An observation of quadratic algebra, dual family of nonlinear coherent states and their non-classical properties, in the generalized isotonic oscillator

TL;DR

This paper constructs and analyzes nonlinear coherent states for the generalized isotonic oscillator, revealing their non-classical properties and deriving various quantum phase space distributions, while also introducing harmonic oscillator type coherent states.

Contribution

It introduces a method to construct nonlinear coherent states for the generalized isotonic oscillator and explores their non-classical properties and phase space representations.

Findings

Non-classicality confirmed via squeezing and phase space functions

Dual states are non-normalizable and do not form a complete set

Harmonic oscillator type coherent states derived from the unitary operator

Abstract

In this paper, we construct nonlinear coherent states for the generalized isotonic oscillator and study their non-classical properties in-detail. By transforming the deformed ladder operators suitably, which generate the quadratic algebra, we obtain Heisenberg algebra. From the algebra we define two non-unitary and an unitary displacement type operators. While the action of one of the non-unitary type operators reproduces the original nonlinear coherent states, the other one fails to produce a new set of nonlinear coherent states (dual pair). We show that these dual states are not normalizable. For the nonlinear coherent states, we evaluate the parameter $A_3$ and examine the non-classical nature of the states through quadratic and amplitude-squared squeezing effect. Further, we derive analytical formula for the $P$-function, $Q$-function and the Wigner function for the nonlinear coherent states. All of them confirm the non-classicality of the nonlinear coherent states. In addition to the above, we obtain the harmonic oscillator type coherent states from the unitary displacement operator.