Gazeau-Klauder coherent states for a harmonic position-dependent mass

TL;DR

This work analyzes a one-dimensional harmonic oscillator with position-dependent mass $m(x)=\frac{m_0}{(1+\alpha x^2)^2}$ by solving the associated Schrödinger-like equation and obtaining an $\alpha$-controlled energy spectrum. It then constructs Gazeau-Klauder coherent states for the discrete spectrum, proves they satisfy Klauder’s conditions (normalization, continuity, resolution of unity, and temporal stability), and investigates their quantum statistics. The GK states exhibit sub-Poissonian photon statistics and a Wigner function with negative regions, confirming nonclassical behavior induced by the PDM deformation. The analysis combines Gegenbauer polynomial eigenfunctions, Mellin transforms, and Meijer G-function weight constructions to establish overcompleteness and temporal dynamics, highlighting the feasibility of GK coherent states in PDM quantum systems and their potential for further nonclassical applications.

Abstract

In this paper, we study the dynamic of position-dependent mass system confined in harmonic oscillator potential. We derive the eigensystems by solving the Schr\''odinger-like equation which describes this system. We construct coherent states a Gazeau-Klauder for this system. We show that these states satisfy the Klauder's mathematical condition to build coherent states. We compute and analyse some statistical properties of these states. We find that these states exhibit sub-Poissonian statistics. We also evaluate quasiprobability distributions such as the Wigner function to demonstrate graphically nonclassical features of these states.

Figures (4)

• Figure 1: PDM versus the position $x$ for different values of $\alpha$.
• Figure 2: The energy for a PDM harmonic oscillator versus the quantum number $n$ for fix values of the parameter $\alpha$.
• Figure 3: Weight function \ref{['weig']} versus the parameter $J$ for different values of $\alpha$.
• Figure 4: The probability distribution \ref{['7']} as a function of quantum number $n$ for different values of the coherent state parameter $J$ and the deformed parameters $\alpha=0.2$, $\alpha=0.3$ for the first two figures, respectively, and $\alpha=0.8$, $\alpha=0.9$ for the second two figures below, respectively.