Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1708.05103 · Advances in Mathematical Physics

Completeness and Nonclassicality of Coherent States for Generalized Oscillator Algebras

Kevin Zelaya, Oscar Rosas-Ortiz, Zurika Blanco-Garcia, Sara Cruz y Cruz

Published Jan 1, 2017• 15 citations
arXivHTMLPDF

Abstract

The purposes of this work are (1) to show that the appropriate generalizations of the oscillator algebra permit the construction of a wide set of nonlinear coherent states in unified form and (2) to clarify the likely contradiction between the nonclassical properties of such nonlinear coherent states and the possibility of finding a classical analog for them since they are P -represented by a delta function. In (1) we prove that a class of nonlinear coherent states can be constructed to satisfy a closure relation that is expressed uniquely in terms of the Meijer G -function. This property automatically defines the delta distribution as the P -representation of such states. Then, in principle, there must be a classical analog for them. Among other examples, we construct a family of nonlinear coherent states for a representation of the su(1,1) Lie algebra that is realized as a deformation of the oscillator algebra. In (2) , we use a beam splitter to show that the nonlinear coherent states exhibit properties like antibunching that prohibit a classical description for them. We also show that these states lack second-order coherence. That is, although the P -representation of the nonlinear coherent states is a delta function, they are not full coherent. Therefore, the systems associated with the generalized oscillator algebras cannot be considered “classical” in the context of the quantum theory of optical coherence.

DOI
10.1155/2017/7168592

Related papers

Completeness and Nonclassicality of Coherent States for Generalized Oscillator Algebras

Kevin Zelaya, Rosas-Ortiz, O., Blanco-Garcia, Z. +1 more

The purposes of this work are to show that the appropriate generalizations of the oscillator algebra permit the construction of a wide set of nonlinear coherent states in unified form and to clarify the likely contradiction between the nonclassical properties of such nonlinear coherent states and the possibility of finding a classical analog for them since they are -represented by a delta function. In we prove that a class of nonlinear coherent states can be constructed to satisfy a closure relation that is expressed uniquely in terms of the Meijer -function. This property automatically defines the delta distribution as the -representation of such states. Then, in principle, there must be a classical analog for them. Among other examples, we construct a family of nonlinear coherent states for a representation of the Lie algebra that is realized as a deformation of the oscillator algebra. In , we use a beam splitter to show that the nonlinear coherent states exhibit properties like antibunching that prohibit a classical description for them. We also show that these states lack second-order coherence. That is, although the -representation of the nonlinear coherent states is a delta function, they are not full coherent. Therefore, the systems associated with the generalized oscillator algebras cannot be considered “classical†in the context of the quantum theory of optical coherence.

•Jan 1, 2017•9 citations

ON A GENERAL FORMALISM OF NONLINEAR CHARGE COHERENT STATES, THEIR QUANTUM STATISTICS AND NONCLASSICAL PROPERTIES

F. EFTEKHARI, M. K. TAVASSOLY

In this paper, we will present a general formalism for constructing the nonlinear charge coherent states which in special case lead to the standard charge coherent states. The su Q(1, 1) algebra as a nonlinear deformed algebra realization of the introduced states is established. In addition, the corresponding even and odd nonlinear charge coherent states have also been introduced. The formalism has the potentiality to be applied to systems either with known "nonlinearity function" f(n) or solvable quantum system with known "discrete nondegenerate spectrum" en. As some physical appearances, a few known physical systems in the two mentioned categories have been considered. Finally, since the construction of nonclassical states is a central topic of quantum optics, nonclassical features and quantum statistical properties of the introduced states have been investigated by evaluating single- and two-mode squeezing, su (1, 1)-squeezing, Mandel parameter and antibunching effect (via g-correlation function) as well as some of their generalized forms we have introduced in the present paper.

arXiv:1011.3605•International Journal of Modern Physics A•Jul 8, 2010•10 citations

SU(1,1) Nonlinear Coherent States

B. Mojaveri, Alireza Dehghani

The idea of construction of the nonlinear coherent states based on the hypergeometric- type operators associated to the Weyl-Heisenberg group [J:P hys:A 45(2012) 095304], are generalized to the similar states for the arbitrary Lie group SU(1, 1). By using of a discrete, unitary and irreducible representation of the Lie algebra su(1, 1) wide range of generalized nonlinear coherent states(GNCS) have been introduced, which admit a resolution of the identity through positive definite measures on the complex plane. We have shown that realization of these states for different values of the deformation pa- rameters r = 1 and 2 lead to the well-known Klauder-Perelomov and Barut-Girardello coherent states associated to the Irreps of the Lie algebra su(1, 1), respectively. It is worth to mention that, like the canonical coherent states, GNCS possess the temporal stability property. Finally, studying some statistical characters implies that they have indeed nonclassical features such as squeezing, anti-bunching effect and sub-Poissonian statistics, too.

arXiv:1212.6888•Dec 31, 2012•0 citations

An observation of quadratic algebra, dual family of nonlinear coherent states and their non-classical properties, in the generalized isotonic oscillator

V. Chithiika Ruby, Murugaian Senthilvelan

In this paper, we construct nonlinear coherent states for the generalized isotonic oscillator and study their non-classical properties in detail. By transforming the deformed ladder operators suitably, which generate the quadratic algebra, we obtain Heisenberg algebra. From the algebra, we define two non-unitary and one unitary displacement type operators. While the action of one of the non-unitary type operators reproduces the original nonlinear coherent states, the other one fails to produce a new set of nonlinear coherent states (dual pair). We show that these dual states are not normalizable. For the nonlinear coherent states, we evaluate the parameter A3 and examine the non-classical nature of the states through quadratic and amplitude-squared squeezing effect. Further, we derive analytical formula for the P-function, Q-function, and the Wigner function for the nonlinear coherent states. All of them confirm the non-classicality of the nonlinear coherent states. In addition to the above, we obtain the harmonic oscillator type coherent states from the unitary displacement operator.

arXiv:1202.5104•Journal of Mathematical Physics•Aug 1, 2012•6 citations

On the generalized intelligent states and certain related nonclassical states of a quantum exactly solvable nonlinear oscillator

V. Chithiika Ruby, Murugaian Senthilvelan

We construct nonlinear coherent states or f -deformed coherent states for a nonpolynomial nonlinear oscillator which can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator (Cariñena et al 2008 J. Phys. A: Math. Theor. 41 085301 ). The deformed annihilation and creation operators which are required to construct the nonlinear coherent states in the number basis are obtained from the solution of the Schrödinger equation. Using these operators, we construct generalized intelligent states, nonlinear coherent states, Gazeau–Klauder coherent states and the even and odd nonlinear coherent states for this newly solvable system. We also report certain nonclassical properties exhibited by these nonlinear coherent states. In addition to the above, we consider the position-dependent mass Schrödinger equation associated with this solvable nonlinear oscillator and construct nonlinear coherent states, Gazeau–Klauder coherent states and the even and odd nonlinear coherent states for it. We also give explicit expressions of all these nonlinear coherent states by considering a mass profile which is often used for studying transport properties in semiconductors.

arXiv:1008.4041•Journal of Physics A Mathematical and Theoretical•Sep 21, 2010•10 citations

Phase-space behavior of physical nonlinear coherent states

Mustapha Ziane, Abdessamad Belfakir, Erik Díaz-Bautista +2 more

Abstract In this paper, we construct the generalized su(1, 1) nonlinear coherent states for particular physical systems. We study the behavior of the Wigner function for coherent states built from differential operators that satisfy either a generalized Heisenberg algebra or a generalized su(1, 1) nonlinear algebra. The systems studied are the harmonic oscillator and those ones associated with Pöschl-Teller potentials. The results show that the Wigner function for the associated nonlinear coherent states is positive for small values of their convergence radius. Furthermore, the Wigner function for these coherent states depends on the corresponding algebraic structure.

Physica Scripta•Oct 22, 2020•0 citations
The purposes of this work are (1) to show that the appropriate generalizations of the oscillator algebra permit the construction of a wide set of nonlinear coherent states in unified form and (2) to clarify the likely contradiction between the nonclassical properties of such nonlinear coherent states and the possibility of finding a classical analog for them since they are P -represented by a delta function. In (1) we prove that a class of nonlinear coherent states can be constructed to satisfy a closure relation that is expressed uniquely in terms of the Meijer G -function. This property automatically defines the delta distribution as the P -representation of such states. Then, in principle, there must be a classical analog for them. Among other examples, we construct a family of nonlinear coherent states for a representation of the su(1,1) Lie algebra that is realized as a deformation of the oscillator algebra. In (2) , we use a beam splitter to show that the nonlinear coherent states exhibit properties like antibunching that prohibit a classical description for them. We also show that these states lack second-order coherence. That is, although the P -representation of the nonlinear coherent states is a delta function, they are not full coherent. Therefore, the systems associated with the generalized oscillator algebras cannot be considered “classical” in the context of the quantum theory of optical coherence.
The purposes of this work are to show that the appropriate generalizations of the oscillator algebra permit the construction of a wide set of nonlinear coherent states in unified form and to clarify the likely contradiction between the nonclassical properties of such nonlinear coherent states and the possibility of finding a classical analog for them since they are -represented by a delta function. In we prove that a class of nonlinear coherent states can be constructed to satisfy a closure relation that is expressed uniquely in terms of the Meijer -function. This property automatically defines the delta distribution as the -representation of such states. Then, in principle, there must be a classical analog for them. Among other examples, we construct a family of nonlinear coherent states for a representation of the Lie algebra that is realized as a deformation of the oscillator algebra. In , we use a beam splitter to show that the nonlinear coherent states exhibit properties like antibunching that prohibit a classical description for them. We also show that these states lack second-order coherence. That is, although the -representation of the nonlinear coherent states is a delta function, they are not full coherent. Therefore, the systems associated with the generalized oscillator algebras cannot be considered “classical†in the context of the quantum theory of optical coherence.
In this paper, we will present a general formalism for constructing the nonlinear charge coherent states which in special case lead to the standard charge coherent states. The su Q(1, 1) algebra as a nonlinear deformed algebra realization of the introduced states is established. In addition, the corresponding even and odd nonlinear charge coherent states have also been introduced. The formalism has the potentiality to be applied to systems either with known "nonlinearity function" f(n) or solvable quantum system with known "discrete nondegenerate spectrum" en. As some physical appearances, a few known physical systems in the two mentioned categories have been considered. Finally, since the construction of nonclassical states is a central topic of quantum optics, nonclassical features and quantum statistical properties of the introduced states have been investigated by evaluating single- and two-mode squeezing, su (1, 1)-squeezing, Mandel parameter and antibunching effect (via g-correlation function) as well as some of their generalized forms we have introduced in the present paper.
The idea of construction of the nonlinear coherent states based on the hypergeometric- type operators associated to the Weyl-Heisenberg group [J:P hys:A 45(2012) 095304], are generalized to the similar states for the arbitrary Lie group SU(1, 1). By using of a discrete, unitary and irreducible representation of the Lie algebra su(1, 1) wide range of generalized nonlinear coherent states(GNCS) have been introduced, which admit a resolution of the identity through positive definite measures on the complex plane. We have shown that realization of these states for different values of the deformation pa- rameters r = 1 and 2 lead to the well-known Klauder-Perelomov and Barut-Girardello coherent states associated to the Irreps of the Lie algebra su(1, 1), respectively. It is worth to mention that, like the canonical coherent states, GNCS possess the temporal stability property. Finally, studying some statistical characters implies that they have indeed nonclassical features such as squeezing, anti-bunching effect and sub-Poissonian statistics, too.
In this paper, we construct nonlinear coherent states for the generalized isotonic oscillator and study their non-classical properties in detail. By transforming the deformed ladder operators suitably, which generate the quadratic algebra, we obtain Heisenberg algebra. From the algebra, we define two non-unitary and one unitary displacement type operators. While the action of one of the non-unitary type operators reproduces the original nonlinear coherent states, the other one fails to produce a new set of nonlinear coherent states (dual pair). We show that these dual states are not normalizable. For the nonlinear coherent states, we evaluate the parameter A3 and examine the non-classical nature of the states through quadratic and amplitude-squared squeezing effect. Further, we derive analytical formula for the P-function, Q-function, and the Wigner function for the nonlinear coherent states. All of them confirm the non-classicality of the nonlinear coherent states. In addition to the above, we obtain the harmonic oscillator type coherent states from the unitary displacement operator.
We construct nonlinear coherent states or f -deformed coherent states for a nonpolynomial nonlinear oscillator which can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator (Cariñena et al 2008 J. Phys. A: Math. Theor. 41 085301 ). The deformed annihilation and creation operators which are required to construct the nonlinear coherent states in the number basis are obtained from the solution of the Schrödinger equation. Using these operators, we construct generalized intelligent states, nonlinear coherent states, Gazeau–Klauder coherent states and the even and odd nonlinear coherent states for this newly solvable system. We also report certain nonclassical properties exhibited by these nonlinear coherent states. In addition to the above, we consider the position-dependent mass Schrödinger equation associated with this solvable nonlinear oscillator and construct nonlinear coherent states, Gazeau–Klauder coherent states and the even and odd nonlinear coherent states for it. We also give explicit expressions of all these nonlinear coherent states by considering a mass profile which is often used for studying transport properties in semiconductors.
Abstract In this paper, we construct the generalized su(1, 1) nonlinear coherent states for particular physical systems. We study the behavior of the Wigner function for coherent states built from differential operators that satisfy either a generalized Heisenberg algebra or a generalized su(1, 1) nonlinear algebra. The systems studied are the harmonic oscillator and those ones associated with Pöschl-Teller potentials. The results show that the Wigner function for the associated nonlinear coherent states is positive for small values of their convergence radius. Furthermore, the Wigner function for these coherent states depends on the corresponding algebraic structure.