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Constructing Barut-Girardello coherent states for the isotonic oscillator in the DOOT approach

Messan Médard Akouetegan, Isiaka Aremua, Mahouton Norbert Hounkonnou

Abstract

In this work, we study the quantum system of the isotonic oscillator from the perspective of the diagonal operator ordering technique (DOOT). Within this framework, we construct the associated Barut-Girardello and Gazeau-Klauder coherent states. We examine their mathematical properties using reproducing kernels and compute the expectation values of observables that characterize the system and its relevant physical features. Further, we perform the quantization of main classical variables in the complex plane. Then, by exploring the thermal behavior of the physical system in the constructed coherent states, we analyze the properties of mixed states described by a canonical density operator. We also obtain the corresponding Glauber-Sudarshan P-representation.

Constructing Barut-Girardello coherent states for the isotonic oscillator in the DOOT approach

Abstract

In this work, we study the quantum system of the isotonic oscillator from the perspective of the diagonal operator ordering technique (DOOT). Within this framework, we construct the associated Barut-Girardello and Gazeau-Klauder coherent states. We examine their mathematical properties using reproducing kernels and compute the expectation values of observables that characterize the system and its relevant physical features. Further, we perform the quantization of main classical variables in the complex plane. Then, by exploring the thermal behavior of the physical system in the constructed coherent states, we analyze the properties of mixed states described by a canonical density operator. We also obtain the corresponding Glauber-Sudarshan P-representation.
Paper Structure (31 sections, 3 theorems, 92 equations, 1 figure)

This paper contains 31 sections, 3 theorems, 92 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The isotonic oscillator quantum Hamiltonian and the underlying $\mathfrak{su}(1,1)$ Lie algebra
  3. The DOOT approach and related coherent states
  4. The $\mathfrak{su}(1,1)$ generators representation in the DOOT framework
  5. Barut-Giradello coherent states
  6. Construction
  7. Overlap of two BGCSs
  8. Continuity in the labeling
  9. Resolution of the unity operator
  10. Gazeau-Klauder coherent states construction
  11. Reproducing kernel
  12. Expectations values
  13. Photon number distribution and density of probability
  14. Photon number distribution
  15. Density probability
  16. ...and 16 more sections

Key Result

Proposition 3.1

The constructed pair of BGCSs satisfies on the Hilbert space $\mathfrak H = span\{\ket{n,\gamma}\}_{n= 0}^{\infty}$ the following resolutions of the identity: where the appropriate weight functions $W_e(|z|^2,\gamma)$ and $W_o(|z|^2,\gamma)$, are obtained through the Mellin transform in the DOOT framework, and provided as: respectively.

Figures (1)

  • Figure 1: The plot of the weight functions $W_e(|z|^2,\gamma)$(a), $W_o(|z|^2,\gamma)$(b), and $\lambda(J)$(c) against $x = |z|$, for different values of the Bargmann index $\gamma$.

Theorems & Definitions (3)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 4.1