Showing papers on "Coherent states in mathematical physics published in 2003"
13 Mar 2003
TL;DR: The history of nonclassical states in quantum physics can be found in this paper, where the authors present a brief review of the state of the art in Quantum Physics and Quantum Optics, from the Jaynes-Cummings Model to collective interactions.
Abstract: 'Nonclassical' States in Quantum Physics: Brief Historical Review. Squeezed States. Parametric Excitation and Generation of Nonclassical States in Linear Media. Even and Odd Coherent States and Tomographic Representation of Quantum Mechanics and Quantum Optics. The Binormial States of Light. Nonclassical States in Kerr Media. From the Jaynes-Cummings Model to Collective Interactions.
251 citations
TL;DR: In this article, a general scheme for constructing vector coherent states, in analogy with the well-known canonical coherent states and their deformed versions, when these latter are expressed as infinite series in powers of a complex variable z, is proposed.
Abstract: A general scheme is proposed for constructing vector coherent states, in analogy with the well-known canonical coherent states, and their deformed versions, when these latter are expressed as infinite series in powers of a complex variable z. In the present scheme, the variable z is replaced by matrix valued functions over appropriate domains. As particular examples, we analyze the quaternionic extensions of the canonical coherent states and the Gilmore–Perelomov and Barut–Girardello coherent states arising from representations of SU(1,1). Possible physical applications are indicated.
44 citations
TL;DR: In this article, a measure of "anticlassicality" of pure and mixed quantum states is introduced as a maximum value of the Hilbert-Schmidt scaling product between the renormalized statistical operators of the state concerned and all displaced thermal states.
42 citations
TL;DR: In this paper, it was shown that the quantum states of Morse potential represent an infinite-dimensional Lie algebra the so-called Morse algebra, and the Barut-Girardello coherent states are constructed as a linear combination of quantum states corresponding to the Morse potential.
34 citations
TL;DR: In this article, a family of generalized coherent states obtained by means of operators of an unitary irreducible representation of the group of affine transformations of the real line was studied.
Abstract: We deal with a family of generalized coherent states obtained by means of operators of an unitary irreducible representation of the group of affine transformations of the real line. We prove that the ranges of the corresponding coherent state transforms coincide with spaces of bound states of the Landau Hamiltonian in the hyperbolic plane. This provides us with a new characterization of hyperbolic Landau states.
23 citations
TL;DR: In this article, the authors continue the development of p-mechanics by introducing the concept of states, which allows us to evaluate classical observables at any point of phase space and simultaneously evaluate quantum probability amplitudes.
Abstract: p-Mechanics is a consistent physical theory which describes both quantum and classical mechanics simultaneously (V. V. Kisil, p-Mechanics as a physical theory. An Introduction, E-print:arXiv:quant-ph/0212101, 2002; International Journal of Theoretical Physics41(1), 63–77, 2002). We continue the development of p-mechanics by introducing the concept of states. The set of coherent states we introduce allows us to evaluate classical observables at any point of phase space and simultaneously to evaluate quantum probability amplitudes. The example of the forced harmonic oscillator is used to demonstrate these concepts.
14 citations
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TL;DR: A survey of the p-mechanical construction can be found in this article, which is a consistent physical theory suitable for a simultaneous description of classical and quantum mechanics Observables in pmechanics are defined to be convolution operators on the Heisenberg group and states are defined as positive linear functionals on p-observables.
Abstract: This is an up-to-date survey of the p-mechanical construction (see funct-an/9405002, quant-ph/9610016, math-ph/0007030, quant-ph/0212101, quant-ph/0303142), which is a consistent physical theory suitable for a simultaneous description of classical and quantum mechanics Observables in p-mechanics are defined to be convolution operators on the Heisenberg group H^n Under irreducible representations of H^n the p-observables generate corresponding observables in classical and quantum mechanics p-States are defined as positive linear functionals on p-observables It is shown that both states and observables can be realised as certain sets of functions/distributions on the Heisenberg group The dynamical equations for both p-observables and p-states are provided The construction is illustrated by the forced and unforced harmonic oscillators Connections with the contextual interpretation of quantum mechanics are discussed Keywords: Classical mechanics, quantum mechanics, Moyal brackets, Poisson brackets, commutator, Heisenberg group, orbit method, deformation quantisation, symplectic group, representation theory, metaplectic representation, Berezin quantisation, Weyl quantisation, Segal--Bargmann--Fock space, coherent states, wavelet transform, Liouville equation, contextual interpretation, interaction picture, forced harmonic oscillator
13 citations
TL;DR: In this paper, the Wigner problem of quantum mechanical commutation relations consistent with the Heisenberg evolution equations of a given shape is studied, where the classical analogy is postulated only for the shape of the time evolution equations and not for a Hamiltonian itself.
Abstract: The Wigner problem, i.e. the investigation of general quantum mechanical commutation relations consistent with the Heisenberg evolution equations of a given shape, is studied. We follow a recently proposed generalization of this idea within which the classical analogy is postulated only for the shape of the time evolution equations and not for a Hamiltonian itself. This links our investigation to the problem of alternative Hamiltonians of classical mechanics and to canonically inequivalent phase-space descriptions of physical systems governed by the same Newton equations of motion. Warned that the time evolution and the other symmetry generators may be given ambiguously even in the formalism of classical mechanics, we do not a priori assume the shape of their quantum analogues. Instead we only require that the set of basic algebraic relations, which quantum mechanical observables are to obey, has a Lie algebra structure. Such a requirement appears to be sufficient to find solutions for simple oscillator-like dynamics. New algebras of quantum mechanical observables are not constructed as a linear envelope of the Heisenberg algebra, and their representations reflect physical results unexpected in the framework of the canonical approach. We illustrate our approach in detail for the example of the one-dimensional harmonic oscillator using the representation of the generalized coherent states.
10 citations
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TL;DR: In this paper, the coherent states for oscillator-like systems connected with the Chebyshev polynomials of the 1-st and 2-nd kind were defined.
Abstract: We define the coherent states for the oscillator-like systems, connected with the Chebyshev polynomials $T_n(x)$ and $U_n(x)$ of the 1-st and 2-nd kind.
9 citations
TL;DR: In this paper, the growth of analytic functions is intimately connected to the completeness of the sequences of these generalized coherent states, and the least density that such sequences must have in order to be overcomplete is calculated.
Abstract: Analytic representations based on generalized coherent states are studied. The growth of the analytic functions is intimately connected to the completeness of the sequences of these generalized coherent states. The least density that such sequences must have in order to be overcomplete is calculated. The results generalize known results on the completeness of von Neumann lattices for the standard coherent states to other sets of coherent states.
7 citations
TL;DR: In this paper, the star product and Moyal bracket are introduced using the coherent states corresponding to quantum systems with non-linear spectra, and two kinds of coherent states are considered.
Abstract: The star product and Moyal bracket are introduced using the coherent states corresponding to quantum systems with non-linear spectra. Two kinds of coherent state are considered. The first kind is the set of Gazeau-Klauder coherent states and the second kind are constructed following the Perelomov-Klauder approach. The particular case of the harmonic oscillator is also discussed.
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TL;DR: In this article, the authors continue the development of p-mechanics by introducing the concept of states, which allow us to evaluate classical observables at any point of phase space simultaneously to evaluating quantum probability amplitudes.
Abstract: p-Mechanics is a consistent physical theory which describes both quantum and classical mechanics simultaneously. We continue the development of p-mechanics by introducing the concept of states. The set of coherent states we introduce allow us to evaluate classical observables at any point of phase space simultaneously to evaluating quantum probability amplitudes. The example of the forced harmonic oscilator is used to demonstrate these concepts.
TL;DR: In this article, the generalized excited even and odd coherent states of the radiation field were constructed by repeatedly applying the pseudo-creation operator b† = νa + μa† to the even and even coherent states m times, respectively.
Abstract: We construct mathematically two new types of quantum state which we call the generalized excited even and odd coherent states of the radiation field They are created by repeatedly applying the pseudo-creation operator b† = νa + μa† to the even and odd coherent states m times, respectively Analytic expressions for the quantum statistical properties are derived, and through numerical computation it is found that these states can exhibit highly nonclassical behaviour depending on the degree of excitation m and other parameters These states may have interesting significance if they can be realized experimentally
TL;DR: In this paper, the duality of the operators becomes a self-duality and the corresponding operators form a usual Heisenberg-Weyl algebra, and it is shown that no wavepackets corresponding to any of the SU(1, 1) coherent states can exactly preserve their shape during time evolution.
Abstract: The two types of SU(1, 1) coherent states of Barut and Girardello and of Perelomov are dual in a sense that the operators in the eigenvalue equation and in the exponentials which create these types of coherent states from the lowest eigenstate (vacuum) form an asymmetric Heisenberg–Weyl algebra. A new type of SU(1, 1) coherent states which takes on an intermediate position between the two dual types of SU(1, 1) coherent states already mentioned is established and investigated. In this new type, the duality of the operators becomes a self-duality and the corresponding operators form a usual Heisenberg–Weyl algebra. Properties of the different SU(1, 1) coherent states are investigated for the realization of SU(1, 1) by one-dimensional quantum-mechanical potential problems leading to a quadratic law of energy-level spacing. Coherent SU(1, 1) phase states are discussed for this realization of SU(1, 1). It is shown that no wavepackets corresponding to any of the SU(1, 1) coherent states can exactly preserve their shape during time evolution.
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TL;DR: In this article, a quantum Heisenberg group is constructed by viewing it as the dual quantum group of the specific non-compact quantum group (A,\Delta) constructed earlier by the author.
Abstract: In this paper, we give a construction of a (C*-algebraic) quantum Heisenberg group. This is done by viewing it as the dual quantum group of the specific non-compact quantum group (A,\Delta) constructed earlier by the author. Our definition of the quantum Heisenberg group is different from the one considered earlier by Van Daele. To establish our object of study as a locally compact quantum group, we also give a discussion on its Haar weight, which is no longer a trace. In the latter part of the paper, we give some additional discussion on the duality mentioned above.
Journal Article•
TL;DR: In this article, the growth of analytic functions is intimately connected to the completeness of the sequences of these generalized coherent states, and the least density that such sequences must have in order to be overcomplete is calculated.
Abstract: Analytic representations based on generalized coherent states are studied. The growth of the analytic functions is intimately connected to the completeness of the sequences of these generalized coherent states. The least density that such sequences must have in order to be overcomplete is calculated. The results generalize known results on the completeness of von Neumann lattices for the standard coherent states to other sets of coherent states.
TL;DR: In this paper, the evolution of a time-dependent quantum system can be described by the dynamics of its generalized coherent state and its phase properties can be investigated by an Hermitian phase operator constructed from a generalized phase state.
TL;DR: In this paper, the Pegg-Barnett formalism of phase operator was used to obtain phase probability distributions of new even and odd nonlinear coherent states, and it was shown that the distributions for the states are rather different.
Abstract: Using the Pegg–Barnett formalism of phase operator, we obtain phase probability distributions of new even and odd nonlinear coherent states. It is shown that the distributions for the states are rather different, and unlike the case of ordinary even and odd coherent states the Pegg–Barnett distribution clearly reflects the different character of quantum interference in the case of the new even and odd coherent states.
TL;DR: In this article, the authors consider the problem of existence of the diagonal representation for operators in the space of a family of generalized coherent states associated with a unitary irreducible representation of a (compact) Lie group.
Abstract: We consider the problem of existence of the diagonal representation for operators in the space of a family of generalized coherent states associated with a unitary irreducible representation of a (compact) Lie group. We show that necessary and sufficient conditions for the possibility of such a representation can be obtained by combining Clebsch–Gordan theory and the reciprocity theorem associated with induced unitary group representations. Applications to several examples involving SU(2), SU(3), and the Heisenberg–Weyl group are presented, showing that there are simple examples of generalized coherent states which do not meet these conditions. Our results are relevant for phase–space description of quantum mechanics and quantum state reconstruction problems.
TL;DR: In this article, a class of shape-invariant bound-state problems which represent two-level systems are examined and a decomposition of identity for these coherent states is given.
Abstract: Algebraic approach to the integrability condition called shape invariance is briefly reviewed. Various applications of shape-invariance available in the literature are listed. A class of shape-invariant bound-state problems which represent two-level systems are examined. These generalize the Jaynes-Cummings Hamiltonian. Coherent states associated with shape-invariant systems are discussed. For the case of quantum harmonic oscillator the decomposition of identity for these coherent states is given. This decomposition of identity utilizes Ramanujan's integral extension of the beta function.
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TL;DR: In this article, a path integral representation of the affine weak coherent state matrix elements of the unitary time-evolution operator has been studied and rigorously established for linear Hamiltonians and the difficulties presented by more general Hamiltonians are addressed.
Abstract: Weak coherent states share many properties of the usual coherent states, but do not admit a resolution of unity expressed in terms of a local integral. They arise e.g. in the case that a group acts on an inadmissible fiducial vector. Motivated by the recent Affine Quantum Gravity Program, the present work studies the path integral representation of the affine weak coherent state matrix elements of the unitary time-evolution operator. Since weak coherent states do not admit a resolution of unity, it is clear that the standard way of constructing a path integral, by time slicing, is predestined to fail. Instead, a well-defined path integral with Wiener measure, based on a continuous-time regularization, is used to approach this problem. The dynamics is rigorously established for linear Hamiltonians, and the difficulties presented by more general Hamiltonians are addressed.
TL;DR: In this article, a new kind of q-deformed charged coherent states is constructed in Fock space of two-mode q-boson system with suq(2) covariance and a resolution of unity for these states is derived.
Abstract: A new kind of q-deformed charged coherent states is constructed in Fock space of two-mode q-boson system with suq(2) covariance and a resolution of unity for these states is derived. We also present a simple way to obtain these coherent states using state projection method.
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TL;DR: In this article, a general procedure for constructing coherent states, which are eigenstates of annihilation operators, related to quantum mechanical potential problems, is presented, which rely on the properties of the orthogonal polynomials, for their derivation.
Abstract: A general procedure for constructing coherent states, which are eigenstates of annihilation operators, related to quantum mechanical potential problems, is presented. These coherent states, by construction are not potential specific and rely on the properties of the orthogonal polynomials, for their derivation. The information about a given quantum mechanical potential enters into these states, through the orthogonal polynomials associated with it and also through its ground state wave function. The time evolution of some of these states exhibit fractional revivals, having relevance to the factorization problem.
01 Jan 2003
TL;DR: In this paper, the authors developed a method of discretization of the continuous theory of coherent states on a general semidirect product Lie group, where the group is assumed to have a unitary representation which is square integrable on some homogeneous space.
Abstract: We develop a method of discretization of the continuous theory of coherent states on a general semidirect product Lie group. The group is assumed to have a unitary representation which is square integrable on some homogeneous space. We show also that the existence of a discrete frame of coherent states in the carrier space of a unitary representation of such a group implies the square integrability of this representation on the label space.
01 Jan 2003
TL;DR: In this article, a recurring theme in Carl Friedrich von Weizsacker's synthesis of physics and philosophy is that at the core of quantum theory there is a non-classical logic.
Abstract: A recurring theme in Carl Friedrich von Weizsacker’s synthesis of physics and philosophy is that at the core of quantum theory there is a non-classical logic. Application of this logic to its own statements leads to the concept of multiple quantization [1]. The reconstruction of quantum theory as done in [2, 3] is an explicit implementation of this idea, starting with the quantization of an abstract classical alternative. The result of this first step is interpreted as three-dimensional momentum space. The second quantization step yields the Weyl, Dirac and Maxwell equations. The third quantization step would then lead to the familiar quantum field theories of matter and gauge fields.
TL;DR: In this article, necessary and sufficient conditions for the existence of the diagonal representation in the context of a family of generalised coherent states associated with any unitary irreducible Lie group representation are presented.
Abstract: "The possibility of describing noncommuting operators in quantum mechanics by classical type functions, and the associated expression of operator multiplication, is of considerable interest. The well known Wigner-Weyl-Moyal theory is an important example. Another is the diagonal representation of operators using standard coherent states. We develop general necessary and sufficient conditions for the existence of the diagonal representation in the context of a family of generalised coherent states associated with any unitary irreducible Lie group representation. Several examples illustrating these conditions, and interesting results in the Heisenberg-Weyl case, are presented".