Deformed Boson Algebras and $\mathcal{W}_{α,β,ν}$-Coherent States: A New Quantum Framework

TL;DR

This work extends coherent state theory by introducing $ \mathcal{W}_{\alpha,\beta,\nu}$-coherent states built on a deformed boson algebra with a generalized factorial $[n]_{\alpha,\beta,\nu}!$. The authors develop the associated deformed oscillator, establish continuity and resolution of unity via a Stieltjes moment problem, and solve for positive weight functions using Mellin transforms, including Mittag-Leffler and Wright special cases with Fox-H representations. They analyze quantum fluctuations and Mandel parameter to reveal parameter-dependent nonclassical features, highlighting sub-Poissonian statistics in certain regimes and Poissonian behavior in the undeformed limit. The framework leverages fractional calculus and special functions to connect classical and quantum regimes, with potential impact on fractional quantum mechanics and quantum optics, while Carleman conditions ensure the uniqueness of the weight measure for a broad parameter range.

Abstract

We introduce a novel class of coherent states, termed $\mathcal{W}^{(\barα,\barν)}(z)$-coherent states, constructed using a deformed boson algebra based on the generalized factorial $[n]_{α,β,ν}!$. This algebra extends conventional factorials, incorporating advanced special functions such as the Mittag-Leffler and Wright functions, enabling the exploration of a broader class of quantum states. The mathematical properties of these states, including their continuity, completeness, and quantum fluctuations, are analyzed. A key aspect of this work is the resolution of the Stieltjes moment problem associated with these states, achieved through the inverse Mellin transformation method. The framework provides insights into the interplay between classical and quantum regimes, with potential applications in quantum optics and fractional quantum mechanics. By extending the theoretical landscape of coherent states, this study opens avenues for further exploration in mathematical physics and quantum technologies.

Figures (6)

• Figure 1: Energy spectra of generalised CS (\ref{['energyev']}) as a function of quantum numbers $n$ for varying deformation parameter $\alpha$. The plots illustrate the influence of the parameter $\alpha$ on the energy eigenvalues $E_n$, with fixed values of $\beta$ and $\nu$. Each curve corresponds to a specific value of $\alpha$, demonstrating the deformation effects in the spectral structure
• Figure 2: Ground states $\langle z|0\rangle$ and First excited states $\langle z|1\rangle$ for different values of the deformation parameters ($\alpha, \beta, \nu$). We set: $m=\omega=\hbar=1$
• Figure 3: Visualisation of quantum fluctuations in quadrature uncertainties of the vacuum state for the generalised deformed states, expressed in units of $\hbar/2$, illustrating the dependence of the uncertainty relation (Equation 3.10) on the parameters $\alpha$, $\beta$, and $\nu$.
• Figure 4: 3D scatter plot illustrating the dependency of the gamma ratio $[1]_{\alpha,\beta,\nu}=\frac{\Gamma(\beta +1)}{\Gamma(\beta +1-\alpha)} \frac{\Gamma(\beta +1-\alpha+\nu)}{\Gamma(1-\alpha+\nu)}$ on the parameters $\alpha$, $\beta$, and $\nu$ within the range $(0, 1)$. The colour scale represents the computed gamma ratio values, highlighting variations across the parameter space.
• Figure 5: Plot of the Mandel parameter $Q_M(x)$ as a function of $x=|z|^2$, in the case of $\alpha =0, \beta=1$ for different values of $\nu$. Mittag-Leffler function $E_{1,\nu+1}(x)$.

Theorems & Definitions (5)

• Remark 1
• Definition 2.1
• Remark 2
• Remark 3
• Remark 4