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arXiv:quant-ph/0604176 · Journal of Physics A Mathematical and General

Geometric approach to nonlinear coherent states using the Higgs model for harmonic oscillator

A Mahdifar, R Roknizadeh, M H Naderi

Published May 16, 2006• 40 citations
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Abstract

In this paper, we investigate the relation between the curvature of the physical space and the deformation function of the deformed oscillator algebra using the nonlinear coherent states approach. For this purpose, we study two-dimensional harmonic oscillators on the flat surface and on a sphere by applying the Higgs model. With the use of their algebras, we show that the two-dimensional oscillator algebra on a surface can be considered as a deformed one-dimensional oscillator algebra where the effect of the curvature of the surface appears as a deformation function. We also show that the curvature of the physical space plays the role of deformation parameter. Then we construct the associated coherent states on the flat surface and on a sphere and compare their quantum statistical properties, including quadrature squeezing and antibunching effect.

DOI
10.1088/0305-4470/39/22/014

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In this paper, we investigate the relation between the curvature of the physical space and the deformation function of the deformed oscillator algebra using the nonlinear coherent states approach. For this purpose, we study two-dimensional harmonic oscillators on the flat surface and on a sphere by applying the Higgs model. With the use of their algebras, we show that the two-dimensional oscillator algebra on a surface can be considered as a deformed one-dimensional oscillator algebra where the effect of the curvature of the surface appears as a deformation function. We also show that the curvature of the physical space plays the role of deformation parameter. Then we construct the associated coherent states on the flat surface and on a sphere and compare their quantum statistical properties, including quadrature squeezing and antibunching effect.
In this paper, we investigate the relation between the curvature of the physical space and the deformation function of the deformed oscillator algebra using non-linear coherent states approach. For this purpose, we study two-dimensional harmonic oscillators on the flat surface and on a sphere by applying the Higgs modell. With the use of their algebras, we show that the two-dimensional oscillator algebra on a surface can be considered as a deformed one-dimensional oscillator algebra where the effect of the curvature of the surface is appeared as a deformation function. We also show that the curvature of the physical space plays the role of deformation parameter. Then we construct the associated coherent states on the flat surface and on a sphere and compare their quantum statistical properties, including quadrature squeezing and antibunching effect.
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In this paper, we construct the Gazeau-Klauder coherent states of a two- dimensional harmonic oscillator on a sphere based on two equivalent approaches. First, we consider the oscillator on the sphere as a deformed (non-degenerate) one-dimensional oscillator. Second, the oscillator on the sphere is considered as the usual (degenerate) two--dimensional oscillator. Then, by investigating the quantum optical properties of the constructed coherent states, we study the effect of the space curvature on the properties of the constructed sphere Gazeau-Klauder coherent states, according to these two approaches.
In this paper we study some basic quantum confinement effects through investigation of a deformed harmonic oscillator algebra. We show that spatial confinement effects on a quantum harmonic oscillator can be represented by a deformation function within the framework of nonlinear coherent states theory. We construct the coherent states associated with the spatially confined quantum harmonic oscillator in a one-dimensional infinite well and examine some of their quantum statistical properties, including sub-Poissonian statistics and quadrature squeezing.
Abstract The two-dimensional Higgs oscillator, defined on the surface of a sphere S 2 , can be mapped via a stereographic projection to a conformally transformed harmonic oscillator on the complex plane. Its quantum mechanical exact solvability provides the foundation for subsequent analyses, including the study of dual SUSY partners and the construction of continuous-variable quantum states. Due to the non-standard form of its Hamiltonian, factorization is complex, and the corresponding ladder operators are deformed. Furthermore, algebraic analysis reveals that the system satisfies the su(1, 1) algebra, which allows for the construction of Perelomov and Barut-Girardello nonlinear coherent states.
In this paper, the Higgs-like approach is used to analyze the quantum dynamics of a harmonic oscillator constrained on a circle. We obtain the Hamiltonian of this system as a function of the Cartesian coordinate of the tangent line through the gnomonic projection and then quantize it in the standard way. We then recast the Hamiltonian in a shape-invariant form and derive the spectrum energy of the confined harmonic oscillator on the circle. With help of the f-deformed oscillator algebra, we construct the coherent states on the circle and investigate their quantum statistical properties. We find that such states show nonclassical features like squeezing and sub-Poissonian statistics even in small curvatures of the circle.