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

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10.1063/1.4739436

### 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 one 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.

© 2012 American Institute of Physics

Received 25 January 2012
Accepted 03 July 2012
Published online 08 August 2012

Article outline:

I. INTRODUCTION

II. DEFORMED OSCILLATORALGEBRA

III. NONLINEAR COHERENT STATES

A. Non-unitary displacement type operators and nonlinear coherent states

B. Unitary operator and coherent states

C. Completeness condition

IV. NON-CLASSICAL PROPERTIES

A. *A* _{3}-parameter

B. Quadrature squeezing

C. Amplitude-squared squeezing

D. Photonstatistical properties

V. QUADRATURE DISTRIBUTION AND QUASI-PROBABILITY FUNCTIONS

A. Phase-parameterized field strength distribution

B. *s*-parameterized quasi-probability function

VI. CONCLUSION

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2012-08-08

2014-04-16

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