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

A Quantum Group Approach to some Exotic States in Quantum Optics

Roger McDermott

Published Jan 1, 1995• 0 citations
Publisher

Abstract

This subject of this thesis is the physical application of deformations of Lie algebras and their use in generalising some exotic quantum optical states. We begin by examining the theory of quantum groups and the q-boson algebras used in their representation theory. Following a review of the properties of conventional coherent states, we describe the extension of the theory to various deformed Heisenberg-Weyl algebras, as well as the q-deformations of su(2) and su(1,1). Using the Deformed Oscillator Algebra of Bonatsos and Daskaloyannis, we construct generalised deformed coherent states and investigate some of their quantum optical properties. We then demonstrate a resolution of unity for such states and suggest a way of investigating the geometric effects of the deformation. The formalism devised by Rembielinski et al is used to consider coherent states of the q-boson algebra over the quantum complex plane. We propose a new unitary operator which is a q-analogue of the displacement operator of conventional coherent state theory: This is used to construct q-displaced vacuum states which are eigenstates of the annihilation operator. Some quantum mechanical properties of these states are investigated and it is shown that they formally satisfy a Heisenberg-type minimum uncertainty relation. After briefly reviewing the theory of conventional squeezed states, we examine the various q-generalisations. We propose a q-analogue of the squeezed vacuum state, and use this in conjunction with the unitary q-displacement operator to construct a general q-squeezed state, parameterised by noncommuting variables.. It is shown that, like their conventional counterparts, such states satisfy the Robertson-Schrodinger Uncertainty Relation. We conclude with a brief discussion about the appearance of noncommuting variables in the states that have been considered.

DOI
10.21954/ou.ro.0000fb71

Related papers

Quantum optical states and Bose-Einstein condensation : a dynamical group approach

Yinqi Feng

The concept of coherent states for a quantum system has been generalized in many different ways. One elegant way is the dynamical group approach. The subject of this thesis is the physical application of some dynamical group methods in quantum optics and Bose-Einstein Condensation(BEC) and their use in generalizing some quantum optical states and BEC states. We start by generalizing squeezed coherent states to the displaced squeezed phase number states and studying the signal-to-quantum noise ratio for these states. Following a review of the properties of Kerr states and the basic theory of the deformation of the boson algebra, we present an algebraic approach to Kerr states and generalize them to the squeezed states of the q-parametrized harmonic oscillator. Using the eigenstates of a nonlinear density-dependent annihilation operator of the deformed boson algebra, we propose general time covariant coherent states for any time-independent quantum system. Using the ladder operator approach similar to that of binomial states, we construct interpolating number-coherent states, intermediate states which are generalizations of some fundamental states in quantum optics. Salient statistical properties and non-classical features of these interpolating numbercoherent states are investigated and the interaction with an atomic system in the framework of the Jaynes-Cummings model and the scheme to produce these states are also studied in detail. After briefly reviewing the realization of Bose-Einstein Condensates and relevant theoretical research using mean-field theory, we present a dynamical group approach to Bose-Einstein condensation and the atomic tunnelling between two condensates which interact via a minimal coupling term. First we consider the spectrum of one Bose-Einstein condensate and show that the mean-field dynamics is characterised by the semi-direct product of the 8U(1,1) and Heisenberg-Weyl groups.We then construct a generalized version of the BEC ground states and weakly excited states. It is shown that our states for BEC provide better fits to the experimental results. Then we investigate the tunnelling between the excitations in two condensates which interact via a minimal coupling term. The dynamics of the two interacting Bose systems is characterised by the 80(3,2) group, which leads to an exactly solvable model. Further we describe the dynamics of the tunnelling of the two coupled condensates in terms of the semi-direct product of 80(3,2) and two independent Heisenberg-Weyl groups. From this we obtain the energy spectrum and eigenstates for the two interacting Bose-Einstein condensates, as well as the Josephson current between the two coupled condensates.

•Jan 1, 2001•0 citations

From operators to groups: a dual perspective on coherent and squeezed states

Arash Azizi

Abstract These lecture notes provide a comprehensive exploration of coherent and squeezed states, which are central to quantum optics and continuous-variable quantum information. We begin with the traditional formalism, where displacement and squeezing operators are defined via the creation and annihilation operators. Coherent states are presented as quantum analogues of classical fields with minimal uncertainty, while squeezed states are introduced as nonclassical resources that reduce quantum noise below the vacuum limit, enabling next-generation quantum technologies. A central feature of these notes is a parallel development from a group-theoretical standpoint. Here, these operators are defined abstractly by their algebraic properties, revealing their structure as bosonic realizations of the Weyl–Heisenberg and $$SU(1,1)$$ S U ( 1 , 1 ) structures, respectively. This advanced perspective not only provides a more rigorous mathematical foundation but also leads to new insights, including a proof of the uniqueness of the single-mode squeezing operator and a no-go theorem for certain multi-mode squeezing configurations. Designed for both pedagogical clarity and advanced study, these notes bridge the foundational operator algebra of quantum optics with the modern group-theoretical structures essential for research in continuous-variable quantum systems.

The European Physical Journal Plus•Jun 9, 2026•0 citations

Generalized q-bosons and their squeezed states

Jacob Katriel, A. I. Solomon

Generalized q-boson operators associated with the simultaneous creation of several q-bosons are introduced. These operators give rise to a realization of the quantum Weyl-Heisenberg algebra and are applied to the construction of corresponding Holstein-Primakoff realizations of the quantum algebras SU q (2) and SU q (1,1). Coherent states of these algebras are defined in the various ways suggested by the equivalent definitions of the harmonic oscillator coherent states, and some of their properties are studied. Particular attention is devoted to the squeezing properties of the quadrature of the electromagnetic field in these states.

Journal of Physics A Mathematical and General•May 1, 1991•86 citations

Quon theories in quantum optics

A. I. Solomon

We start from the viewpoint of quantum canonical transformations, analogues of the usual classical canonical transformations. We show that in the quantum case, such transformations lead naturally to the definition of some important quantum optics states, including the vacuum, coherent states, squeezed vacuum and squeezed states. We then go on to consider quons, operators satisfying deformed versions of the canonical commutation relations. Extending the idea of a canonical transformation to deformed quantum commutators gives quon analogues of these states, quon states. We finally show that this approach provides us with defining relations for some quantum groups.

AIP conference proceedings•Jan 1, 2000•0 citations

The quantum multiboson algebra and generalized coherent states of SUq(1,1)

Zhe Chang

The author gives a generalization of the deformed oscillator and shows that it satisfies the quantum Weyl-Heisenberg group. The nonlinear transformations from the usual deformed oscillator operators to the deformed k-boson operators form an Abelian group. The generalized Holstein-Primakoff realizations in terms of these multiboson operators are used to define the quantum group-theoretic coherent states of SU q (1,1).

Journal of Physics A Mathematical and General•Jun 21, 1992•10 citations

Coherent and squeezed states of quantum Heisenberg algebras

Nibaldo Alvarez-Moraga

Starting from deformed quantum Heisenberg Lie algebras some realizations are given in terms of the usual creation and annihilation operators of the standard harmonic oscillator. Then the associated algebra eigenstates are computed and give rise to new classes of deformed coherent and squeezed states. They are parametrized by deformed algebra parameters and suitable redefinitions of them as paragrassmann numbers. Some properties of these deformed states are also analysed.

arXiv:math-ph/0503055•Journal of Physics A Mathematical and General•Mar 3, 2005•4 citations
This subject of this thesis is the physical application of deformations of Lie algebras and their use in generalising some exotic quantum optical states. We begin by examining the theory of quantum groups and the q-boson algebras used in their representation theory. Following a review of the properties of conventional coherent states, we describe the extension of the theory to various deformed Heisenberg-Weyl algebras, as well as the q-deformations of su(2) and su(1,1). Using the Deformed Oscillator Algebra of Bonatsos and Daskaloyannis, we construct generalised deformed coherent states and investigate some of their quantum optical properties. We then demonstrate a resolution of unity for such states and suggest a way of investigating the geometric effects of the deformation. The formalism devised by Rembielinski et al is used to consider coherent states of the q-boson algebra over the quantum complex plane. We propose a new unitary operator which is a q-analogue of the displacement operator of conventional coherent state theory: This is used to construct q-displaced vacuum states which are eigenstates of the annihilation operator. Some quantum mechanical properties of these states are investigated and it is shown that they formally satisfy a Heisenberg-type minimum uncertainty relation. After briefly reviewing the theory of conventional squeezed states, we examine the various q-generalisations. We propose a q-analogue of the squeezed vacuum state, and use this in conjunction with the unitary q-displacement operator to construct a general q-squeezed state, parameterised by noncommuting variables.. It is shown that, like their conventional counterparts, such states satisfy the Robertson-Schrodinger Uncertainty Relation. We conclude with a brief discussion about the appearance of noncommuting variables in the states that have been considered.
The concept of coherent states for a quantum system has been generalized in many different ways. One elegant way is the dynamical group approach. The subject of this thesis is the physical application of some dynamical group methods in quantum optics and Bose-Einstein Condensation(BEC) and their use in generalizing some quantum optical states and BEC states. We start by generalizing squeezed coherent states to the displaced squeezed phase number states and studying the signal-to-quantum noise ratio for these states. Following a review of the properties of Kerr states and the basic theory of the deformation of the boson algebra, we present an algebraic approach to Kerr states and generalize them to the squeezed states of the q-parametrized harmonic oscillator. Using the eigenstates of a nonlinear density-dependent annihilation operator of the deformed boson algebra, we propose general time covariant coherent states for any time-independent quantum system. Using the ladder operator approach similar to that of binomial states, we construct interpolating number-coherent states, intermediate states which are generalizations of some fundamental states in quantum optics. Salient statistical properties and non-classical features of these interpolating numbercoherent states are investigated and the interaction with an atomic system in the framework of the Jaynes-Cummings model and the scheme to produce these states are also studied in detail. After briefly reviewing the realization of Bose-Einstein Condensates and relevant theoretical research using mean-field theory, we present a dynamical group approach to Bose-Einstein condensation and the atomic tunnelling between two condensates which interact via a minimal coupling term. First we consider the spectrum of one Bose-Einstein condensate and show that the mean-field dynamics is characterised by the semi-direct product of the 8U(1,1) and Heisenberg-Weyl groups.We then construct a generalized version of the BEC ground states and weakly excited states. It is shown that our states for BEC provide better fits to the experimental results. Then we investigate the tunnelling between the excitations in two condensates which interact via a minimal coupling term. The dynamics of the two interacting Bose systems is characterised by the 80(3,2) group, which leads to an exactly solvable model. Further we describe the dynamics of the tunnelling of the two coupled condensates in terms of the semi-direct product of 80(3,2) and two independent Heisenberg-Weyl groups. From this we obtain the energy spectrum and eigenstates for the two interacting Bose-Einstein condensates, as well as the Josephson current between the two coupled condensates.
Abstract These lecture notes provide a comprehensive exploration of coherent and squeezed states, which are central to quantum optics and continuous-variable quantum information. We begin with the traditional formalism, where displacement and squeezing operators are defined via the creation and annihilation operators. Coherent states are presented as quantum analogues of classical fields with minimal uncertainty, while squeezed states are introduced as nonclassical resources that reduce quantum noise below the vacuum limit, enabling next-generation quantum technologies. A central feature of these notes is a parallel development from a group-theoretical standpoint. Here, these operators are defined abstractly by their algebraic properties, revealing their structure as bosonic realizations of the Weyl–Heisenberg and SU(1,1)SU(1,1)SU(1,1) S U ( 1 , 1 ) structures, respectively. This advanced perspective not only provides a more rigorous mathematical foundation but also leads to new insights, including a proof of the uniqueness of the single-mode squeezing operator and a no-go theorem for certain multi-mode squeezing configurations. Designed for both pedagogical clarity and advanced study, these notes bridge the foundational operator algebra of quantum optics with the modern group-theoretical structures essential for research in continuous-variable quantum systems.
Generalized q-boson operators associated with the simultaneous creation of several q-bosons are introduced. These operators give rise to a realization of the quantum Weyl-Heisenberg algebra and are applied to the construction of corresponding Holstein-Primakoff realizations of the quantum algebras SU q (2) and SU q (1,1). Coherent states of these algebras are defined in the various ways suggested by the equivalent definitions of the harmonic oscillator coherent states, and some of their properties are studied. Particular attention is devoted to the squeezing properties of the quadrature of the electromagnetic field in these states.
We start from the viewpoint of quantum canonical transformations, analogues of the usual classical canonical transformations. We show that in the quantum case, such transformations lead naturally to the definition of some important quantum optics states, including the vacuum, coherent states, squeezed vacuum and squeezed states. We then go on to consider quons, operators satisfying deformed versions of the canonical commutation relations. Extending the idea of a canonical transformation to deformed quantum commutators gives quon analogues of these states, quon states. We finally show that this approach provides us with defining relations for some quantum groups.