SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms

Atakishiyev, Natig M.; Kibler, M.; Wolf, Kurt Bernardo

doi:10.3390/sym2031461

Cited by 12 publications

(17 citation statements)

References 58 publications

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“…Quite recently, in connection with the generalized oscillator algebra reported in [35], a similar set of operators (with a negative number κ instead of the operatorκ op ) has been introduced in [36]. There, the authors show that the parameter κ defines the dimension of the representation d = 1 − 1/κ and gives rise to a phase factor in the relations (90).…”

Section: Spectrum Generating Algebra Of 'Diagonal' Hierarchiesmentioning

confidence: 99%

SU(1,1) and SU(2) app

2016

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It is shown that each one of the Lie algebras su(1, 1) and su(2) determine the spectrum of the radial oscillator. States that share the same orbital angular momentum are used to construct the representation spaces of the non-compact Lie group SU (1, 1). In addition, three different forms of obtaining the representation spaces of the compact Lie group SU (2) are introduced, they are based on the accidental degeneracies associated with the spherical symmetry of the system as well as on the selection rules that govern the transitions between different energy levels. In all cases the corresponding generalized coherent states are constructed and the conditions to squeeze the involved quadratures are analyzed.

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Section: Spectrum Generating Algebra Of 'Diagonal' Hierarchiesmentioning

confidence: 99%

SU(1,1) and SU(2) app

2016

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“…In the Pegg-Barnett formalism, all quantities are initially defined and calculated on the finite dimensional Hilbert space H N , and then the N → ∞ limit has to be taken. The justification of this formalism has been argued and tested experimentally, which leads to physically admissible results for various quantum states of the radiation field so far [20,21,22,23].…”

Section: Pegg-barnett's Phase Operatormentioning

confidence: 99%

Angles in fuzzy disc and angular noncommutative solitons

Kobayashi

Asakawa

2013

J. High Energ. Phys.

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The fuzzy disc, introduced by the authors of [1], is a disc-shaped region in a noncommutative plane, and is a fuzzy approximation of a commutative disc. In this paper we show that one can introduce a concept of angles to the fuzzy disc, by using the phase operator and phase states known in quantum optics. We gave a description of the fuzzy disc in terms of operators and their commutation relations, and studied properties of angular projection operators. A similar construction for the fuzzy annulus is also given. As an application, we constructed fan-shaped soliton solutions of a scalar field theory on the fuzzy disc. We also applied this concept to the theory of noncommutative gravity we proposed in [2]. In addition, possible connections to some systems in physics are suggested.

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“…Furthermore, the cases κ > 0 and κ < 0 describe the Pöschl-Teller system (described by the su(1, 1) algebra) and the Morse system (described by the su(2) algebra), respectively [25,27,28].…”

Section: Generalized Weyl-heisenberg Algebramentioning

confidence: 99%

Generalized coherent states for polynomial Weyl-Heisenberg algebras

Kibler¹,

Daoud²

2012

AIP Conference Proceedings

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It is the aim of this paper to show how to construct à la Perelomov and à la Barut-Girardello coherent states for a polynomial Weyl-Heisenberg algebra. This algebra depends on r parameters. For some special values of the parameter corresponding to r = 1, the algebra covers the cases of the su(1,1) algebra, the su(2) algebra and the ordinary Weyl-Heisenberg or oscillator algebra. For r arbitrary, the generalized Weyl-Heisenberg algebra admits finite or infinite-dimensional representations depending on the values of the parameters. Coherent states of the Perelomov type are derived in finite and infinite dimensions through a Fock-Bargmann approach based on the use of complex variables. The same approach is applied for deriving coherent states of the Barut-Girardello type in infinite dimension. In contrast, the construction of à la Barut-Girardello coherent states in finite dimension can be achieved solely at the price to replace complex variables by generalized Grassmann variables. Finally, some preliminary developments are given for the study of Bargmann functions associated with some of the coherent states obtained in this work.

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SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms

Cited by 12 publications

References 58 publications

SU(1,1) and SU(2) app

SU(1,1) and SU(2) app

Angles in fuzzy disc and angular noncommutative solitons

Generalized coherent states for polynomial Weyl-Heisenberg algebras

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