Quantum deformations and<i>q</i>-boson operators

Jarvis, Peter; Lohe, M. A.

doi:10.1088/1751-8113/49/43/431001

“…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%

A Squeezed Review on Coherent States and Nonclassicality for Non-Hermitian Systems with Minimal Length

Dey

¹

,

Fring

²

,

Hussin

³

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. CONTENTS* dey@iisermohali.ac.in; sanjibdey4@gmail.com † a.fring@city.ac.uk ‡ veronique.hussin@umontreal.ca A. Squeezed states 11 B. Schrödinger cat states 12 C. Photon-added coherent states 13 VI. Applications 13 VII. Concluding remarks 13 References 14 arXiv:1801.01139v1 [quant-ph] 3 Jan 2018 III. NON-HERMTIAN SYSTEMS IN MINIMAL LENGTH SCENARIOHaving introduced the general features of coherent states and nonclassicality, in this section, let us discuss a system on which our article is based on, i.e. some non-Hermitian models based on quantum mechanics in NC space and, then, we discuss how these models can be applied to physical systems like coherent states. A. Noncommutative quantum mechanicsThe original proposal of space-time noncommutativity is very old and was introduced in the pioneering days of quan-

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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

View full text Add to dashboard Cite

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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“…The q-deformation is also used to fit the theoretical results to the experimental ones in nuclear and atomic physics [48][49][50][51][52][53], and Sviratcheva et al revealed the significant role of q-deformation in understanding higherorder effects in the many-body interactions [54]. Various dynamic equations of NLSE type with q-deformed potentials have been proposed to model q-deformed processes in nuclear and atomic physics (see i.e.…”

Section: Introductionmentioning

confidence: 99%

Self-localized Solitons of a q-Deformed Quantum System

Bayindir,

Altintas,

Ozaydin

2020

Preprint

0

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Beyond a pure mathematical interest, q-deformation is promising for the modeling and interpretation of various physical phenomena. In this paper, we numerically investigate the existence and properties of the self-localized soliton solutions of the nonlinear Schrödinger equation (NLSE) with a q-deformed Rosen-Morse potential. By implementing a Petviashvili method (PM), we obtain the self-localized one and two soliton solutions of the NLSE with a q-deformed Rosen-Morse potential. In order to investigate the temporal behavior and stabilities of these solitons, we implement a Fourier spectral method with a 4 th order Runge-Kutta time integrator. We observe that the self-localized one and two solitons are stable and remain bounded with a pulsating behavior and minor changes in the sidelobes of the soliton waveform. Additionally, we investigate the stability and robustness of these solitons under noisy perturbations. A sinusoidal monochromatic wave field modeled within the frame of the NLSE with a q-deformed Rosen-Morse potential turns into a chaotic wavefield and exhibits rogue oscillations due to modulation instability triggered by noise, however, the selflocalized solitons of the NLSE with a q-deformed Rosen-Morse potential are stable and robust under the effect of noise. We also show that soliton profiles can be reconstructed after a denoising process performed using a Savitzky-Golay filter.

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Quantum deformations andq-boson operators

Cited by 8 publications

References 71 publications

A Squeezed Review on Coherent States and Nonclassicality for Non-Hermitian Systems with Minimal Length

A Squeezed Review on Coherent States and Nonclassicality for Non-Hermitian Systems with Minimal Length

Coherent States and Their Applications

Self-localized Solitons of a q-Deformed Quantum System

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