Showing papers on "Coherent states in mathematical physics published in 2013"
TL;DR: In this article, the most general displaced number coherent states, based on the Heisenberg, su(2) and su(1, 1) Lie algebras symmetries, are constructed.
Abstract: The most general displaced number ?coherent? states, based on the Heisenberg, su(2) and su(1, 1) Lie algebras symmetries, are constructed. They depend on two parameters, and can be converted into the well-known photon-added, two variable Glauber coherent states and displaced number states respectively, depending on which of the parameters is equal to zero. The relations of the Weyl?Heisenberg algebra guarantee a corresponding resolution of the identity conditions. A discussion of the statistical properties of these states is included. Significant are their squeezing properties, which can be raised by increasing the energy and angular momentum quantum numbers n and m. The maximum squeezing is obtained for Bext = 0. Depending on the particular choice of parameters in the above scenarios, we are able to determine the status of compliance with Poissonian statistics. In the limiting case, we obtain a major result about the non-classical properties of the Glauber minimum uncertainty coherent states. In other words, in addition to the requirement to minimize uncertainty conditions, they carry non-classical features too. Finally, a theoretical framework is proposed to generate them.
23 citations
TL;DR: In this article, the authors explore the group theoretical underpinning of non-commutative quantum mechanics for a system moving on the two-dimensional plane and show that the pertinent groups for the system are the twofold central extension of the Galilei group in (2 + 1)-space-time dimensions.
Abstract: We explore the group theoretical underpinning of noncommutative quantum mechanics for a system moving on the two-dimensional plane. We show that the pertinent groups for the system are the two-fold central extension of the Galilei group in (2 + 1)-space-time dimensions and the two-fold extension of the group of translations of R 4 . This latter group is just the standard Weyl-Heisenberg group of standard quantum mechanics with an additional central extension. We also look at a further extension of this group and discuss its significance to noncommutative quantum mechanics. We build unitary irreducible representations of these various groups and construct the associated families of coherent states. A coherent state quantization of the underlying phase space is then carried out, which is shown to lead to exactly the same commutation relations as usually postulated for this model of noncommutative quantum mechanics.
20 citations
TL;DR: In this article, supersymmetric coherent states that are eigenstates of a general four-parameter family of annihilation operators are defined as operators in Fock space that transform a definite number of particles into a subspace with one particle removed.
Abstract: This study presents supersymmetric coherent states that are eigenstates of a general four-parameter family of annihilation operators. The elements of this family are defined as operators in Fock space that transform a subspace of a definite number of particles into a subspace with one particle removed. The emphasis is on classifying parameter space in various regions according to the uncertainty bounds of the corresponding coherent states. Specifically, the uncertainty in position-momentum is analyzed, with specific focus on characterizing regions of minimum uncertainty states, regions where the uncertainties are bounded from above, and where they grow unbound.
18 citations
TL;DR: In this paper, the authors present the full characterization of phase-randomized or phase-averaged coherent states, a class of states exploited in communication channels and in decoy state-based quantum key distribution protocols.
Abstract: We present the full characterization of phase-randomized or phase-averaged coherent states, a class of states exploited in communication channels and in decoy state-based quantum key distribution protocols. We report on the suitable formalism to analytically describe the main features of these states and on their experimental investigation, that results in agreement with theory. In particular, we consider a recently proposed non-Gaussianity measure based on the quantum fidelity, that we compare with previous ones, and we use the mutual information to investigate the amount of correlations one can produce by manipulating this class of states.
16 citations
TL;DR: In this paper, the α-deformed Weyl-heisenberg algebra is used to obtain the su(2)- and su(1, 1)-algebras whenever α has specific values.
Abstract: At first, we introduce α-deformed algebra as a kind of generalization of the Weyl–Heisenberg algebra so that we get the su(2)- and su(1, 1)-algebras whenever α has specific values. After that, we construct coherent states of this algebra. Third, a realization of this algebra is given in the system of a harmonic oscillator confined at the center of a potential well. Then, we introduce two-boson realization of the α-deformed Weyl–Heisenberg algebra and use this representation to write α-deformed coherent states in terms of the two modes number states. Following these points, we consider mean number of excitations (we call them in general photons) and Mandel parameter as statistical properties of the α-deformed coherent states. Finally, the Fubini–Study metric is calculated for the α-coherent states manifold.
13 citations
TL;DR: In this paper, a coherent state representation of the Jacobi group with two parameters, the part coming from the Heisenberg group and the part from the positive discrete series representation, is presented.
Abstract: The coherent state representation of the Jacobi group $G^J_1$ is indexed with two parameters, $\mu (=\frac{1}{\hbar})$, describing the part coming from the Heisenberg group, and $k$, characterizing the positive discrete series representation of $\text{SU}(1,1)$. The Ricci form, the scalar curvature and the geodesics of the Siegel-Jacobi disk $\mathcal{D}^J_1$ are investigated. The significance in the language of coherent states of the transform which realizes the fundamental conjecture on the Siegel-Jacobi disk is emphasized. The Berezin kernel, Calabi's diastasis, the Kobayashi embedding, and the Cauchy formula for the Sigel-Jacobi disk are presented.
11 citations
TL;DR: In this article, the truncated coherent state associated with a particular form of Morse potential is constructed as a finite-dimensional quantum system, which will be called by us as quasi-dual of Gazeau-Klauder coherent states, and various nonclassical properties like sub-Poissonian statistics, antibunching effect, normal and amplitude-squared squeezing are examined numerically.
9 citations
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TL;DR: In this article, the authors clarified the relations between certain new coherent states for loop quantum gravity and the analytically continued heat kernel coherent states, highlighting the underlying general construction, the presence of a modified heat equation as well as the way in which the properties of the heat kernels are automatically inherited by these new states.
Abstract: We clarify the relations between certain new coherent states for loop quantum gravity and the analytically continued heat kernel coherent states, highlighting the underlying general construction, the presence of a modified heat equation as well as the way in which the properties of the heat kernels are automatically inherited by these new states.
8 citations
TL;DR: In this paper, the notions of qubit and density operators are described in the framework of coherent states, and a qubit is expressed as a coherent state, and thus a sequence of qubits becomes the tensor product of the coherent states.
Abstract: In the quantum information theory operates with qubits and N-qubits that can be express through coherent states. Density operator admits a representation in terms of coherent states formalism. Consequently, in this paper the notions of qubit and density operators are described in the framework of coherent states. We have expressed a qubit as a coherent state, and thus a sequence of qubits becomes the tensor product of the coherent states. For the ensembles of qubits, it could be used the density operator, in order to describe the informational content of the ensemble. The coherent states representation may play an important role in the quantum information theory and the use of this formalism is not only theoretical, but also, due to its applications, of some practical relevance.
5 citations
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TL;DR: In this paper, the entanglement of multi-qubit fermionic coherent states described by anticommutative Grassmann numbers is investigated, and it is shown that it is possible to construct some entangled pure states, consisting of GHZ, W, Bell and biseparable states, by tensor product of fermion coherent states.
Abstract: In this paper we investigate the entanglement of multi-qubit fermionic coherent states described by anticommutative Grassmann numbers. Choosing an appropriate weight function, we show that it is possible to construct some entangled pure states, consisting of GHZ, W, Bell and biseparable states, by tensor product of fermion coherent states. Moreover a comparison with maximal entangled bosonic coherent states is presented and it is shown that in some cases they have fermionic counterpart which are maximal entangled after integration with suitable weight functions.
4 citations
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TL;DR: In this article, a dual pair of nonlinear coherent states (NCS) in the context of changes of bases in the underlying Hilbert space for a model pertaining to the condensed matter physics, which obeys a $f$-deformed Heisenberg algebra was constructed.
Abstract: This work addresses a construction of a dual pair of nonlinear coherent states (NCS) in the context of changes of bases in the underlying Hilbert space for a model pertaining to the condensed matter physics, which obeys a $f$-deformed Heisenberg algebra. The existence and properties of reproducing kernel in the NCS Hilbert space are studied and discussed; the probability density and its dynamics in the basis of constructed coherent states are provided. A Glauber-Sudarshan $P$-representation of the density matrix and relevant issues related to the reproducing kernel properties are presented. Moreover, a NCS quantization of classical phase space observables is performed and illustrated in a concrete example of $q$-deformed coherent states. Finally, an exposition of quantum optical properties is given.
TL;DR: In this paper, a new family of coherent states called the "mother coherent states" are defined on the whole Hilbert space of the Fock basis vectors, i.e., they are the eigenstates of the lowering operator.
Abstract: In this paper, we shall define a new family of coherent states which we shall call the “mother coherent states,” bearing in mind the fact that these states are independent from any parameter (the Bargmann index, the rotational quantum number J, and so on). So, these coherent states are defined on the whole Hilbert space of the Fock basis vectors. The defined coherent states are of the Barut-Girardello kind, i.e., they are the eigenstates of the lowering operator. For these coherent states we shall calculate the expectation values of different quantum observables, the corresponding Mandel parameter, the Husimi's distribution function and also the P- function. Finally, we shall particularize the obtained results for the three-dimensional harmonic and pseudoharmonic oscillators.
TL;DR: In this article, the authors extended the study of quartic polynomials and reported classes of genuine quantum states |?4,n? with interesting properties, which follow directly from f4?=?0 for x?p.
Abstract: In a previous work, based upon complementarity and inspired by a standard derivation of the x?p uncertainty inequality (via a non-negative quadratic polynomial f2), we explored one possible extension, through a non-negative quartic polynomial f4, for non-commuting quantum variables. That work led to new quantum inequalities (expressed through the discriminant of the equation f4?=?0) and to connections with quantum optics. Here, we extend the study of quartic polynomials and report classes of genuine quantum states |?4,n? with interesting properties, which follow directly from f4?=?0 for x?p. We explore |?4,n? when they are generated from the coherent and the squeezed coherent states of the quantum harmonic oscillator and we display their relationships to states previously proposed by other authors through different constructions. In quantum optics, the |?4,n? associated to the coherent states are just displaced Fock states of light, which have already been generated experimentally.
01 Jan 2013
TL;DR: In this paper, the basic notions of group representations, with some emphasis on unitary irreducible representations of compact groups, are reviewed, and a number of recent generalizations of the classical theory of coherent states are decribed.
Abstract: We begin by quickly reviewing the basic notions of group representations, with some emphasis on unitary irreducible representations of compact groups. Then we turn to square integrable representations, the most natural generalizations of the latter. These representations are the mathematical background of the classical theory of coherent states (CS). In the next section, we examine in detail the most popular examples of this construction: (i) The continuous wavelet transform in one and two dimensions (corresponding, respectively, to the ax + b group of the line and the similitude group of the plane); and (ii) The Short-Time Fourier or Gabor transform (corresponding to the Weyl-Heisenberg group), leading to the familiar canonical coherent states. Finally, we decribe a number of recent generalizations of the classical theory, among them the so-called covariant coherent states.
03 Apr 2013
TL;DR: Supersymmetric quantum mechanics (SUSY-QM) has been developed as an elegant analytical approach to one-dimensional problems as mentioned in this paper, where the SUSY charge operators not only allow the factorization of a onedimensional Hamiltonian but also form a Lie algebra structure.
Abstract: Supersymmetric quantum mechanics (SUSY-QM) has been developed as an elegant analytical approach to one-dimensional problems. The SUSY-QM formalism generalizes the ladder operator approach used in the treatment of the harmonic oscillator. In analogy with the harmonic oscillator Hamiltonian, the factorization of a one-dimensional Hamiltonian can be achieved by introducing “charge operators”. For the one-dimensional harmonic oscillator, the charge operators are the usual raising and lowering operators. The SUSY charge operators not only allow the factorization of a one-dimensional Hamiltonian but also form a Lie algebra structure. This structure leads to the generation of isospectral SUSY partner Hamiltonians. The eigenstates of the various partner Hamiltonians are connected by application of the charge operators. As an analytical approach, the SUSY-QM approach has been utilized to study a number of quantum mechanics problems including the Morse oscillator ([16]) and the radial hydrogen atom equation ([24]). In addition, SUSY-QM has been applied to the discovery of new exactly solvable potentials, the development of a more accurate WKB approximation, and the improvement of large N expansions and variational methods ([7, 11]). Developments and applications of one-dimensional SUSY-QM can be found in relevant reviews and books ([7, 9, 11, 15, 26, 32, 33]). Recently, SUSY-QM has been developed as a computational tool to provide much more accurate excitation energies using the standard Rayleigh-Ritz variational method ([5, 19, 20]).
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TL;DR: In this paper, coherent-squeezed states satisfy the minimal uncertainty of Heisenberg under some condition imposed on the parameter space, so that we can study the metric from the view point of uncertainty principle.
Abstract: In this paper we treat coherent-squeezed states of Fock space once more and study some basic properties of them from a geometrical point of view.
Since the set of coherent-squeezed states $\{\ket{\alpha, \beta}\ |\ \alpha, \beta \in \fukuso\}$ makes a real 4-dimensional surface in the Fock space ${\cal F}$ (which is of course not flat), we can calculate its metric.
On the other hand, we know that coherent-squeezed states satisfy the minimal uncertainty of Heisenberg under some condition imposed on the parameter space $\{\alpha, \beta\}$, so that we can study the metric from the view point of uncertainty principle. Then we obtain a surprising simple form (at least to us).
We also make a brief review on Holonomic Quantum Computation by use of a simple model based on nonlinear Kerr effect and coherent-squeezed operators.
TL;DR: In this paper, the authors used the methods of constructions of and deformed coherent states in order to construct the coherent states for down conversion processes, which can be understood as a quasi-exactly solvable model of quantum mechanics.
Abstract: We use the methods of constructions of and deformed coherent states in order to construct the coherent states for down conversion processes. The down conversion process can be understood as a quasi-exactly solvable model of quantum mechanics. After the reduction of the Hamiltonian, we use the Turbiner polynomials approach, and the eigenvalues of the Hamiltonian for low number of photons are calculated and the approximation formula is found out. After the discussion on the time evolution and the entanglement, the coherent states are constructed as the eigenstates of the reduced annihilation operator.
TL;DR: In this article, the authors constructed the coherent states for a particle in the D-dimensional maximally superintegrable Smorodinsky-Winternitz potential by mapping the system into 2D harmonic oscillators, and then constructing coherent states of them by evaluating the transition amplitudes.
Abstract: In this study, we construct the coherent states for a particle in the D-dimensional maximally superintegrable Smorodinsky-Winternitz potential. We, first, map the system into 2D harmonic oscillators, second, construct the coherent states of them by evaluating the transition amplitudes. Third, in the Cartesian and the hyperspherical coordinates, we find the coherent states and the stationary states of the original sytem by reduction.