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arXiv:2109.02251 · International Journal of Geometric Methods in Modern Physics

Coherent states on a circle: The Higgs-like approach

Ali Mahdifar, Ehsan Amooghorban

Published May 4, 2022• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s0219887822501407

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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.
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.
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.
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Abstract In this article, we introduce generalized Gilmore-Perelomov-type coherent states of a quantum harmonic oscillator on a circle and demonstrate that their underlying algebra is a deformed su(1,1). The effects of the circle’s spatial curvature are inherently encoded in these states, making them suitable for investigating curvature-induced phenomena in quantum optical processes at laboratory scales. Furthermore, we propose an experimental scheme to realize these states based on the intensity-dependent version of the Jaynes–Cummings model appearing in a laser-driven trapped ion. Employing the nonlinear coherent states of the atomic center of mass motion of a trapped ion, it is possible to control the laser-ion interaction in such a way that Gilmore-Perelomov-type circle coherent states are obtained as motional dark states of the system. Finally, by examining quadrature squeezing, we demonstrate that unlike standard coherent states which exhibit no squeezing, these curvature-dependent states display curvature-tunable squeezing effects. This property makes them particularly valuable for applications in high-precision metrology.
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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.