Showing papers on "Coherent states in mathematical physics published in 1995"
TL;DR: A survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications, can be found in this paper.
Abstract: We present a survey of the theory of coherent states (CS); and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.
173 citations
TL;DR: In this paper, a decomposition of the level-oneq-deformed Fock space representations of U q(sl n ) is given, and it is shown that the action of U′ q(SL n ) on these Fock spaces is centralized by a Heisenberg algebra, which arises from the center of the affine Hecke algebra.
Abstract: A decomposition of the level-oneq-deformed Fock space representations ofU q(sl n ) is given. It is found that the action ofU′ q(sl n ) on these Fock spaces is centralized by a Heisenberg algebra, which arises from the center of the affine Hecke algebraĤ N in the limitN → ∞. Theq-deformed Fock space is shown to be isomorphic as aU′ q(sl n )-Heisenberg-bimodule to the tensor product of a level-one irreducible highest weight representation ofU′ q(sl n ) and the Fock representation of the Heisenberg algebra. The isomorphism is used to decompose theq-wedging operators, which are intertwiners between theq-deformed Fock spaces, into constituents coming fromU′ q(sl n ) and from the Heisenberg algebra.
109 citations
79 citations
TL;DR: In this article, the authors studied the dynamical algebra associated with a family of isospectral oscillator Hamiltonians through the analysis of its representation in the basis of energy eigenstates.
Abstract: The dynamical algebra associated with a family of isospectral oscillator Hamiltonians is studied through the analysis of its representation in the basis of energy eigenstates. It is shown that this representation becomes similar to that of the standard Heisenberg algebra, and it is dependent on a parameter omega >or=0. We call it the distorted Heisenberg algebra, where omega is the distortion parameter. The corresponding coherent states for an arbitrary omega are derived, and some particular examples are discussed in detail. A prescription to produce the squeezing, by adequately selecting the initial state of the system, is given.
73 citations
01 Jan 1995
TL;DR: In this paper, a family of nonlinear Schroedinger Equations are used to represent the transformation of spin1/2 Particles in a curved spacetime with absolute time, and the results of these transformations are used for geometric quantization.
Abstract: Quantization, Field Theory, and Representation Theory: On Quantum Mechanics in a Curved Spacetime with Absolute Time (D. Canarutto et al.). Massless Spinning Particles on the Antide Sitter Spacetime (S. De Bievre, S. Mehdi). A Family of Nonlinear Schroedinger Equations: Linearizing Transformations and Resulting Structure (H.D. Doebner et al.). Modular Structures in Geometric Quantization (G.G. Emch). Diffeomorphism Groups and Anyon Fields (G.A. Goldin, D.H. Sharp). On a Full Quanization of the Torus (M.J. Gotay). Differential Forms on the Skyrmion Bundle (C. Gross). Explicitly Covariant Algebraic Representations for Transitional Currents of Spin1/2 Particles (M.I. Krivoruchenko). The Quantum Su(2,2)Harmonic Oscillator (W. Mulak). GeometricStochastic Quantization and Quantum Geometry (E. Prugovecki). Prequantization (D.J. Simms). Classical Yang-Mills and Dirac Fields in the Minkowski Space and in a Bag (J. Sniatycki). Symplectic Induction, Unitary Induction and BRST Theory (Summary) (G.M. Tuynman). Coherent States, Complex and Poisson Structures: Spin Coherent States for the Poincare Group (S.T. Ali, J.P. Gazeau). Coherent States and Global Differential Geometry (S. Berceanu). Natural Transformations of Lagrangians into pforms on the Tangent Bundle (J. Debecki). SL(2,IR)Coherent States and Itegrable Systems in Classical and Quantum Physics (J.P. Gazeau). Symplectic and Lagrangian Realization of Poisson Manifolds (M. Giordano et al.). From the Poincare-Cartan Form to a Gerstehhaber Algebra of Poisson Brackets in Field Theory (I.V. Kanatchikov). Geometric Coherent States, Membranes, and Star Products (M. Karasev). Integral Representation of Eigenfunctions and Coherent States for the Zeeman Effect (M. Karasev, E. Novikova). QDeformations and Quantum Groups, Noncommutative Geometry: Quantum Coherent States and the Method of Orbits (B. Jurco, P. tovicek). On the Deformation of Commutation Relations (W. Marcinek). The qdeformed Quantum Mechanics in the Coherent States Map Approach (V. Maximov, A. Odzijewicz). Quantization by Quadratic Polynomials in Creation and Annihilation Operators (W. Slowikowski). On Dirac Type Brackets (Yu.M. Vorobjev, R. Flores Espinoza). Quantum Trigonometry and Phasespace Propensity (K. Wodkiewicz, B.G. Englert). Noncommutative Space-Time Impled by Spin (S. Zadrzewski). Miscellaneous Problems of Quantum Dynamics: Spectrum of the Dirac Operator on the SU(2) Manifold as Energy Spectrum for the Polyaniline Macromolecule (H. Makaruk). On Geometric Methods in the Description of Quantum Fluids (R. Owczarek). Galactic Dynamics in the Siegel Halfplane (G. Rosensteel). Graded Contractions of so(4,2) (J. Tolar, P. Travnicek). The Berry Phase and the Geometry of Coset Spaces (E.A. Tolkachev, A.A. Tregubovich). Index.
46 citations
TL;DR: In this article, a modified Holstein-Primakoff realization of the SU(1,1) Lie algebra has been studied for photon states with simple phase-state representations.
Abstract: Statistical and phase properties and number-phase uncertainty relations are systematically investigated for photon states associated with the Holstein-Primakoff realization of the SU(1,1) Lie algebra. Perelomov's SU(1,1) coherent states and the eigenstates of the SU(1,1) lowering generator (the Barut-Girardello states) are discussed. A recently developed formalism, based on the antinormal ordering of exponential phase operators, is used for studying phase properties and number-phase uncertainty relations. This study shows essential differences between properties of the Barut-Girardello states and the SU(1,1) coherent states. The philophase states, defined as states with simple phase-state representations, relate the quantum description of the optical phase to the properties of the SU(1,1) Lie group. A modified Holstein-Primakoff realization is derived, and eigenstates of the corresponding lowering generator are discussed. These stares are shown to contract, in a proper limit, to the familiar Glauber coherent states.
31 citations
TL;DR: In this paper, a new family of stationary coherent states for the two-dimensional harmonic oscillator is presented, which are coherent in the sense that they minimize an uncertainty relation for observables related to the orientation and the eccentricity of an ellipse.
Abstract: A new family of stationary coherent states for the two-dimensional harmonic oscillator is presented. These states are coherent in the sense that they minimize an uncertainty relation for observables related to the orientation and the eccentricity of an ellipse. The wavefunction of these states is particularly simple and well localized on the corresponding classical elliptical trajectory. As the number of quanta increases, the localization on the classical invariant structure is more pronounced. These coherent states give a useful tool to compare classical and quantum mechanics and form a convenient basis to study weak perturbations.
29 citations
TL;DR: The quantum stochastic calculus on (Boson) Fock space has developed into a new field of mathematics keeping a profound contact with physical applications as discussed by the authors, which is highlighted in the excellent books by Meyer [21] and by Parthasarathy [26].
Abstract: As is highlighted in the excellent books by Meyer [21] and by Parthasarathy [26] quantum stochastic calculus on (Boson) Fock space has developed into a new field of mathematics keeping a profound contact with physical applications. Since Hudson and Parthasarathy [12] first formulated quantum stochastic integrals of Ito type in 1984 a crucial role has been played by three basic quantum stochastic processes:
27 citations
TL;DR: A class of coherent states defined in terms of the excitation and deexcitation of pairs of photons is studied with reference to its nonclassical and other quantum-statistical properties.
Abstract: A class of coherent states defined in terms of the excitation and deexcitation of pairs of photons is studied with reference to its nonclassical and other quantum-statistical properties. These states supplement the other well-known two-mode states such as Caves-Schumaker states and pair coherent states and can be produced by dissipative processes involving emission and absorption of photons in pairs.
26 citations
Journal Article•
TL;DR: In this article, the von Neumann type subsystems of $q$-deformed coherent states are considered and the completeness of such subsystems is proved, which is the case for all coherent states.
Abstract: The von Neumann type subsystems of $q$-deformed coherent states are considered. The completeness of such subsystems is proved.
26 citations
TL;DR: In this paper, two kinds of coherent states are constructed in the context of the Calogero-Sutherland singular oscillator and the motion of the peaks of the wavefunctions of these coherent states is compared with the classical trajectory.
Abstract: Two kinds of coherent states are constructed in the context of the Calogero-Sutherland singular oscillator. The motion of the peaks of the wavefunctions of these coherent states are compared with the classical trajectory. It is found that while the wavefunction for one kind of coherent states is always singly peaked, that for the other acquires multiple peaks close to the classical turning point near the origin. The two coherent states are found to exhibit a kind of complementarity.
TL;DR: A path integral written in terms of the group theoretic coherent states by using the Kahler structure of the coherent state manifold with the particular emphasis on the boundary fixing term derivation is considered in this paper.
Abstract: A path integral written in terms of the group theoretic coherent states by using the Kahler structure of the coherent state manifold with the particular emphasis on the boundary‐fixing term derivation is considered herein. The path integral for a propagator of the system with Hamiltonian linear in the SU(2)/SU(1,1) generators is shown to be diagonalized by an appropriate motion in the phase space.
TL;DR: In this paper, the authors define multimode coherent states as linear superpositions of suitable composition states, which is equivalent to the definition of coherent states in general as eigenvectors of a generalized annihilation operator.
Abstract: We define multimode (entangled) coherent states as properly chosen linear superpositions of suitable composition states. Our definition is equivalent to the definition of coherent states as eigenvectors of a corresponding generalized annihilation operator. In certain limit cases we discuss the statistical properties of the states defined.
TL;DR: In this paper, the master field becomes the Boltzmann field in the free Fock space and the quantum semigroup SUq(2) becomes a central element of the quantum group bialgebra.
Abstract: In recent works by Singer, Douglas, Gopakumar and Gross an application of results of Voiculescu from noncommutative probability theory to constructions of the master field for large-N matrix field theories have been suggested. It turns out that this master field becomes the Boltzmann field in the free Fock space. In this note we consider interrelations between the master field and quantum semigroups. We define the master field algebra and observe that it is isomorphic to the algebra of functions on the quantum semigroup SUq(2) for q=0. The master field becomes a central element of the quantum group bialgebra. The quantum Haar measure on the SUq(2) for any q gives the Wigner semicircle distribution for the master field. Coherent states on SUq(2) become coherent states in the master field theory.
TL;DR: In this article, the notion of atomic coherent states is extended to the multilevel case, and it is used to define a holomorphic representation for atomic states and operators in the theory of cascade superfluorescence and superradiant lasing.
Abstract: The notion of atomic coherent states is extended to the multilevel case. Since the representation based on coherent states is convenient in treating collective interactions of atoms with photons, and since many optical processes involve atoms of three or more levels, it is expected that this extension will play a role in the theory of such processes as cascade superfluorescence and superradiant lasing. Like their bosonic counterparts, atomic coherent states may be used to define a holomorphic representation for atomic states and operators. We discuss this in detail and give examples to illustrate applications.
TL;DR: A phase space representation based on pair coherent states rather than the standard harmonic-oscillator coherent states is derived and the utility of the resulting "bi-pair coherent states" in the context of four-mode interactions in quantum optics is discussed.
Abstract: We introduce and study the properties of a class of coherent states for the group SU(1,1)×SU(1,1) and derive explicit expressions for these using the Clebsch-Gordan algebra for the SU(1,1) group. We also derive a phase space representation based on pair coherent states rather than the standard harmonic-oscillator coherent states. We discuss the utility of the resulting "bi-pair coherent states" in the context of four-mode interactions in quantum optics.
Journal Article•
TL;DR: For a time-dependent harmonic oscillator with an inverse squared singular term, the generalized invariant using the Lie algebra of $SU(2)$ and the number-type eigenstates and the coherent states were constructed group-theoretically for both the time-independent and the time dependent harmonic oscillators with the singular term.
Abstract: For a time-dependent harmonic oscillator with an inverse squared singular term, we find the generalized invariant using the Lie algebra of $SU(2)$ and construct the number-type eigenstates and the coherent states using the spectrum-generating Lie algebra of $SU(1,1)$ We obtain the evolution operator in both of the Lie algebras The number-type eigenstates and the coherent states are constructed group-theoretically for both the time-independent and the time-dependent harmonic oscillators with the singular term It is shown that the squeeze operator transforms unitarily the time-dependent basis of the spectrum-generating Lie algebra of $SU(1,1)$ for the generalized invariant, and thereby evolves the initial vacuum into a final coherent vacuum
TL;DR: In this paper, the photo-count distribution and its higher-order factorial moments are expressed as contour integrals of the generating function of the photon-number probabilities, and applied to derive the photo count statistics of a superposition of two arbitrary coherent states.
Posted Content•
TL;DR: Analogs of ordinary Gaussian coherent states on bosonic Fock spaces are constructed for the case of free Fock space, which appear to be natural mathematical structures suitable for description of large N matrix models.
Abstract: Analogs of ordinary Gaussian coherent states on bosonic Fock spaces are constructed for the case of free Fock spaces, which appear to be natural mathematical structures suitable for description of large N matrix models.
TL;DR: In this paper, a special form of the GNS representation in terms of generalized creation and annihilation operators living in the infinite tensor product of Hilbert spaces is given leading to a generalization of the Fock space.
Abstract: States on free *‐algebras expressed in terms of ordered partitions are studied. To this class belong twisted limit states that are derived from a convolution type q‐ analog of the quantum central limit theorem (qclt) for twisted free *‐bialgebras Cq, studied previously in the case of quantum groups. A special form of the GNS representation in terms of generalized creation and annihilation operators, A(v) and A(v*), respectively, living in the infinite tensor product of Hilbert spaces is given leading to a generalization of the Fock space. It is also shown that another family of states of similar combinatorics can be obtained from the convolution series.
TL;DR: In this paper, the wave packet generalizes the conventional coherent states of minimal uncertainty and does not change its shape for times of order Ω(1/hbar) for integrable systems.
Abstract: Quantum dynamics of integrable systems is discussed. Localized wave packets generalizing the conventional coherent states of minimal uncertainty are constructed. The wave packet moves along a certain trajectory and does not change its shape for times of order $\frac{1}{\hbar}$.
Posted Content•
TL;DR: In this paper, the geometric phase associated with the generalized coherent states was studied and a possible experimental detection of the phases was proposed in such a way that the geometric phases can be discriminated from the dynamical phase.
Abstract: We study characteristic aspects of the geometric phase which is associated with the generalized coherent states. This is determined by special orbits in the parameter space defining the coherent state, which is obtained as a solution of the variational equation governed by a simple model Hamiltonian called the ”resonant Hamiltonian”. Three typical coherent states are considered: SU(2), SU(1,1) and Heisenberg-Weyl. A possible experimental detection of the phases is proposed in such a way that the geometric phases can be discriminated from the dynamical phase.
01 Jan 1995
TL;DR: In this paper, the authors studied q-quantum mechanics in one degree of freedom and discussed the holomorphic representation of the q-deformed Heisenberg-Weyl algebra and its realization by covariant Berezin symbols.
Abstract: We study q-quantum mechanics in one degree of freedom. Among other things, we discuss the holomorphic representation of the q-deformed Heisenberg-Weyl algebra and its realization by covariant Berezin symbols.