Maths-type q-deformed coherent states for q&gt;1

Quesne, Christiane; Penson, K. A.; Tkachuk, V. M.

doi:10.1016/s0375-9601(03)00732-1

Cited by 49 publications

(45 citation statements)

References 26 publications

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“…Note that by replacing in (31) the parameter q by its inverse q = q −1 , we recover the well known orthonormal basis ([ j ] q !) −1/2 z j of the classical Arïk-Coon type space with q = q −1 > 1 [25].…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

A q-deformation of true-polyanalytic Bargmann transforms when q -1 >1

Moize¹,

Mouayn²

2022

Comptes Rendus. Mathématique

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We combine continuous q −1 -Hermite Askey polynomials with new 2D orthogonal polynomials introduced by Ismail and Zhang as q-analogs for complex Hermite polynomials to construct a new set of coherent states depending on a nonnegative integer parameter m. Our construction leads to a new qdeformation of the m-true-polyanalytic Bargmann transform on the complex plane. In the analytic case m = 0, the obtained coherent states transform can be associated with the Arïk-Coon oscillator for q = q −1 > 1. These result may be used to introduce a q-deformed Ginibre-type point process.

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

A q-deformation of true-polyanalytic Bargmann transforms when q -1 >1

Moize¹,

Mouayn²

2022

Comptes Rendus. Mathématique

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“…In the introduction, we have mentioned the role of super-Poisson states respectively, sub-Poisson states in quantum optics. A mathematical model for distinguishing between super-Poisson states and sub-Poisson states is given by the Mandel functional, see for instance [5,6]. The characterization of Mandel's functional is associated with Jacobi operators in l 2 (N 0 ).…”

Section: Discussion Of Applications To Coherent State Theorymentioning

confidence: 99%

Using supermodels in quantum optics

Garbers¹,

Ruffing²

2006

Adv. Differ Equ.

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Starting from supersymmetric quantum mechanics and related supermodels within Schrödinger theory, we review the meaning of self-similar superpotentials which exhibit the spectrum of a geometric series. We construct special types of discretizations of the Schrödinger equation on time scales with particular symmetries. This discretization leads to the same type of point spectrum for the referred Schrödinger difference operator than in the self-similar superpotential case, hence exploiting an isospectrality situation. A discussion is opened on the question of how the considered energy sequence and its generalizations serve the understanding of coherent states in quantum optics.

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“…An intimate connection between the gravitation and the existence of the fundamental length scale was proposed in [109]. The minimal length has found to exist in string theory [110], loop quantum gravity [111], path integral quantum gravity [112], special relativity [113], doubly special relativity [114], coherent states [22][23][24][25][26][27][28][115][116][117][118][119][120][121][122],, etc. Furthermore, some thought experiments [109] in the spirit of black hole physics suggest that any theory of quantum gravity must be equipped with a minimum length scale [123], due to the fact that the energy required to probe any region of space below the Plank length is greater than the energy required to create a mini black hole in that region of space.…”

Section: A Noncommutative Quantum Mechanicsmentioning

confidence: 99%

Coherent States and Their Applications

2018

Springer Proceedings in Physics

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It was at the dawn of the historical developments of quantum mechanics when Schrödinger, Kennard and Darwin proposed an interesting type of Gaussian wave packets, which do not spread out while evolving in time. Originally, these wave packets are the prototypes of the renowned discovery, which are familiar as "coherent states" today. Coherent states are inevitable in the study of almost all areas of modern science, and the rate of progress of the subject is astonishing nowadays. Nonclassical states constitute one of the distinguished branches of coherent states having applications in various subjects including quantum information processing, quantum optics, quantum superselection principles and mathematical physics. On the other hand, the compelling advancements of non-Hermitian systems and related areas have been appealing, which became popular with the seminal paper by Bender and Boettcher in 1998. The subject of non-Hermitian Hamiltonian systems possessing real eigenvalues are exploding day by day and combining with almost all other subjects rapidly, in particular, in the areas of quantum optics, lasers and condensed matter systems, where one finds ample successful experiments for the proposed theory. For this reason, the study of coherent states for non-Hermitian systems have been very important. In this article, we review the recent developments of coherent and nonclassical states for such systems and discuss their applications and usefulness in different contexts of physics. In addition, since the systems considered here originate from the broader context of the study of minimal uncertainty relations, our review is also of interest to the mathematical physics community.

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Maths-type q-deformed coherent states for q>1

Cited by 49 publications

References 26 publications

A q-deformation of true-polyanalytic Bargmann transforms when q -1 >1

A q-deformation of true-polyanalytic Bargmann transforms when q -1 >1

Using supermodels in quantum optics

Coherent States and Their Applications

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