Showing papers on "Coherent states in mathematical physics published in 2001"
TL;DR: In this article, a general construction of diffeomorphism covariant coherent states for quantum gauge theories is presented, which is the first paper in a series of articles entitled ''Gauge field theory coherent states (GCS)' which aims to connect nonperturbative quantum general relativity with the low-energy physics of the standard model.
Abstract: In this paper we outline a rather general construction of diffeomorphism covariant coherent states for quantum gauge theories. By this we mean states ψ(A,E), labelled by a point (A,E) in the classical phase space, consisting of canonically conjugate pairs of connections A and electric fields E, respectively, such that: (a) they are eigenstates of a corresponding annihilation operator which is a generalization of A-iE smeared in a suitable way; (b) normal ordered polynomials of generalized annihilation and creation operators have the correct expectation value; (c) they saturate the Heisenberg uncertainty bound for the fluctuations of Â,E; and (d) they do not use any background structure for their definition, that is, they are diffeomorphism covariant. This is the first paper in a series of articles entitled `Gauge field theory coherent states (GCS)' which aims to connect non-perturbative quantum general relativity with the low-energy physics of the standard model. In particular, coherent states enable us for the first time to take into account quantum metrics which are excited everywhere in an asymptotically flat spacetime manifold as is needed for semiclassical considerations. The formalism introduced in this paper is immediately applicable also to lattice gauge theory in the presence of a (Minkowski) background structure on a possibly infinite lattice.
241 citations
TL;DR: In this paper, a particle trapped in an infinite square-well and also in Poschl-Teller potentials of the trigonometric type is shown to share a common SU(1,1) symmetry.
Abstract: This article is a direct illustration of a construction of coherent states which has been recently proposed by two of us (JPG and JK). We have chosen the example of a particle trapped in an infinite square-well and also in Poschl–Teller potentials of the trigonometric type. In the construction of the corresponding coherent states, we take advantage of the simplicity of the solutions, which ultimately stems from the fact they share a common SU(1,1) symmetry a la Barut-Girardello. Many properties of these states are then studied, both from mathematical and from physical points of view.
238 citations
TL;DR: In this article, the authors apply the methods outlined in the previous paper of this series to the particular set of states obtained by choosing the complexifier to be a Laplace operator for each edge of a graph.
Abstract: In this article we apply the methods outlined in the previous paper of this series to the particular set of states obtained by choosing the complexifier to be a Laplace operator for each edge of a graph. The corresponding coherent state transform was introduced by Hall for one edge and generalized by Ashtekar, Lewandowski, Marolf, Mourao and Thiemann to arbitrary, finite, piecewise analytic graphs. However, both of these works were incomplete with respect to the following two issues : (a) The focus was on the unitarity of the transform and left the properties of the corresponding coherent states themselves untouched. (b) While these states depend in some sense on complexified connections, it remained unclear what the complexification was in terms of the coordinates of the underlying real phase space. In this paper we complement these results : First, we explicitly derive the com- plexification of the configuration space underlying these heat kernel coherent states and, secondly, prove that this family of states satisfies all the usual properties : i) Peakedness in the configuration, momentum and phase space (or Bargmann-Segal) representation. ii) Saturation of the unquenched Heisenberg uncertainty bound. iii) (Over)completeness. These states therefore comprise a candidate family for the semi-classical anal- ysis of canonical quantum gravity and quantum gauge theory coupled to quantum gravity. They also enable error-controlled approximations to difficult analytical cal- culations and therefore set a new starting point for numerical canonical quantum general relativity and gauge theory. The text is supplemented by an appendix which contains extensive graphics in order to give a feeling for the so far unknown peakedness properties of the states constructed.
216 citations
170 citations
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TL;DR: In this article, the authors introduce several basic theorems of coherent states and generalized coherent states based on Lie algebras and give some applications of them to quantum information theory for graduate students or non-experts who are interested in both Geometry and Quantum Information Theory.
Abstract: The purpose of this paper is to introduce several basic theorems of coherent states and generalized coherent states based on Lie algebras su(2) and su(1,1), and to give some applications of them to quantum information theory for graduate students or non--experts who are interested in both Geometry and Quantum Information Theory In the first half we make a general review of coherent states and generalized coherent states based on Lie algebras su(2) and su(1,1) from the geometric point of view and, in particular, prove that each resolution of unity can be obtained by the curvature form of some bundle on the parameter space In the latter half we apply a method of generalized coherent states to some important topics in Quantum Information Theory, in particular, swap of coherent states and cloning of coherent ones We construct the swap operator of coherent states by making use of a generalized coherent operator based on su(2) and show an "imperfect cloning" of coherent states, and moreover present some related problems In conclusion we state our dream, namely, a construction of {\bf Geometric Quantum Information Theory}
68 citations
TL;DR: In this article, the authors define coherent states for SU(3) using six bosonic creation and annihilation operators, which satisfy continuity properties and possess a variety of group theoretic properties.
Abstract: We define coherent states for SU(3) using six bosonic creation and annihilation operators. These coherent states are explicitly characterized by six complex numbers with constraints. For the completely symmetric representations (n,0) and (0,m), only three of the bosonic operators are required. For mixed representations (n,m), all six operators are required. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties. We introduce an explicit parametrization of the group SU(3) and the corresponding integration measure. Finally, we discuss the path integral formalism for a problem in which the Hamiltonian is a function of SU(3) operators at each site.
50 citations
TL;DR: In this article, two coherent states associated with the cyclic group of order λ$-extended oscillator algebra are introduced, which satisfy a unity resolution relation in the Fock space and give rise to Bargmann representations of the latter.
Abstract: Two new types of coherent states associated with the $C_{\lambda}$-extended oscillator, where $C_{\lambda}$ is the cyclic group of order $\lambda$, are introduced. They satisfy a unity resolution relation in the $C_{\lambda}$-extended oscillator Fock space (or in some subspace thereof) and give rise to Bargmann representations of the latter, wherein the generators of the $C_{\lambda}$-extended oscillator algebra are realized as differential-operator-valued matrices.
37 citations
TL;DR: In this paper, the nearest Hilbert-Schmidt distance of pure states to coherent states is discussed as a quantitative measure of nonclassicality of states in quantum optics or quantum mechanics of harmonic oscillators.
Abstract: The nearest Hilbert-Schmidt distance of (pure) states to coherent states is discussed as a quantitative measure of nonclassicality of states in quantum optics or quantum mechanics of harmonic oscillators. Fock states, squeezed coherent states and finite superpositions of coherent states (Schrodinger cat states) are considered as examples. Some other approaches to nonclassicality of states in quantum optics are shortly discussed.
27 citations
TL;DR: In this paper, the authors discussed the canonical quantization of (1+1)-dimensional Yang-Mills theory on a spacetime cylinder from the point of view of coherent states, or equivalently, the Segal-Bargmann transform.
Abstract: This paper discusses the canonical quantization of (1+1)-dimensional Yang–Mills theory on a spacetime cylinder from the point of view of coherent states, or equivalently, the Segal–Bargmann transform. Before gauge symmetry is imposed, the coherent states are simply ordinary coherent states labeled by points in an infinite-dimensional linear phase space. Gauge symmetry is imposed by projecting the original coherent states onto the gauge-invariant subspace, using a suitable regularization procedure. We obtain in this way a new family of "reduced" coherent states labeled by points in the reduced phase space, which in this case is simply the cotangent bundle of the structure group K. The main result explained here, obtained originally in a joint work of the author with B. Driver, is this: The reduced coherent states are precisely those associated to the generalized Segal–Bargmann transform for K, as introduced by the author from a different point of view. This result agrees with that of K. Wren, who uses a different method of implementing the gauge symmetry. The coherent states also provide a rigorous way of making sense out of the quantum Hamiltonian for the unreduced system. Various related issues are discussed, including the complex structure on the reduced phase space and the question of whether quantization commutes with reduction.
26 citations
TL;DR: In this paper, group symmetries can be used to reconstruct quantum states in the context of the two-mode SU(1,1) states of the radiation field, where the input field passes through a nondegenerate parametric amplifier and one measures the probability of finding the output state with a certain number (usually zero) of photons in each mode.
Abstract: We show how group symmetries can be used to reconstruct quantum states The method we propose is presented in the context of the two-mode SU(1,1) states of the radiation field In our scheme for SU(1,1) states, the input field passes through a nondegenerate parametric amplifier and one measures the probability of finding the output state with a certain number (usually zero) of photons in each mode The density matrix in the Fock basis is retrieved from the measured data by the least-squares method after singular value decomposition of the design matrix followed by Tikhonov regularization Several illustrative examples involving the reconstruction of a pair coherent state, a Perelomov coherent state, and a coherent superposition of pair coherent states are considered
22 citations
TL;DR: The relationships between certain important nonclassical states of the quantized field and the coherent states associated with the SU(2) and SU(1,1) Lie groups and associated Lie algebras is briefly reviewed.
Abstract: The relationships between certain important nonclassical states of the quantized field and the coherent states associated with the SU(2) and SU(1,1) Lie groups and associated Lie algebras is briefly reviewed. As an example of the utility of group theoretical methods in quantum optics, a method for generating maximally entangled photonic states is discussed. These states may be of great importance for achieving Heisenberg-limited interferometry and in beating the diffraction limit in lithography.
TL;DR: In this paper, the Segal-Bargmann representation space of the one-mode interacting Fock space was given and the generalized coherent vector associated with the Szego-Jacobi parameters was defined.
Abstract: In this paper, we shall give the Segal–Bargmann representation space of the one-mode interacting Fock space. For this purpose, we shall define the generalized coherent vector associated with the Szego–Jacobi parameters and introduce the associated integral transformation.
TL;DR: In this paper, the coherent states of the Morse potential were connected with the representations of the affine group of the real line and some of its extensions, and a generalized Wigner function on this phase space was constructed.
Abstract: The coherent states of the Morse potential that have been obtained earlier from supersymmetric quantum mechanics, are shown to be connected with the representations of the affine group of the real line and some of its extensions. This relation is similar to the one between the Heisenberg-Weyl group and the coherent states of the harmonic oscillator. The states that minimize the uncertainty product of the generators of the affine Lie algebra are shown to contain all the coherent states of the Morse oscillator plus the intelligent states of the Morse Hamiltonians with different shape parameter s. The representations of the central extension of the affine group denoted by G0 and its further extension will be shown to define the phase space relevant to the problem by choosing an appropriate orbit of the coadjoint representation of . This allows one to construct a generalized Wigner function on this phase space, which is again essentially in the same relation with the affine group, as the ordinary Wigner function with the Heisenberg-Weyl group.
TL;DR: In this paper, a quantum decomposition of generators of a discrete group with a suitable length function is introduced and a sequence of one-mode interacting Fock spaces associated with a filtration of the group is constructed.
Abstract: We introduce a kind of quantum decomposition of generators of a discrete group with a suitable length function and construct a sequence of one-mode interacting Fock spaces associated with a filtration of the group. We show stochastic convergence of such a sequence of interacting Fock spaces and obtain a general stochastic limit theorems on discrete groups. As an application to free groups, we see that the Haagerup states give rise to a Gaussian–Poisson transform through its coherent expression.
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TL;DR: In this article, the authors show an interesting relation between coherent states and the Bell states in the case of spin 1/2, which was suggested by Fivel, and try to generalize this relation to get several generalized Bell states.
Abstract: In the first half we show an interesting relation between coherent states and the Bell states in the case of spin 1/2, which was suggested by Fivel. In the latter half we treat generalized coherent states and try to generalize this relation to get several generalized Bell states. Our method is based on a geometry and our task may give a hint to open a deep relation between a coherence and an entanglement.
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TL;DR: In this paper, it was shown that quantum computation circuits with coherent states as the logical qubits can be constructed using very simple linear networks, conditional measurements and coherent superposition resource states.
Abstract: We show that quantum computation circuits with coherent states as the logical qubits can be constructed using very simple linear networks, conditional measurements and coherent superposition resource states.
TL;DR: In this paper, the authors studied the properties of all level k coherent states in the context of string theory on a group manifold (WZWN) models and provided the construction of states (i) and compared the two sets and discuss their properties.
TL;DR: In this article, the authors consider the time evolution of generalized coherent states based on non-standard fiducial vectors, and show that only for a restricted class of such vectors does the associated classical motion determine the quantum evolution of the states.
Abstract: I consider the time evolution of generalized coherent states based on non-standard fiducial vectors, and show that only for a restricted class of such vectors does the associated classical motion determine the quantum evolution of the states. I discuss some consequences of this for path integral representations.
TL;DR: In this article, it is shown that the diagonal matrix elements of all-order coherent states for the quantized electromagnetic field have to constitute a Poisson distribution with respect to the photon number.
TL;DR: In this paper, the nonlinear SU(3) charged and hypercharged bosonic coherent state in three-mode Fock space was introduced and recast into a compact exponential form.
Abstract: We introduce the nonlinear SU(3) charged and hypercharged bosonic coherent state in three-mode Fock space. It can be further recast into a compact exponential form. The fermionic case is also discussed.
TL;DR: In this article, the authors consider arithmetical aspects of analysis on Fock spaces (Boson Fock space, Fermion Fockspace, and Boson-Fermion space) with applications to analytic number theory.
Abstract: We consider arithmetical aspects of analysis on Fock spaces (Boson Fock space, Fermion Fock space, and Boson-Fermion Fock space) with applications to analytic number theory.
TL;DR: In this paper, the main properties of standard quantum mechanical coherent states and the two generalizations of Klauder and Perelomov are reviewed and necessary and sufficient conditions for existence of a diagonal coherent stable representation for all Hilbert-Schmidt operators are obtained.
Abstract: The main properties of standard quantum mechanical coherent states and the two generalizations of Klauder and of Perelomov are reviewed. For a system of generalized coherent states in the latter sense, necessary and sufficient conditions for existence of a diagonal coherent stable representation for all Hilbert-Schmidt operators are obtained. The main ingredients are Clebsch-Gordan theory and induced representation theory.
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TL;DR: In this article, it was shown that the quantum theory can be formulated on homogeneous spaces of generalized coherent states in a manner that accounts for interference, entanglement, and the linearity of dynamics without using the superposition principle.
Abstract: It is shown that the quantum theory can be formulated on homogeneous spaces of generalized coherent states in a manner that accounts for interference, entanglement, and the linearity of dynamics without using the superposition principle. The coherent state labels, which are essentially instructions for preparing states, make it unnecessary to identify properties with projectors in Hilbert space. This eliminates the so called "eigenvalue-eigenstate" link, and the theory thereby escapes the measurement problem. What the theory allows us to predict is the distribution in the outcomes of tests of relations between coherent states. It is shown that quantum non-determinism can be attributed to a hidden variable (noise) in the space of relations without violating the no-go theorems (e.g. Kochen-Specker). It is shown that the coherent state vacuum is distorted when entangled states are generated. The non-locality of the vacuum permits this distortion to be felt everywhere without the transmission of a signal and thereby accounts for EPR correlations in a manifestly covariant way.
TL;DR: In this paper, it was shown that the Wigner distribution function of quantum optical coherent states, or of a superposition of such states, can be produced and measured with a classical optical set-up using classical coherent light fields.
Abstract: The paper shows that the Wigner distribution function of quantum optical coherent states, or of a superposition of such states, can be produced and measured with a classical optical set-up using classical coherent light fields. This measurement cannot be done directly in quantum optics since the quantum phase space variables correspond to non-commuting operators. As an example, the Wigner distribution function of Schrodinger cat states of light has been measured. It is also shown that the possibility of measuring the Wigner distribution function of quantum coherent states with classical coherent fields is unique in the sense that it cannot be extended to other quantum states, not even to the incoherent limit of the superposition of coherent states.
TL;DR: In this paper, a new kind of two-mode bosonic realization of SU(1,1) Lie algebra is constructed, on the basis of which the SU( 1,1)-generalized coherent states in the twomode Fock space are derived.
Abstract: We have constructed a new kind of two-mode bosonic realization of SU(1,1) Lie algebra, on the basis of which the SU(1,1) generalized coherent states in the two-mode Fock space are derived. These two-mode SU(1,1) coherent states, which are called uncorrelated two-mode SU(1,1) coherent states, include three special cases. For these states, we study the mean photon number distribution and their non-classical properties, which are photon anti-bunching, violations of Cauchy-Schwarz inequality and two-mode squeezing.
TL;DR: In this paper, the authors presented a simulation of the BB84 protocol, using the continuum coherent states, in a fibre interferometer commonly used in quantum cryptography, and observed the fluctuations of the mean photon number in the pulses that arrive at Bob, introduced by the statistical property of the simulation.
Abstract: The continuum states formalism is suitable for field quantization in optical fibre; however, they are harder to use than discrete states. On the other hand, a Hermitian phase operator can be defined only in a finite dimensional space. We approximated a coherent continuum state by a finite tensor product of coherent states, each one defined in a finite dimensional space. Using this, in the correct limit, we were able to obtain some statistical properties of the photon number and phase of the continuum coherent states from the probability density functions of the individual, finite dimensional, coherent states. Then, we performed a simulation of the BB84 protocol, using the continuum coherent states, in a fibre interferometer commonly used in quantum cryptography. We observed the fluctuations of the mean photon number in the pulses that arrive at Bob, which occurs in the practical system, introduced by the statistical property of the simulation.
TL;DR: In this article, the even and odd binomial states are eigenstates of the definite nonlinear combination of radiation field operators, which generalizes the nonlinear coherent state theory in quantum optics.
Abstract: We show that the even- and odd-binomial states are eigenstates of the definite nonlinear combination of radiation field operators, which generalizes the nonlinear coherent state theory in quantum optics.
TL;DR: In this paper, the SUq(2) algebra is extended by introducing additional raising and lowering operators and constructing their coherent states, and a new algebra of coherent states and the commutation relations between the extended operators are investigated.
Abstract: The SUq(2) algebra is extended by introducing additional raising and lowering operators and constructing their coherent states. This new algebra of coherent states and the commutation relations between the extended operators are investigated and a resolution of unity is proposed.
TL;DR: In this article, the Perelomov coherent states of SU(1,1) are labeled by elements of the quotient ofSU(1) by its rotation subgroup.
Abstract: The Perelomov coherent states ofSU(1,1) are labeled by elements of the quotient ofSU(1,1) by its rotation subgroup. Taking advantage of the fact that this quotient is isomorphic to the affine group of the real line, we are able to parameterize the coherent states by elements of that group. Such a formulation permits to find new properties of theSU(1,1) coherent states and to relate them to affine wavelets.
TL;DR: In this paper, two kinds of nonlinear displaced Fock states (NLDFS) are introduced, which are composed by nonlinear displacing operator on the nonlinear Fock state Df(α)| n>f and.
Abstract: As a generalization of the nonlinear coherent states we construct nonlinear displaced Fock states (NLDFS). In contrast to the ordinary displaced Fock state, two kinds of NLDFS are introduced, which are composed by nonlinear displacing operator on the nonlinear Fock states Df(α)| n>f and . The over-completeness relation for NLDFS is obtained by virtue of the generalized technique of integration within an ordered product of operators. Generation of some kind of NLDFS is briefly discussed.