Showing papers on "Coherent states in mathematical physics published in 1996"
TL;DR: In this article, the coherent states for a quantum particle on a circle are introduced and the Bargmann representation within the actual treatment provides the representation of the algebra, where U is unitary, which is a direct consequence of the Heisenberg algebra.
Abstract: The coherent states for a quantum particle on a circle are introduced. The Bargmann representation within the actual treatment provides the representation of the algebra , where U is unitary, which is a direct consequence of the Heisenberg algebra , but it is more adequate for the study of circular motion.
140 citations
TL;DR: In this paper, a weak resolution of the identity in terms of the Perelomov SU(1,1) coherent states is presented which is valid even when the Bargmann index k is smaller than.
Abstract: We consider two analytic representations of the SU(1,1) Lie group: the representation in the unit disc based on the SU(1,1) Perelomov coherent states and the Barut - Girardello representation based on the eigenstates of the SU(1,1) lowering generator. We show that these representations are related through a Laplace transform. A `weak' resolution of the identity in terms of the Perelomov SU(1,1) coherent states is presented which is valid even when the Bargmann index k is smaller than . Various applications of these results in the context of the two-photon realization of SU(1,1) in quantum optics are also discussed.
73 citations
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TL;DR: In this paper, a canonical basis of the Fock space representation of affine Lie algebras is defined, and the entries of the transition matrix between this basis and the natural basis of fock space are analogues of decomposition numbers of the $v$-Schur algesas specialized to a n-th root of unity.
Abstract: We define a canonical basis of the $q$-deformed Fock space representation of the affine Lie algebra $\glchap_n$. We conjecture that the entries of the transition matrix between this basis and the natural basis of the Fock space are $q$-analogues of decomposition numbers of the $v$-Schur algebras for $v$ specialized to a $n$th root of unity.
71 citations
TL;DR: In this article, a general scheme for the wedge construction of q-deformed Fock spaces using the theory of perfect crystals is presented, and a tensor product of an irreducible Uq (An(1))-module and a bosonic Fock space is given.
Abstract: In [S], [KMS] the semi-infinite wedge construction of level 1Uq (An(1)) Fock spaces and their decomposition into the tensor product of an irreducibleUq (An(1))-module and a bosonic Fock space were given. Here a general scheme for the wedge construction ofq-deformed Fock spaces using the theory of perfect crystals is presented.
61 citations
TL;DR: In this article, the quantum Weyl-Heisenberg algebra was used to formalize the coherent states of the von Neumann lattice. But the quantum mechanics formalism for lattice systems has not yet been studied.
Abstract: By resorting to the Fock--Bargmann representation, we incorporate the quantum Weyl--Heisenberg ($q$-WH) algebra into the theory of entire analytic functions. The main tool is the realization of the $q$--WH algebra in terms of finite difference operators. The physical relevance of our study relies on the fact that coherent states (CS) are indeed formulated in the space of entire analytic functions where they can be rigorously expressed in terms of theta functions on the von Neumann lattice. The r\^ole played by the finite difference operators and the relevance of the lattice structure in the completeness of the CS system suggest that the $q$--deformation of the WH algebra is an essential tool in the physics of discretized (periodic) systems. In this latter context we define a quantum mechanics formalism for lattice systems.
51 citations
TL;DR: In this article, the eigenfunctions of the Hamiltonian for the hydrogen atom in a homogeneous magnetic field are expressed in terms of Bessel coherent states and the irreducible representations of this quadratic algebra are realized on hypergeometric states.
Abstract: Global formulas for eigenfunctions and solutions to the Cauchy problem, including the path integral representation, are obtained using the coherent states technique. The reduction of coherent states via symmetry groups is studied for a transformation from “Bessel” to “hypergeometric” states. The eigenfunctions of the Hamiltonian for the hydrogen atom in a homogeneous magnetic field are expressed in terms of Bessel coherent states. For a small field, after quantum averaging, the Hamiltonian is represented in terms of generators with quadratic commutation relations. The irreducible representations of this quadratic algebra are realized on hypergeometric states. The notion of deformed hypergeometric states is also introduced for this quadratic algebra as an analog of squeezed Gaussian packets of the Heisenberg algebra. The asymptotic equations of eigenfunctions with respect to a small field and a large leading quantum number are derived using these states and their “deaveraging.” Some explicit formulas for the Zeeman splitting of the spectrum are obtained up to the fourth order with respect to the field, as well as for lower and upper levels in the cluster, including the case of “incidence on the center.”
30 citations
TL;DR: The coherent states of a Hamiltonian linear in SU(1,1) operators are constructed by defining them, in analogy with the harmonic-oscillator coherent states, as the minimum-uncertainty states with equal variance in two observables.
Abstract: The coherent states of a Hamiltonian linear in SU(1,1) operators are constructed by defining them, in analogy with the harmonic-oscillator coherent states, as the minimum-uncertainty states with equal variance in two observables. The proposed approach is thus based on a physical characteristic of the harmonic-oscillator coherent states which is in contrast with the existing ones which rely on the generalization of the mathematical methods used for constructing the harmonic-oscillator coherent states. The set of states obtained by following the proposed method contains not only the known SU(1,1) coherent states but also a different class of states. \textcopyright{} 1996 The American Physical Society.
24 citations
CERN1
TL;DR: In this article, the authors introduced the concept of coherent states and their properties for simple quantum compact groups, such as Al, Bl, Cl and Dl, and showed that the coherent state can be interpreted as a holomorphic function on this orbit with values in the carrier Hilbert space of an irreducible representation of the corresponding quantized enveloping algebra.
Abstract: Coherent states are introduced and their properties are discussed for simple quantum compact groupsAl, Bl, Cl andDl. The multiplicative form of the canonical element for the quantum double is used to introduce the holomorphic coordinates on a general quantum dressing orbit. The coherent state is interpreted as a holomorphic function on this orbit with values in the carrier Hilbert space of an irreducible representation of the corresponding quantized enveloping algebra. Using Gauss decomposition, the commutation relations for the holomorphic coordinates on the dressing orbit are derived explicitly and given in a compactR-matrix formulation (generalizing this way theq-deformed Grassmann and flag manifolds). The antiholomorphic realization of the irreducible representations of a compact quantum group (the analogue of the Borel-Weil construction) is described using the concept of coherent state. The relation between representation theory and non-commutative differential geometry is suggested.
23 citations
TL;DR: An alternative derivation of the disentanglement formulas for exponential operators, also known as the Baker-Campbell-Housdorff formulas is given for several Lie algebras in this article.
Abstract: An alternative derivation of the disentanglement formulas for exponential operators, also known as the Baker–Campbell–Housdorff formulas is given for several Lie algebras. The method is especially suited to Lie algebras having no nontrivial center. We also look at some results that can be obtained by the use of these formulas in the case of squeezed coherent states.
22 citations
TL;DR: In this article, a survey on classical Heisenberg groups and algebras, q-deformed heisenberg algesas, and their representations and applications is presented.
Abstract: This paper is a survey on classical Heisenberg groups and algebras, q-deformed Heisenberg algebras, q-oscillator algebras, their representations and applications. Describing them, we tried, for the reader's convenience, to explain where the q-deformed case is close to the classical one, and where there are principal differences. Different realizations of classical Heisenberg groups, their geometrical aspects, and their representations are given. Moreover, relations of Heisenberg groups to other linear groups are described. Intertwining operators for different (Schrodinger, Fock, compact) realizations of unitary irreducible representations of Heisenberg groups are given in explicit form. Classification of irreducible representations and representations of the q-oscillator algebra is derived for the cases when q is not a root of unity and when q is a root of unity. The Fock representation of the q-oscillator algebra is studied in detail. In particular, q-coherent states are described. Spectral properties of some operators of the Fock representations of q-oscillator algebras are given. Some of applications of Heisenberg groups and algebras, q-Heisenberg algebras and q-oscillator algebras are briefly described.
20 citations
TL;DR: In this article, the authors constructed coherent states for quons as eigenstates of the annihilation operator and showed that these states form a complete set with respect to a measure, and their properties are studied.
Abstract: Coherent states for quons are constructed as eigenstates of the annihilation operator and their properties are studied. In particular, it is shown that these states form a complete set with respect to a measure.
TL;DR: In this paper, the authors give a construction of coherent states for strictly isospectral Hamiltonians by exploiting the fact that these are related by a unitary transformation, and hence the corresponding coherent states must be related by the same unitary transform.
TL;DR: A set of coherent states which are associated with quantum systems governed by a trilinear boson Hamiltonian are introduced and the resolution of the identity is derived and the related analytic representation in the complex plane is developed.
Abstract: We introduce a set of coherent states which are associated with quantum systems governed by a trilinear boson Hamiltonian. These states are produced by the action of a nonunitary displacement operator on a reference state and can be equivalently defined by some eigenvalue equations. The system prepared initially in the reference state will evolve into the coherent state during the first instants of the interaction process. Some properties of the coherent states are discussed. In particular, the resolution of the identity is derived and the related analytic representation in the complex plane is developed. It is shown that this analytic representation coincides with a double representation based on the Glauber coherent states of the pump mode and on the SU(1,1) Perelomov coherent states of the signal-idler system. Entanglement between the field modes and photon statistics of the coherent states are studied. Connections between the coherent states and the long-time evolution induced by the trilinear Hamiltonian are considered.
TL;DR: In this paper, the authors studied the contraction of the Lie algebras of the Poincare group SU(1,1) at both the classical level and the quantum level.
Abstract: The group SU(1,1) is a deformation of the Poincare group. This relationship is studied both at the classical level (coadjoint orbits) and at the quantum level (unitary representations). The contraction of the Lie algebras is written in such a way that the limit of coadjoint orbits, and hence of the classical mechanics, appears clearly. At the quantum level the representations are written on holomorphic functions Hilbert spaces and the contraction is realized by restricting these functions. It is shown that this restriction is a continuous operator. Moreover, using suitable coherent states, it is proved that the contraction extends to the representation of the whole enveloping algebras of the groups, hence it allows us to define the contraction of the quantum mechanics observables.
TL;DR: In this article, the authors construct coherent states for the full PoincarA© group in one space and one time dimension, for representations corresponding to mass m > 0 and arbitrary integral or half-integral spin.
Abstract: In the spirit of some earlier work on building coherent states for the PoincarA© group in one space and one time dimension, we construct here analogous families of states for the full PoincarA© group, for representations corresponding to mass m > 0 and arbitrary integral or half-integral spin. Each family of coherent states is defined by an affine section in the group and constitutes a frame. The sections, in their turn, are determined by particular velocity vector fields, the latter always appearing in dual pairs. Geometrically, each family of coherent states is related to the choice of a Riemannian structure on the forward mass hyperboloid or, equivalently, to the choice of a certain parallel bundle in the relativistic phase space. The large variety of coherent states obtained tempts us to believe that there is rich scope here for application to spin-dependent problems in atomic and nuclear physics, as well as to image reconstruction problems, using the discretized versions of these frames. © 1996 IOP Publishing Ltd.
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TL;DR: In this paper, the authors consider the problem of quantum group covariant q-commuting operators acting on standard bosonic/fermionic Fock spaces and show that the answer is positive in some simple cases.
Abstract: Can one represent quantum group covariant q-commuting "creators, annihilators" $A^+_i,A^j$ as operators acting on standard bosonic/fermionic Fock spaces? We briefly address this general problem and show that the answer is positive (at least) in some simplest cases.
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TL;DR: In this article, the eigenstate of general complex linear combination of SU(1,1) generators (su^c(1-1) algebraic coherent states (ACS)) is constructed and discussed.
Abstract: Eigenstates of general complex linear combination of SU(1,1) generators (su^c(1,1) algebraic coherent states (ACS)) are constructed and discussed. In case of quadratic boson representation ACS can exhibit strong both linear and quadratic amplitude squeezing. ACS for a given Lie group algebra contain the corresponding Perelomov CS with maximal symmetry.
TL;DR: Even and odd phase coherent states associated with the Hermetian phase operator theory are introduced in this article in terms of the creation operation of the phase quanta defined in a finite-dimensional phase state space.
Abstract: Even and odd phase coherent states associated with the Hermetian phase operator theory are introduced in terms of the creation operation of the phase quanta defined in a finite-dimensional phase state space. Some mathematical and physical properties of these quantum states are studied in some detail. It is shown that the even phase coherent states together with the odd ones build an overcomplete Hilbert space. Even and odd coherent-state formalism of the Pegg - Barnett phase operator is given in terms of the projection operator in the even and odd phase coherent-state space. The number - phase uncertainty relation is investigated for these quantum states. It is shown that even and odd phase coherent states are not minimum uncertainty and intelligent states for the number and phase operators.
ENEA1
TL;DR: Coupled-harmonic-oscillator Hamiltonians, modelling different physical processes, are discussed along with their eigenfunctions, which are shown to be generalized harmonicoscillators with many indices and variables.
Abstract: Coupled-harmonic-oscillator Hamiltonians, modelling different physical processes, are discussed along with their eigenfunctions, which are shown to be generalized harmonic-oscillator functions with many indices and variables. We introduce the relevant coherent states and analyse their peculiar properties. The physical and mathematical consequences of our study are finally considered.
TL;DR: In this paper, a computer program from the CUPS project is described which demonstrates the action of the annihilation operator on these states, constructs coherent states which behave like classical electromagnetic fields, and shows how such states can be squeezed.
Abstract: Students first meet the wave‐particle paradox through the photon and wave descriptions of light. Yet, in basic courses on quantum mechanics, they study matter particles only, because the mathematics of the quantized radiation field is usually considered too advanced. An oscillating electromagnetic field is formally similar to a harmonic oscillator, whose energy eigenstates can represent states of well‐defined photon number. Using a computer program from the CUPS project, an approach will be described which demonstrates the action of the annihilation operator on these states, constructs coherent states which behave like classical electromagnetic fields, and shows how such states can be squeezed. All of these have practical relevance in modern optics. This is just one example of the computer making a hitherto unapproachable subject accessible to ordinary undergraduates. Computers have already changed how much of quantum mechanics is taught. As more such possibilities are realized, the teaching of the whole ...
TL;DR: In this article, a new type of q-coherent states with M components is introduced, and some properties of the qcoherent state are discussed, while the cycle representations of quantum algebra SUq(2) are obtained by means of the two-mode q-Coherent states.
Abstract: In this letter, a new type of q-coherent states with M components is introduced. Some properties of the q-coherent state are discussed. The cycle representations of quantum algebra SUq(2) are obtained by means of the two-mode q-coherent states.
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TL;DR: In this paper, certain non-linear relations between the generators of the (q-deformed) Heisenberg algebra are found and some of these relations are invariant under quantization and $q$-deformation.
Abstract: Certain non-linear relations between the generators of the (q-deformed) Heisenberg algebra are found. Some of these relations are invariant under quantization and $q$-deformation.
01 Jan 1996
TL;DR: In this article, generalized coherent states for general potentials, constructed through a controlling mechanism, can also be obtained applying on a reference state suitable operators, and an explicit example is supplied.
Abstract: Generalized coherent states for general potentials, constructed through a controlling mechanism, can also be obtained applying on a reference state suitable operators. An explicit example is supplied.
TL;DR: In this article, the multicomponent coherent states associated with the Lie algebra SO(4) are presented, and an inhomogeneous differential realization of SO (4) in this multic-component coherent state space is obtained.
Abstract: The multicomponent coherent states associated with the Lie algebra SO(4) are presented. An inhomogeneous differential realization of SO(4) in this multicomponent coherent state space is obtained.
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TL;DR: In this paper, the roll of first class constraints for coherent states has been considered, and the authors consider the roll-of-first-class constraints for the first class constraint.
Abstract: Coherent states possess a regularized path integral and gives a natural relation between classical variables and quantum operators. Recent work by Klauder and Whiting has included extended variables, that can be thought of as gauge fields, into this formalism. In this paper, I consider the next step, and look at the roll of first class constraints.
01 Jan 1996
TL;DR: In this paper, a class of coherent states for the group SU(1, 1) X SU( 1, 1), and explicit expressions for these using the Clebsch-Gordan algebra for the SU (1,1) group were derived.
Abstract: We introduce and study the properties of a class of coherent states for the group SU(1,1) X SU(1,1) and derive explicit expressions for these using the Clebsch-Gordan algebra for the SU(1,1) group. We restrict ourselves to the discrete series representations of SU(1,1). These are the generalization of the 'Barut Girardello' coherent states to the Kronecker Product of two non-compact groups. The resolution of the identity and the analytic phase space representation of these states is presented. This phase space representation is based on the basis of products of 'pair coherent states' rather than the standard number state canonical basis. We discuss the utility of the resulting 'bi-pair coherent states' in the context of four-mode interactions in quantum optics.
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TL;DR: In this paper, the geometric phases of the coherent states were introduced in a way analogous to that used in the classical polarization optics, and generalized coherent states of the SU(2)_p group of the light fields were considered.
Abstract: Polarization coherent states (PCS) are considered as generalized coherent states of $SU(2)_p$ group of the polarization invariance of the light fields. The geometric phases of PCS are introduced in a way, analogous to that used in the classical polarization optics.