dependent algebra with nonstandard commutators.
• They exploit an SU(1,1) algebraic framework that enables exact diagonalization of the Hamiltonian using tilting and Bogoliubov transformations.
• The deformation parameter μ induces modified photon statistics, notably affecting the second-order correlation while leaving the Mandel parameter unchanged.

Searching arXiv for the cited Dunkl-state papers and closely related SU(1,1)/Dunkl work. Dunkl squeezed states are squeezed quantum states constructed from Dunkl-deformed ladder operators rather than the ordinary Heisenberg–Weyl bosonic operators. In the one-dimensional setting relevant to the Dunkl parametric amplifier, the deformation is controlled by a real parameter $μ$ and a reflection operator $R$, so that the basic commutator becomes $[a_μ,a_μ^\dagger]=1+2μR$ instead of $[a,a^\dagger]=1$ (Guillén et al., 18 Feb 2026). Within this framework, quadratic combinations of the Dunkl creation and annihilation operators generate an $su(1,1)$ Lie algebra, which permits exact algebraic diagonalization of the Hamiltonian and the explicit construction of squeezed-vacuum and squeezed-number states (Guillén et al., 18 Feb 2026). The resulting states retain the analytic tractability of standard $SU(1,1)$ squeezing while exhibiting deformation-dependent parity structure and modified photon statistics, most notably through an explicit $μ$-dependence of the second-order correlation function $g^{(2)}(0)$ (Guillén et al., 18 Feb 2026).

1. Dunkl deformation and the $su(1,1)$ algebraic setting

The Dunkl formalism on $\mathbb{R}$ replaces the ordinary derivative by the Dunkl derivative

$R$0

where $R$1 is the deformation parameter and $R$2 is the reflection operator with $R$3 (Guillén et al., 18 Feb 2026). This construction inserts parity directly into the kinematics.

The associated ladder operators are defined by

$R$4

They satisfy

$R$5

in contrast with standard bosons, which obey $R$6 and commute with $R$7 (Guillén et al., 18 Feb 2026).

The quadratic combinations

$R$8

satisfy the clean $R$9 commutation relations

$[a_μ,a_μ^\dagger]=1+2μR$0

(Guillén et al., 18 Feb 2026). On Dunkl Fock states $[a_μ,a_μ^\dagger]=1+2μR$1, with $[a_μ,a_μ^\dagger]=1+2μR$2, one introduces the Dunkl number

$[a_μ,a_μ^\dagger]=1+2μR$3

which encodes the even–odd asymmetry generated by the reflection sector (Guillén et al., 18 Feb 2026).

This algebraic structure places Dunkl squeezing naturally in the non-compact $[a_μ,a_μ^\dagger]=1+2μR$4 setting rather than in the ordinary oscillator algebra. Earlier work on the two-dimensional Dunkl oscillator established the same general strategy: the radial Hamiltonian can be factorized in terms of $[a_μ,a_μ^\dagger]=1+2μR$5 generators, and the corresponding unitary irreducible representations admit Perelomov coherent states with Bargmann index

$[a_μ,a_μ^\dagger]=1+2μR$6

(Salazar-Ramírez et al., 2016). That result situates Dunkl squeezed states within a broader family of Dunkl-deformed $[a_μ,a_μ^\dagger]=1+2μR$7 coherent constructions.

2. Dunkl parametric amplifier and exact diagonalization

The Dunkl version of the degenerate parametric amplifier is obtained by replacing $[a_μ,a_μ^\dagger]=1+2μR$8 with $[a_μ,a_μ^\dagger]=1+2μR$9 in the standard Hamiltonian:

$[a,a^\dagger]=1$0

Using

$[a,a^\dagger]=1$1

one obtains, up to a constant,

$[a,a^\dagger]=1$2

or equivalently, with $[a,a^\dagger]=1$3,

$[a,a^\dagger]=1$4

(Guillén et al., 18 Feb 2026).

A central structural property is parity conservation:

$[a,a^\dagger]=1$5

so the even sector $[a,a^\dagger]=1$6 and odd sector $[a,a^\dagger]=1$7 decouple (Guillén et al., 18 Feb 2026). This decoupling is not a secondary feature but part of the defining architecture of Dunkl squeezing.

The spectral problem is solved exactly by two algebraic methods. In the $[a,a^\dagger]=1$8 tilting transformation, one introduces the squeezing displacement

$[a,a^\dagger]=1$9

and chooses $su(1,1)$0 together with

$su(1,1)$1

This yields

$su(1,1)$2

and therefore the exact eigenvalues

$su(1,1)$3

(Guillén et al., 18 Feb 2026).

The generalized Bogoliubov transformation gives the same spectrum. One defines

$su(1,1)$4

with $su(1,1)$5 and $su(1,1)$6, so that $su(1,1)$7. Then

$su(1,1)$8

(Guillén et al., 18 Feb 2026). The agreement between the tilting and generalized Bogoliubov approaches shows that the Dunkl-deformed parametric amplifier remains exactly solvable despite the parity-modified commutation relations.

3. Construction of Dunkl squeezed-vacuum and squeezed-number states

The canonical Dunkl squeezed vacuum is

$su(1,1)$9

where $SU(1,1)$0 is the Dunkl Fock vacuum satisfying $SU(1,1)$1 (Guillén et al., 18 Feb 2026). In the even sector, with Bargmann index

$SU(1,1)$2

its expansion is

$SU(1,1)$3

The coefficients therefore carry explicit $SU(1,1)$4-dependence through $SU(1,1)$5 (Guillén et al., 18 Feb 2026).

More generally, the Dunkl squeezed-number eigenstates of the parametric amplifier are the Perelomov number-coherent states

$SU(1,1)$6

and admit analogous series expansions in $SU(1,1)$7 with $SU(1,1)$8-dependent weights through the Dunkl numbers $SU(1,1)$9 (Guillén et al., 18 Feb 2026). The parity jump by two quanta reflects the quadratic $μ$0 structure.

The connection with earlier $μ$1 coherent-state theory is explicit. For the two-dimensional Dunkl oscillator, the Perelomov coherent states are

$μ$2

with $μ$3 and $μ$4 (Salazar-Ramírez et al., 2016). This is structurally the same Perelomov mechanism used in the one-dimensional Dunkl parametric amplifier, although the physical realization differs.

That earlier work also gives a closed-form radial wavefunction,

$μ$5

with

$μ$6

which illustrates how Dunkl-$μ$7 squeezing can be represented analytically in coordinate space (Salazar-Ramírez et al., 2016). This suggests that the algebraic formalism of Dunkl squeezed states is compatible with explicit wavefunction realizations, not only operator constructions.

4. Statistical diagnostics: Mandel parameter and second-order correlation

The statistical properties of Dunkl squeezed states in the parametric amplifier are analyzed through the Mandel parameter

$μ$8

and the second-order correlation function

$μ$9

(Guillén et al., 18 Feb 2026).

For the squeezed vacuum $g^{(2)}(0)$0 one finds

$g^{(2)}(0)$1

and

$g^{(2)}(0)$2

hence

$g^{(2)}(0)$3

A notable result is that $g^{(2)}(0)$4 is independent of $g^{(2)}(0)$5 and reproduces the standard super-Poissonian result (Guillén et al., 18 Feb 2026). In this sense, the Dunkl deformation does not alter the Mandel diagnosis of the squeezed vacuum, despite changing the underlying algebra.

The second-order correlation is different. For arbitrary squeezed-number states, the exact result is

$g^{(2)}(0)$6

(Guillén et al., 18 Feb 2026). For the vacuum,

$g^{(2)}(0)$7

Thus $g^{(2)}(0)$8 depends explicitly on $g^{(2)}(0)$9, and the deformation tunes photon bunching beyond the standard case (Guillén et al., 18 Feb 2026).

This separation between a universal Mandel parameter and a deformation-sensitive $su(1,1)$0 is one of the main distinctive features of Dunkl squeezed states. It means that two standard diagnostics of nonclassical statistics respond differently to the same parity deformation.

5. Quadrature squeezing in the broader Dunkl-$su(1,1)$1 literature

In the two-dimensional Dunkl oscillator, the natural $su(1,1)$2 quadratures are

$su(1,1)$3

and for the Perelomov coherent state with parametrization $su(1,1)$4 one has

$su(1,1)$5

$su(1,1)$6

(Salazar-Ramírez et al., 2016). Squeezing in the $su(1,1)$7 quadrature is characterized by

$su(1,1)$8

Maximum squeezing in $su(1,1)$9 occurs at $\mathbb{R}$0, for which

$\mathbb{R}$1

(Salazar-Ramírez et al., 2016).

Time evolution is also algebraically simple in that model. Since $\mathbb{R}$2, the propagator is

$\mathbb{R}$3

and one finds

$\mathbb{R}$4

so that

$\mathbb{R}$5

Equivalently, the squeezing angle evolves by $\mathbb{R}$6 (Salazar-Ramírez et al., 2016). The radial quadrature therefore becomes alternately squeezed and anti-squeezed in time with period $\mathbb{R}$7.

These results do not describe the parametric-amplifier Hamiltonian itself, but they clarify the general role of the $\mathbb{R}$8 displacement operator in Dunkl systems. They show that Dunkl squeezing is not restricted to photon-number statistics; it also has a quadrature interpretation in which a squeezing parameter controls noise reduction and a phase controls the squeezing direction.

6. Nonlinear extensions: Dunkl Kerr dynamics and interference-induced squeezing

A distinct nonlinear realization appears in the Dunkl anharmonic oscillator, or Kerr medium, with Hamiltonian

$\mathbb{R}$9

(Ojeda-Guillén et al., 24 Apr 2026). In the Heisenberg picture,

$R$00

so that

$R$01

(Ojeda-Guillén et al., 24 Apr 2026). The field quadratures are

$R$02

with time-dependent forms obtained by substituting the evolved ladder operators (Ojeda-Guillén et al., 24 Apr 2026).

Using the $R$03 generators,

$R$04

and the paper computes exact variances for an initial superposition of even and odd Dunkl coherent states (Ojeda-Guillén et al., 24 Apr 2026). Squeezing is defined by

$R$05

The only time dependence enters through cosine sums involving the parity-dependent “next-nearest-neighbor” energy gaps

$R$06

$R$07

(Ojeda-Guillén et al., 24 Apr 2026).

The paper states that maximal $R$08-squeezing occurs when the interference sum is as negative as possible and that the simplest nontrivial alignment time is

$R$09

(Ojeda-Guillén et al., 24 Apr 2026). For integer $R$10, both cosine families give $R$11 at this time, producing the deepest destructive interference and the strongest $R$12-squeezing at $R$13; for half-integer $R$14, both give $R$15, so there is less squeezing exactly at $R$16 but a pair of nearby dips (Ojeda-Guillén et al., 24 Apr 2026). The standard Kerr medium is recovered when $R$17, in which case the two energy-difference families collapse into one and the usual Kerr quadrature variance is regained (Ojeda-Guillén et al., 24 Apr 2026).

This suggests that Dunkl squeezing in nonlinear media is governed not only by ordinary squeezing amplitudes but also by parity-dependent phase shifts between even and odd sectors. In the terminology of the paper, the deformation generates interference-induced squeezed states around $R$18 and modulates fractional revivals and perfect state reconstructions at half-periods for specific deformation values (Ojeda-Guillén et al., 24 Apr 2026).

7. Standard limit, parity structure, and conceptual significance

The Dunkl-deformed constructions reduce continuously to the ordinary parametric amplifier as $R$19. In that limit,

$R$20

the $R$21 index becomes $R$22, and all $R$23-dependent shifts disappear (Guillén et al., 18 Feb 2026). One recovers the ordinary squeezed-vacuum expansion, the vacuum Mandel result

$R$24

and

$R$25

together with the well-known squeezed-number statistics (Guillén et al., 18 Feb 2026).

The conceptual significance of Dunkl squeezed states lies in the fact that the deformation preserves exact solvability while introducing a parity-resolved algebraic structure. In the parametric-amplifier problem, the even and odd sectors decouple because $R$26 (Guillén et al., 18 Feb 2026). In the Kerr problem, the same parity structure induces distinct interference patterns and half-cycle rephasing phenomena (Ojeda-Guillén et al., 24 Apr 2026). The literature therefore presents Dunkl squeezing as a deformation of standard bosonic squeezing in which parity is not an auxiliary symmetry but an explicit dynamical ingredient.

Within the available results, one robust conclusion is that different observables encode the deformation differently. The vacuum Mandel parameter remains universal, whereas $R$27 acquires a clear $R$28-dependence in the parametric amplifier (Guillén et al., 18 Feb 2026). In nonlinear evolution, quadrature squeezing itself can become parity-controlled through the separation of even and odd energy gaps (Ojeda-Guillén et al., 24 Apr 2026). A plausible implication is that Dunkl squeezed states provide a systematic framework for studying squeezed light in parity-deformed algebras while retaining the operator techniques of $R$29 coherent-state theory developed for Dunkl oscillators more generally (Salazar-Ramírez et al., 2016).