Hybrid Superpositions in Quantum Physics

• Hybrid Superpositions are quantum states that coherently combine distinct dual families of nonlinear coherent states, exhibiting novel algebraic and interference properties.
• They are constructed via two principal methods—algebraic combinations preserving f-deformed structure and normalized superpositions revealing altered nonclassical signatures.
• These states offer practical avenues for quantum state engineering in optics and metrology by enabling tailored control over antibunching, squeezing, and other quantum observables.

A hybrid superposition in quantum physics refers to a quantum state that coherently combines constituents with distinct algebraic, physical, or operational properties—often involving generalized or nonlinear coherent states, states arising from dual algebraic constructions, or diverse nonclassical signatures. A central thrust of current research is the systematic generation and characterization of such superpositions and the investigation of their nonclassical behavior, especially how quantum interference modifies statistical and squeezing properties. This article focuses on the rigorous mathematical framework for hybrid superpositions of the dual family of nonlinear coherent states (CSs), methods of combining them, their impact on nonclassical observables, physical realizations in quantum optics, and the role of quantum interference.

1. Dual Families of Nonlinear Coherent States

Nonlinear CSs generalize canonical coherent states by deforming bosonic ladder operators with a nonlinearity function $f(n)$, where $n = a^\dagger a$ is the number operator. The f-deformed annihilation and creation operators are: $A = a f(n), \qquad A^\dagger = f(n) a^\dagger$ Nonlinear CSs are eigenstates of $A$: $A|\alpha, f\rangle = \alpha |\alpha, f\rangle$ Their Fock-basis expansion is

$$|\alpha, f\rangle = \mathcal{N}_f \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}[f(n)]!} |n\rangle$$

where $[f(n)]! = f(1)f(2)\ldots f(n)$, $[f(0)]! = 1$. The dual family is defined as

$$|\tilde{\alpha}, f\rangle = \tilde{\mathcal{N}}_f \sum_{n=0}^{\infty} \frac{\alpha^n [f(n)]!}{\sqrt{n!}} |n\rangle$$

The two families are generally nonorthogonal, with overlap: $$\langle \alpha, f | \tilde{\alpha}, f \rangle = \mathcal{N}_f \tilde{\mathcal{N}}_f e^{|\alpha|^2}$$ This duality structure is suited for algebraic manipulations and, crucially, sets the stage for novel superpositions.

2. Types of Hybrid Superpositions: Algebraic Constructions

Two principal forms of hybrid superpositions of the dual CS families are introduced:

Type 1: Combination of Unnormalized Duals (Algebraic Combination)

Neglect normalization, add the states, then normalize: $$|\alpha, f\rangle_{s1} = \mathcal{N}_{s1} \sum_{n=0}^{\infty} \frac{\alpha^n \left[1 + ([f(n)]!)^2\right]}{\sqrt{n!}[f(n)]!} |n\rangle$$ Or explicitly as a weighted superposition: $$|\alpha, f\rangle_{s1} = c_1 |\alpha, f\rangle + c_2 |\tilde{\alpha}, f\rangle$$ with $c_1 = \mathcal{N}_{s1} \mathcal{N}_f$, $c_2 = \mathcal{N}_{s1} \tilde{\mathcal{N}}_f$.

A key property is that this combination remains in the class of f-deformed CSs with an effective nonlinearity: $$f_{s1}(n) = \frac{1 + ([f(n-1)]!)^2}{1 + ([f(n)]!)^2} f(n)$$ This preserves the algebraic structure, allowing for further Hamiltonian engineering.

Type 2: Superposition of Normalized Duals

First normalize each dual, then superpose and normalize again: $$|\alpha, f\rangle_{s2} = \mathcal{N}_{s2} \sum_{n=0}^\infty \frac{\alpha^n G(n,|\alpha|^2)}{\sqrt{n!}[f(n)]!} |n\rangle$$ with

$$G(n,|\alpha|^2) = \mathcal{N}_f + \tilde{\mathcal{N}}_f ([f(n)]!)^2$$

$\mathcal{N}_{s2}$ is determined by normalization. In this form, the superposition need not be an eigenstate of a simple deformed annihilation operator—i.e., the f-deformed structure may not survive.

3. Nonclassical Properties of Superposed States

The essential nonclassical signatures investigated are:

• Antibunching/Sub-Poissonian Statistics: Quantified by the second-order correlation function

$$g^{(2)}(0) = \frac{\langle a^{\dagger 2} a^2 \rangle}{\langle a^\dagger a \rangle^2}$$

Sub-Poissonian values $g^{(2)}(0) < 1$ signal quantum antibunching.

• Quadrature Squeezing: In quadratures

$$X = \frac{a + a^\dagger}{\sqrt{2}}, \qquad Y = \frac{a - a^\dagger}{i\sqrt{2}}$$

Squeezing is indicated by

$(\Delta X)^2 < \frac{1}{2}$

More broadly, the inequalities

$$I_1 = \langle a^2 \rangle + \langle a^{\dagger 2} \rangle - \langle a \rangle^2 - \langle a^\dagger \rangle^2 - 2\langle a^\dagger a \rangle + 2 < 0$$

are derived for general analysis.

• Amplitude Squared Squeezing: Given by

$$X_2 = \frac{a^2 + a^{\dagger 2}}{2},\qquad Y_2 = \frac{a^2 - a^{\dagger 2}}{2i}$$

Squeezing is present when $(\Delta X_2)^2$ or $(\Delta Y_2)^2$ falls below a limit set by Heisenberg uncertainty.

Numerical analysis reveals that, for physical models with Hydrogen-like spectra (and, separately, Gazeau-Klauder CSs with the Pöschl–Teller potential), the two types of superpositions display altered and sometimes suppressed nonclassical features compared to their dual constituents. Notably, superpositions may convert sub-Poissonian into super-Poissonian behavior and shift the domain of squeezing in phase space.

4. Physical Realizations: Hydrogen-like and GKCS Systems

Concrete realizations of hybrid superpositions leverage systems with known nonlinearity functions:

Hydrogen-like Spectrum

Energy spectrum is $e_n = 1 - 1/(n+1)^2$, with $f(n) = \sqrt{\frac{n+2}{n+1}}$. Application of the above superpositions with this $f(n)$ in the Fock expansion leads to specific, testable predictions for $g^{(2)}(0)$, quadrature squeezing, and amplitude squared squeezing.

Gazeau-Klauder Coherent States (GKCSs) and Pöschl–Teller Potential

GKCSs are of the form

$$|z,y\rangle_{GK} = \mathcal{N}_{GK} \sum_{n=0}^{\infty} \frac{z^n e^{-i e_n y}}{\sqrt{\rho(n)}} |n\rangle$$

Superpositions are constructed analogously, with the energy spectrum and nonlinearity fixed for the Pöschl–Teller potential by $e_n = n(n + A + \kappa)$ and $f(n) = \sqrt{n+\kappa}$. The resultant "hybrid" states demonstrate that quantum interference, as controlled by the form of the superposition, can strongly influence observables such as squeezing and photon statistics.

5. Quantum Interference and Impact on Nonclassicality

The hybrid superpositions introduced allow direct control over quantum interference between dual families of nonlinear coherent states. The alteration of nonclassical indicators—such as transitioning from sub- to super-Poissonian statistics or shifting the presence of squeezing to different quadratures—stems from interference between probability amplitudes in the Fock basis, as encapsulated by the structure of the expansion coefficients in the two types of superpositions. In both the Hydrogen-like and GKCS cases, numerical results confirm that these interference effects often reduce or entirely suppress signatures present in the constituent (original) states.

For type-1 combinations, persistence of the f-deformed algebra and the possibility to reconstruct an effective Hamiltonian with modified nonlinearity is a marked feature. For type-2 (superposition of normalized states), such algebraic correspondence is generally lost, and new nonclassical behaviors emerge idiosyncratically.

6. Significance for State Engineering and Quantum Technologies

The controlled formation of hybrid superpositions provides a flexible platform for the design of quantum states with tunable nonclassicality, valuable in quantum optics and quantum information processing. The first type of combination, preserving modified f-deformed algebraic structure, is especially relevant for theoretical investigations aiming to connect operator algebras and physical observables, and for reconstructing tailored Hamiltonians. The sensitivity of nonclassicality indicators to quantum interference in the hybrid states indicates potential for engineering states that are either robust to or specifically sensitive to external perturbations—a property exploitable in quantum metrology and communication.

7. Mathematical Summary

Key equations underpinning hybrid superpositions of the dual CS family include:

Component	Formula	Notes
Nonlinear annihilation operator	$A = a f(n)$	$n = a^\dagger a$
Nonlinear CS expansion	$$|\alpha, f\rangle = \mathcal{N}_f \sum \frac{\alpha^n}{\sqrt{n!}[f(n)]!} |n\rangle$$
Dual state expansion	$$|\tilde{\alpha}, f\rangle = \tilde{\mathcal{N}}_f \sum \frac{\alpha^n [f(n)]!}{\sqrt{n!}} |n\rangle$$
First-type superposition (combination)	$$|\alpha, f\rangle_{s1} = \mathcal{N}_{s1} \sum \frac{\alpha^n (1 + ([f(n)]!)^2)}{\sqrt{n!}[f(n)]!} |n\rangle$$	Remains in f-deformed family, with $f_{s1}$
Effective nonlinearity function	$$f_{s1}(n) = \frac{1 + ([f(n-1)]!)^2}{1 + ([f(n)]!)^2} f(n)$$	Modified algebraic structure
Second-type superposition (normalized)	$$|\alpha, f\rangle_{s2} = \mathcal{N}_{s2} \sum \frac{\alpha^n G(n,|\alpha|^2)}{\sqrt{n!}[f(n)]!} |n\rangle$$	$G(n,|\alpha|^2) = N_f + \tilde{N}_f ([f(n)]!)^2$
Antibunching criterion	$$g^{(2)}(0) = \langle a^{\dagger 2} a^2 \rangle / \langle a^\dagger a \rangle^2$$	$g^{(2)}(0) < 1$ signals nonclassicality

The systematic analysis of these structures offers a clear path to deriving the nonclassical content of hybrid superpositions and manipulating their properties in both theory and experiment.

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Hybrid superpositions in dual families of nonlinear coherent states thus constitute a rigorous and versatile approach for engineering quantum interference and tailoring nonclassicality in physically realizable systems. The interplay between algebraic method (type-1 combination), operational superposition (type-2), and observable consequences (squeezing, antibunching) is essential for understanding and advancing quantum state engineering (Abbasi et al., 2010).