Showing papers on "Coherent states in mathematical physics published in 2000"
Book•
01 Jan 2000
TL;DR: A survey of the theory of coherent states, wavelets, and some of their generalizations can be found in the second edition of the book as discussed by the authors, where the authors present a new chapter on coherent state quantization and the related probabilistic aspects.
Abstract: This second edition is fully updated, covering in particular new types of coherent states (the so-called Gazeau-Klauder coherent states, nonlinear coherent states, squeezed states, as used now routinely in quantum optics) and various generalizations of wavelets (wavelets on manifolds, curvelets, shearlets, etc.). In addition, it contains a new chapter on coherent state quantization and the related probabilistic aspects. As a survey of the theory of coherent states, wavelets, and some of their generalizations, it emphasizes mathematical principles, subsuming the theories of both wavelets and coherent states into a single analytic structure. The approach allows the user to take a classical-like view of quantum states in physics.Starting from the standard theory of coherent states over Lie groups, the authors generalize the formalism by associating coherent states to group representations that are square integrable over a homogeneous space; a further step allows one to dispense with the group context altogether. In this context, wavelets can be generated from coherent states of the affine group of the real line, and higher-dimensional wavelets arise from coherent states of other groups. The unified background makes transparent an entire range of properties of wavelets and coherent states. Many concrete examples, such as coherent states from semisimple Lie groups, Gazeau-Klauder coherent states,coherent states forthe relativity groups, and several kinds of wavelets, are discussed in detail. The book concludes with a palette of potentialapplications, from the quantum physically oriented,likethe quantum-classical transition or the construction of adequate states in quantum information, to the most innovative techniques to be used in data processing.Intended as an introduction to current research for graduate students and others entering the field, the mathematical discussion is self-contained. With its extensive references to the research literature, the first edition of the book is already a proven compendium for physicists and mathematicians active in the field, and with full coverage of the latest theory and results the revised second edition is even more valuable.
768 citations
TL;DR: In this article, the Hilbert-Schmidt distance between two arbitrary normalizable states is discussed as a measure of the similarity of the states and the connection to other definitions of the non-classicality of states are discussed.
Abstract: The Hilbert—Schmidt distance between two arbitrary normalizable states is discussed as a measure of the similarity of the states. Unitary transformations of both states with the same unitary operator (e.g. the displacement of both states in the phase plane by the same displacement vector and squeezing of both states) do not change this distance. The nearest distance of a given state to the whole set of coherent states is proposed as a quantitative measure of non-classicality of the state which is identical when considering the coherent states as the most classical ones among pure states and the deviations from them as non-classicality. The connection to other definitions of the non-classicality of states is discussed. The notion of distance can also be used for the definition of a neighbourhood of considered states. Inequalities for the distance of states to Fock states are derived. For given neighbourhoods, they restrict common characteristics of the state as the dispersion of the number operato...
123 citations
TL;DR: The coherent states for a particle on a sphere are introduced in this article, where the expectation values of the position and angular momentum in the coherent states are regarded as the best possible approximation of the classical phase space.
Abstract: The coherent states for a particle on a sphere are introduced. These states are labelled by points of the classical phase space, i.e. the position on the sphere and the angular momentum of a particle. As with the coherent states for a particle on a circle discussed in Kowalski et al (1996 J. Phys. A: Math. Gen. 29 4149), we deal with a deformation of the classical phase space related to quantum fluctuations. The expectation values of the position and the angular momentum in the coherent states are regarded as the best possible approximation of the classical phase space. The correctness of the introduced coherent states is illustrated by an example of the rotator.
58 citations
TL;DR: In this article, a unified approach for finding coherent states (CSs) of polynomially deformed algebras such as the quadratic and Higgs was presented, which is relevant for various multiphoton processes in quantum optics.
Abstract: We present a general unified approach for finding the coherent states (CSs) of polynomially deformed algebras such as the quadratic and Higgs algebras, which are relevant for various multiphoton processes in quantum optics. We give a general procedure to map these deformed algebras to appropriate Lie algebras. This is used, for the noncompact cases, to obtain the annihilation operator eigenstates, by finding the canonical conjugates of these operators. Generalized CSs, in the Perelomov sense, also follow from this construction. This allows us to explicitly construct CSs associated with various quantum optical systems.
55 citations
TL;DR: In this article, the exponential form of the two-mode nonlinear coherent states is given, and the parity coherent states are introduced as examples of two modes of coherent states, and they are superpositions of two corresponding coherent states.
37 citations
TL;DR: In this paper, the mathematical foundation of quantum tomography using the theory of square-integrable representations of unimodular Lie groups is described using the notion of Lie groups.
Abstract: The paper is devoted to the mathematical foundation of quantum tomography using the theory of square-integrable representations of unimodular Lie groups.
35 citations
TL;DR: In this paper, the authors constructed Gaussian Klauder coherent states for the harmonic oscillator, the planar rotor, and the particle in a box, and showed that these properties are of utility in understanding quantum-classical correspondence.
Abstract: Gaussian Klauder coherent states are constructed for the harmonic oscillator, the planar rotor, and the particle in a box. The standard harmonic oscillator coherent states are given by expansions in the eigenstates of the Hamiltonian in terms of a complex parameter \ensuremath{\alpha}. When the complex modulus of \ensuremath{\alpha} is large, these states are identical in behavior with a particular choice of Gaussian Klauder coherent state. When the angular momentum of a planar rotor is large compared with Planck's constant, the angle distribution associated with a Gaussian Klauder coherent state for this case remains sharply localized for many rotations. Similarly, for the particle in a box, it is possible to choose parameters in the Gaussian Klauder coherent state so that a localized particle bounces back and forth at constant velocity between the walls of the box for many periods without significant delocalization. Buried in this behavior is the Fourier series for a triangle wave. These examples show how Gaussian Klauder coherent states are of utility in understanding quantum-classical correspondence.
28 citations
TL;DR: In this article, the C λ -extended oscillator spectrum generating algebra is shown to be a polynomial deformation of su(1,1) and its coherent states are constructed.
24 citations
TL;DR: In this paper, the Bures fidelity for thermal states of a diagonalizable quadratic Hamiltonian in multi-mode Fock space was studied, where two photons are generated and one studies two or more modes.
Abstract: Fidelity, as a measure of the distinguishability of states, is an important concept in quantum mechanics, quantum optics and quantum information theory. Recently, the explicit expressions of fidelity for single-mode squeezed states have been given. However, in experimental studies, especially in non-degenerate parametric down-conversion, two photons are generated and one studies two- or more-mode systems. In this paper we study the Bures fidelity for thermal states of a diagonalizable quadratic Hamiltonian in multi-mode Fock space. To the best of our knowledge, no one has yet attempted to give an explicit general formula of fidelity of mixed states in multi-mode systems.
22 citations
TL;DR: In this paper, the kinematics of affine variables have been investigated in the context of quantum gravity, and the coherent-state overlap function has been realized in terms of suitable path-integral formulations.
Abstract: Affine variables, which have the virtue of preserving the positive-definite character of matrix-like objects, have been suggested as replacements for the canonical variables of standard quantization schemes, especially in the context of quantum gravity. We develop the kinematics of such variables, discussing suitable coherent states, their associated resolution of unity, polarizations, and finally the realization of the coherent-state overlap function in terms of suitable path-integral formulations.
18 citations
TL;DR: In this article, an interpretation of the f-oscillator is provided as corresponding to a special nonlinearity of vibration for which the frequency of the oscillation depends on the energy.
Abstract: The notion of f-oscillators generalizing q-oscillators is discussed. For the classical and quantum cases, an interpretation of the f-oscillator is provided as corresponding to a special nonlinearity of vibration for which the frequency of the oscillation depends on the energy. The f-coherent states generalizing the q-coherent states are constructed. Applied to quantum optics, the photon distribution function and photon number means and dispersions are calculated for the f-coherent states as well as the Wigner-Moyal function and Q-function. As an example, it is shown how this nonlinearity may affect the Planck's distribution formula.
01 Jan 2000
TL;DR: In this paper, it is shown that the standard SU(1,1) and SU(2) coherent states are the unique states which minimize the second order characteristic inequality for the three generators.
Abstract: The three ways of generalization of canonical coherent states are briefly reviewed and compared with the emphasis laid on the (minimum) uncertainty way. The characteristic uncertainty relations, which include the Schroedinger and Robertson inequalities, are extended to the case of several states. It is shown that the standard SU(1,1) and SU(2) coherent states are the unique states which minimize the second order characteristic inequality for the three generators. A set of states which minimize the Schroedinger inequality for the Hermitian components of the su_q(1,1) ladder operator is also constructed. It is noted that the characteristic uncertainty relations can be written in the alternative complementary form.
TL;DR: The ladder operator formalism of a general quantum state for su(1, 1) Lie algebra is obtained in this paper, where the state bears the generally deformed oscillator algebraic structure.
Abstract: The ladder operator formalism of a general quantum state for su(1, 1) Lie algebra is obtained. The state bears the generally deformed oscillator algebraic structure. It is found that the Perelomov's coherent state is a su(1, 1) nonlinear coherent state. The expansion and the exponential form of the nonlinear coherent state are given. We obtain the matrix elements of the su(1, 1) displacement operator in terms of the hypergeometric functions and the expansions of the displaced number states and Laguerre polynomial states are followed. Finally some interesting su(1, 1) optical systems are discussed.
TL;DR: In this article, the authors introduce orthogonal even nonlinear coherent states and study their statistical properties in terms of orthogonality and non-coherence, and show that their properties are similar to ours.
Abstract: We introduce orthogonal even nonlinear coherent states and study theirstatistical properties.
TL;DR: In this paper, the authors introduced the concept of number-difference-phase squeezing, which is a special combination of generalized SU(1, 1) coherent state for the Noh, Fougers and Mandel (NFM) operational phase operator.
Abstract: For the two-mode photon number-difference operator D = a†a -b†b and the Noh, Fougers and Mandel (NFM) operational phase operator , we introduce the concept of number-difference-phase squeezing. We then find a new minimum-uncertainty state for number-difference-phase squeezing, which turns out to be a special combination of generalized SU(1,1) coherent state. As a by-product, a new orthonormal and complete representation in two-mode Fock space made up of R†n|m, m> is found.
TL;DR: In this paper, the ladder operator formalisms of two general states defined in the even/odd Fock space were obtained for one-and two-photon quantum states, respectively.
Abstract: We show that various kinds of one-photon quantum states studied in the field of quantum optics admit ladder operator formalisms and have the generally deformed oscillator (GDO) algebraic structure. The two-photon case is also considered. We obtain the ladder operator formalisms of two general states defined in the even/odd Fock space. The two-photon states may also have a GDO algebraic structure. Some interesting examples of one- and two-photon quantum states are given.
TL;DR: In this paper, the superposition coherent states (SCS) were studied and their quantum statistical properties, the fluctuations of field and squeezing have been discussed in detail, and the squeezing regions in phase space for these states were described.
Abstract: Special kinds of generalized superposition states, superposition coherent states, are studied in this paper. These states can be produced by superposing a pair of coherent states |a〉 and |−a〉. Their quantum statistical properties, the fluctuations of field and squeezing have been discussed in detail. These properties are dependent on superposition phase. We also describe the squeezing regions in phase space for these states.
TL;DR: In this paper, the intelligent states associated with the Holstein-Primakoff realization of (2) were studied and the explicit expressions of these states in terms of the Gauss hypergeometric functions were derived and their statistical properties were investigated in detail.
Abstract: We study the intelligent states associated with the Holstein-Primakoff realization of (2). The explicit expressions of these states in terms of the Gauss hypergeometric functions are derived and their statistical properties are investigated in detail. It is shown that in some special or asymptotic cases these states turn out to be such important states as the binomial states, number states, Glauber coherent states, squeezed coherent states, etc in quantum optics.
Posted Content•
TL;DR: In this article, a unified approach for finding coherent states of polynomially deformed algebras such as the quadratic and Higgs, which are relevant for various multiphoton processes in quantum optics is presented.
Abstract: We present a general unified approach for finding the coherent states of polynomially deformed algebras such as the quadratic and Higgs algebras, which are relevant for various multiphoton processes in quantum optics. We give a general procedure to map these deformed algebras to appropriate Lie algebras. This is used, for the non compact cases, to obtain the annihilation operator coherent states, by finding the canonical conjugates of these operators.
Generalized coherent states, in the Perelomov sense also follow from this construction. This allows us to explicitly construct coherent states associated with various quantum optical systems.
TL;DR: In this paper, a new number-difference-phase coherent state analogue in two-mode Fock space was proposed, which possesses non-orthonormal and overcompleteness properties.
Abstract: We propose a new number-difference-phase coherent state analogue in two-mode Fock space by introducing a new operator . The coherent state analogue is the eigenvector of A and possesses non-orthonormal and overcompleteness properties. It is constructed on certain superposition states in the radius direction.