Family Of Coherent States Research Articles

SU(1,1)-displaced coherent states, photon counting, and squeezing

We revisit the Perelomov SU(1,1)-displaced coherent states as possible quantum states of light. We disclose interesting statistical aspects of these states in relation to photon counting and squeezing. In the non-displaced case, we discuss the efficiency of the photodetector as inversely proportional to the parameterϰof the discrete series of unitary irreducible representations of SU(1,1). In the displaced case, we study the counting and squeezing properties of the states in terms ofϰand the number of photons in the original displaced state. We finally examine the quantization of a classical radiation field based on these families of coherent states. The procedure yields displacement operators that might allow to prepare such states in the way proposed by Glauber for standard coherent states.

Coherent states for a system of an electron moving in a plane: case of discrete spectrum

In this work, we construct different classes of coherent states related to a quantum system, recently studied in [1], of an electron moving in a plane in uniform external magnetic and electric fields which possesses both discrete and continuous spectra. The eigenfunctions are realized as an orthonormal basis of a suitable Hilbert space appropriate for building the related coherent states. These latter are achieved in the context where we consider both spectra purely discrete obeying the criteria that a family of coherent states must satisfy.

Generalized Susskind–Glogower coherent states

Susskind-Glogower coherent states, whose Fock expansion coefficients include Bessel functions, have recently attracted considerable attention for their optical properties. Nevertheless, identity resolution is still an open question, which is an essential mathematical property that defines an overcomplete basis in the Fock space and allows a coherent state quantization map. In this regard, the modified Susskind-Glogower coherent states have been introduced as an alternative family of states that resolve the identity resolution. In the present manuscript, the quantization map related to the modified Susskind-Glogower coherent states is exploited, which naturally leads to a particular representation of the $\mathfrak{su}(1,1)$ Lie algebra in its discrete series. The latter provides evidence about further generalizations of coherent states, built from the Susskind-Glogower ones by extending the indexes of the Bessel functions of the first kind and, alternatively, by employing the modified Bessel functions of the second kind. In this form, the new families of Susskind-Glogower-I and Susskind-Glogower-II coherent states are introduced. The corresponding quantization maps are constructed so that they lead to general representations of elements of the $\mathfrak{su}(1,1)$ and $\mathfrak{su}(2)$ Lie algebras as generators of the SU$(1,1)$ and SU$(2)$ unitary irreducible representations respectively. For completeness, the optical properties related to the new families of coherent states are explored and compared with respect to some well-known optical states.

Polymer quantization of connection theories: Graph coherent states

We present the construction of a new family of coherent states for quantum theories of connections obtained following the polymer quantization. The realization of these coherent states is based on the notion of graph change, in particular the one induced by the quantum dynamics in Yang-Mills and gravity quantum theories. Using a Fock-like canonical structure that we introduce, we derive the new coherent states that we call the graph coherent states. These states take the form of an infinite superposition of basis network states with different graphs. We further discuss the properties of such states and certain extensions of the proposed construction.

Coherent states for the quantum complete rigid rotor

Abstract Motivated by the possibility to describe orientations of quantum triaxial rigid rotors, such as molecules, with respect to both internal (body-fixed) and external (laboratory) frames, we go through the theory of coherent states and design the appropriate family of coherent states on T ∗ SO ( 3 ) , the classical phase space of the freely rotating rigid body (the Euler top). We pay particular attention to the resolution of identity property in order to establish the explicit relation between the parameters of the coherent states and classical phase-space variables, actions and angles.

Noncommutative coherent states and related aspects of Berezin–Toeplitz quantization**Dedicated by the first and the third authors to the memory of the second author, with gratitude for his friendship and for all they learnt from him.

In this paper, we construct noncommutative coherent states using various families of unitary irreducible representations (UIRs) of , a connected, simply connected nilpotent Lie group, which was identified as the kinematical symmetry group of noncommutative quantum mechanics for a system of two degrees of freedom in an earlier paper. Similarly described are the degenerate noncommutative coherent states arising from the degenerate UIRs of . We then compute the reproducing kernels associated with both these families of coherent states and study the Berezin–Toeplitz quantization of the observables on the underlying 4-dimensional phase space, analyzing in particular the semi-classical asymptotics for both these cases.

Action-angle coherent states for quantum systems with cylindric phase space

Quantum versions of cylindric phase space, like for the motion of a particle on a circle, are obtained through different families of coherent states. The latter are built from various probability distributions of the action variable. The method is illustrated with Gaussian distributions and uniform distributions on intervals, and resulting quantizations are explored.

The Schrödinger representation and its relation to the holomorphic representation in linear and affine field theory

We establish a precise isomorphism between the Schrödinger representation and the holomorphic representation in linear and affine field theory. In the linear case, this isomorphism is induced by a one-to-one correspondence between complex structures and Schrödinger vacua. In the affine case we obtain similar results, with the role of the vacuum now taken by a whole family of coherent states. In order to establish these results we exhibit a rigorous construction of the Schrödinger representation and use a suitable generalization of the Segal-Bargmann transform. Our construction is based on geometric quantization and applies to any real polarization and its pairing with any Kähler polarization.

Some orthogonal polynomials arising from coherent states

We explore in this paper some orthogonal polynomials which are naturally associated with certain families of coherent states, often referred to as nonlinear coherent states in the quantum optics literature. Some examples turn out to be known orthogonal polynomials but in many cases we encounter a general class of new orthogonal polynomials for which we establish several qualitative results.

Coherent State Quantization and Moment Problem

Berezin-Klauder-Toeplitz (“anti-Wick”) or “coherent state” quantization of the complex plane, viewed as the phase space of a particle moving on the line, is derived from the resolution of the unity provided by the standard (or gaussian) coherent states. The construction of these states and their attractive properties are essentially based on the energy spectrum of the harmonic oscillator, that is on natural numbers. We follow in this work the same path by considering sequences of non-negative numbers and their associated “non-linear” coherent states. We illustrate our approach with the 2-d motion of a charged particle in a uniform magnetic field. By solving the involved Stieltjes moment problem we construct a family of coherent states for this model. We then proceed with the corresponding coherent state quantization and we show that this procedure takes into account the circle topology of the classical motion.

ON CONSTRUCTION OF COHERENT STATES ASSOCIATED WITH SEMIDIRECT PRODUCTS

Most of the interesting groups encountered in the physics literature are semidirect product groups. These groups are of the general form G = H ×τ K, where H and K are locally compact groups. In this paper, we consider the quasi regular representation on such a group G and investigate when it is possible to construct a family of coherent states associated to this representation.

Coherent states and Bayesian duality

We demonstrate how large classes of discrete and continuous statistical distributions can be incorporated into coherent states, using the concept of a reproducing kernel Hilbert space. Each family of coherent states is shown to contain, in a sort of duality, which resembles an analogous duality in Bayesian statistics, a discrete probability distribution and a discretely parametrized family of continuous distributions. It turns out that nonlinear coherent states, of the type widely studied in quantum optics, are a particularly useful class of coherent states from this point of view, in that they contain many of the standard statistical distributions. We also look at vector coherent states and multidimensional coherent states as carriers of mixtures of probability distributions and joint probability distributions.

Open boundaries for the nonlinear Schrödinger equation

We present a new algorithm, the time dependent phase space filter (TDPSF) which is used to solve time dependent nonlinear Schrödinger equations (NLS). The algorithm consists of solving the NLS on a box with periodic boundary conditions (by any algorithm). Periodically in time we decompose the solution into a family of coherent states. Coherent states which are outgoing are deleted, while those which are not are kept, reducing the problem of reflected (wrapped) waves. Numerical results are given, and rigorous error estimates are described. The TDPSF is compatible with spectral methods for solving the interior problem. The TDPSF also fails gracefully, in the sense that the algorithm notifies the user when the result is incorrect. We are aware of no other method with this capability.

MODULI OF QUANTA

The classical phase of the matrix model of 11-dimensional M-theory is complex, infinite-dimensional Hilbert space. As a complex manifold, the latter admits a continuum of nonequivalent, complex-differentiable structures that can be placed in 1-to-1 correspondence with families of coherent states in the Hilbert space of quantum states. The moduli space of nonbiholomorphic complex structures on classical phase space turns out to be an infinite-dimensional symmetric space. We argue that each choice of a complex differentiable structure gives rise to a physically different notion of an elementary quantum.

Coherent States for the Legendre Oscillator

Two families of coherent states (the eigenstates of the annihilation operator and the Klauder–Gazeau temporally stable coherent states) are constructed for the Legendre oscillator defined in this paper. Bibliography: 14 titles.

Coherent states for the unitary symplectic group

We present an explicit construction of coherent states for an arbitrary irreducible representation of the unitary symplectic group USp(4). Three different families of coherent states are obtained, corresponding to the subgroups U(1) × U(1), U(2) and SU(2) × SU(2). The symplectic structure on the manifold of coherent states is obtained, and canonical coordinates are used to express the classical limit of quantum observables. One of the families is seen to provide a trivial classical limit.

Combinatorial Coherent States via Normal Ordering of Bosons

We construct and analyze a family of coherent states built on sequences of integers originating from the solution of the boson normal ordering problem. These sequences generalize the conventional combinatorial Bell numbers and are shown to be moments of positive functions. Consequently, the resulting coherent states automatically satisfy the resolution of unity condition. In addition they display such non-classical fluctuation properties as super-Poissonian statistics and squeezing.

Angle operators on weighted Bergman spaces

Among the multiplication operators on weighted Bergman Hilbert spaces are those where the multiplying function (operator symbol) depends only on the angular polar coordinate in the unit disk: we call these “angle operators.” As these Hilbert spaces carry a CCR representation unitarily equivalent to the Schrödinger representation, angle operators are associated with quantum phase in the same way as are Toeplitz operators, for example. We determine the matrix elements of the angle operators with respect to the natural orthonormal basis on each of these spaces, and also with respect to the appropriate family of coherent states. By using a method of comparison with the corresponding results for Toeplitz operators, asymptotic expressions for the expectations and variances in these two families of states are obtained for the angle operators whose symbols are the polar angle function and its two complex exponentials. Notable is the fact that the asymptotic limit of the variance of the polar angle operator in the natural basis family is π2/3, which many authors take to be a requirement for a quantum phase operator.

TRICOMI COHERENT STATES

We construct explicit families of coherent states whose parametrization permits an arbitrarily small deviation from conventional coherent states. They possess the resolution of unity with a positive weight function given as the solution of a Stieltjes power-moment problem, with moments analytically expressible through Tricomi's confluent hypergeometric function. These states are in general squeezed, can be either sub- or super-Poissonian in nature, and induce a deformation of the metric of the complex plane.

NON-HERMITIAN OSCILLATOR-LIKE HAMILTONIANS AND λ-COHERENT STATES REVISITED

Previous λ-deformed non-Hermitian Hamiltonians with respect to the usual scalar product of Hilbert spaces dealing with harmonic oscillator-like developments are (re)considered with respect to a new scalar product in order to take into account their property of self-adjointness. The corresponding deformed λ-states lead to new families of coherent states according to the DOCS, AOCS and MUCS points of view.