On completeness of coherent states in noncommutative spaces with the generalised uncertainty principle

Dey, Sanjib

doi:10.1007/978-3-319-69164-0_21

Cited by 5 publications

(5 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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: Beam Splitter Entanglementmentioning

confidence: 99%

“…Keeping aside the rapid progress of the subject of coherent states itself, in this article we intend to provide a concise review on the developments of coherent states for non-Hermitian systems primarily based on [22][23][24][25][26][27][28]. Specifically, the systems that we study here originate form the noncommutative (NC) quantum mechanical structure with minimal length associated with the generalized uncertainty principle.…”

mentioning

confidence: 99%

See 1 more Smart Citation

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

Dey

Fring

Hussin

2018

Springer Proceedings in Physics

Self Cite

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. 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-

show abstract

Section: Beam Splitter Entanglementmentioning

confidence: 99%

mentioning

confidence: 99%

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

Dey

Fring

Hussin

2018

Springer Proceedings in Physics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Also, it is worth mentioning that in order to claim a well-defined quantum optical state, one has to study all the necessary properties of the states individually. For example, the construction of coherent states requires the resolution of identity; see, for instance [99], for further details. Nevertheless, the method of generalization emerging from a nonlinear function f (n) consisting of the creation and annihilation operators is widely known as the nonlinear generalization [100][101][102][103] and, nowadays, the endeavor is widely established; see, for example [20,22,[104][105][106].…”

Section: Nonlinear Generalizationmentioning

confidence: 99%

An introductory review on resource theories of generalized nonclassical light

Dey

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Quantum resource theory is perhaps the most revolutionary framework that quantum physics has ever experienced. It plays vigorous roles in unifying the quantification methods of a requisite quantum effect as wells as in identifying protocols that optimize its usefulness in a given application in areas ranging from quantum information to computation. Moreover, the resource theories have transmuted radical quantum phenomena like coherence, nonclassicality and entanglement from being just intriguing to being helpful in executing realistic thoughts. A general quantum resource theoretical framework relies on the method of categorization of all possible quantum states into two sets, namely, the free set and the resource set. Associated with the set of free states there is a number of free quantum operations emerging from the natural constraints attributed to the corresponding physical system. Then, the task of quantum resource theory is to discover possible aspects arising from the restricted set of operations as resources. Along with the rapid growth of various resource theories corresponding to standard harmonic oscillator quantum optical states, significant advancement has been expedited along the same direction for generalized quantum optical states. Generalized quantum optical framework strives to bring in several prosperous contemporary ideas including nonlinearity, PT -symmetric non-Hermitian theories, q-deformed bosonic systems, etc., to accomplish similar but elevated objectives of the standard quantum optics and information theories. In this article, we review the developments of nonclassical resource theories of different generalized quantum optical states and their usefulness in the context of quantum information theories.

show abstract

“…and |q| = 1 (for which R w = ∞ as we have already seen), we can then take ρ(r) = π −1 exp(−r 2 ) or, equivalently ρ(t 1/2 ) = π −1 e −t , by standard identities from integral calculus. See [10] for other examples in a non-commutative setting. Of course, after successfully finding a function ρ solving the moment problem (4.6), one still has to check that the possibly stronger condition (4.2) holds.…”

Section: Resolution Of the Identitymentioning

confidence: 99%

Coherent States for the Manin Plane via Toeplitz Quantization

Durdevich,

Sontz

2019

Preprint

View full text Add to dashboard Cite

In the theory of Toeplitz quantization of algebras, as developed by the second author, coherent states are defined as eigenvectors of a Toeplitz annihilation operator. These coherent states are studied in the case when the algebra is the generically non-commutative Manin plane. In usual quantization schemes one starts with a classical phase space, then quantizes it in order to produce annihilation operators and then their eigenvectors and eigenvalues. But we do this in the opposite order, namely the set of the eigenvalues of the previously defined annihilation operator is identified as a generalization of a classical mechanical phase space. We introduce the resolution of the identity, upper and lower symbols as well as a coherent state quantization, which in turn quantizes the Toeplitz quantization. We thereby have a curious combination of quantization schemes which might be a novelty. We proceed by identifying a generalized Segal-Bargmann space SB of square-integrable, anti-holomorphic functions as the image of a coherent state transform. Then SB has a reproducing kernel function which allows us to define a secondary Toeplitz quantization, whose symbols are functions. Finally, this is compared with the coherent states of the Toeplitz quantization of a closely related noncommutative space known as the paragrassmann algebra.

show abstract

On completeness of coherent states in noncommutative spaces with the generalised uncertainty principle

Cited by 5 publications

References 31 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

An introductory review on resource theories of generalized nonclassical light

Coherent States for the Manin Plane via Toeplitz Quantization

Contact Info

Product

Resources

About