Wick's theorem for q-deformed boson operators

In this paper combinatorial aspects of normal ordering arbitrary words in the creation and annihilation operators of the q-deformed boson are discussed. In particular, it is shown how by introducing appropriate q-weights for the associated ``Feynman diagrams'' the normally ordered form of a general expression in the creation and annihilation operators can be written as a sum over all q-weighted Feynman diagrams, representing Wick's theorem in the present context.Comment: 9 page

Representation-theoretic derivation of the Temperley-Lieb-Martin algebras

Explicit expressions for the Temperley-Lieb-Martin algebras, i.e., the quotients of the Hecke algebra that admit only representations corresponding to Young diagrams with a given maximum number of columns (or rows), are obtained, making explicit use of the Hecke algebra representation theory. Similar techniques are used to construct the algebras whose representations do not contain rectangular subdiagrams of a given size.Comment: 12 pages, LaTeX, to appear in J. Phys.

Geometric approach to nonlinear coherent states using the Higgs model for harmonic oscillator

In this paper, we investigate the relation between the curvature of the physical space and the deformation function of the deformed oscillator algebra using non-linear coherent states approach. For this purpose, we study two-dimensional harmonic oscillators on the flat surface and on a sphere by applying the Higgs modell. With the use of their algebras, we show that the two-dimensional oscillator algebra on a surface can be considered as a deformed one-dimensional oscillator algebra where the effect of the curvature of the surface is appeared as a deformation function. We also show that the curvature of the physical space plays the role of deformation parameter. Then we construct the associated coherent states on the flat surface and on a sphere and compare their quantum statistical properties, including quadrature squeezing and antibunching effect.Comment: 12 pages, 7 figs. To be appeared in J. Phys.

Sensitivity of spin structures of small spin-1 condensates against a magnetic field studied beyond the mean field theory

The spin structures of small spin-1 condensates ($N\leq 1000$) under a magnetic field $B$ has been studied beyond the mean field theory (MFT). Instead of the spinors, the many body spin-eigenstates have been obtained. We have defined and calculated the spin correlative probabilities to extract information from these eigenstates. The correlation coefficients and the fidelity susceptibility have also been calculated. Thereby the details of the spin-structures responding to the variation of $B$ can be better understood. In particular, from the correlation coefficients which is the ratio of the 2-body probability to the product of two 1-body probabilities, strong correlation domains (SCD) of $B$ are found. The emphasis is placed on the sensitivity of the condensates against $B$. No phase transitions in spin-structures are found. However, abrupt changes in the derivatives of observables (correlative probabilities) are found in some particular domains of $B$. In these domains the condensates are highly sensitive to $B$. The effect of temperature is considered. The probabilities defined in the paper can work as a bridge to relate theories and experiments. Therefore, they can be used to discriminate various spin-structures and refine the interactions.Comment: 18 pages, 8 figure

An algebraic approach to the Tavis-Cummings problem

An algebraic method is introduced for an analytical solution of the eigenvalue problem of the Tavis-Cummings (TC) Hamiltonian, based on polynomially deformed su(2), i.e. su_n(2), algebras. In this method the eigenvalue problem is solved in terms of a specific perturbation theory, developed here up to third order. Generalization to the N-atom case of the Rabi frequency and dressed states is also provided. A remarkable enhancement of spontaneous emission of N atoms in a resonator is found to result from collective effects.Comment: 13 pages, 7 figure

Multidimensional Isotropic and Anisotropic Q-Oscillator Models

q-oscillator models are considered in two and higher dimensions and their symmetries are explored. New symmetries are found for both isotropic and anisotropic cases. Applications to the spectra of triatomic molecules and superdeformed nuclei are discussed.Comment: 12 Pages, LATEX, no figures, (Submitted to J. PHYS. A

The qq-boson-fermion realizations of quantum suprealgebra Uq(gl(2/1))U_q(gl(2/1))

We show that our construction of realizations for Lie algebras and quantum algebras can be generalized to quantum superalgebras, too. We study an example of quantum superalgebra $U_q(gl(2/1))$ and give the boson-fermion realization with respect to one pair od q-deformed boson operator and 2 pairs of fermions.Comment: 8 page

Generally Deformed Oscillator, Isospectral Oscillator System and Hermitian Phase Operator

The generally deformed oscillator (GDO) and its multiphoton realization as well as the coherent and squeezed vacuum states are studied. We discuss, in particular, the GDO depending on a complex parameter q (therefore we call it q-GDO) together with the finite dimensional cyclic representations. As a realistic physical system of GDO the isospectral oscillator system is studied and it is found that its coherent and squeezed vacuum states are closely related to those of the oscillator. It is pointed out that starting from the q-GDO with q root of unity one can define the hermitian phase operators in quantum optics consistently and algebraically. The new creation and annihilation operators of the Pegg-Barnett type phase operator theory are defined by using the cyclic representations and these operators degenerate to those of the ordinary oscillator in the classical limit q->1.Comment: 21 pages, latex, no figure

Hierarchical Dobinski-type relations via substitution and the moment problem

We consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson ([a,a*]=1) monomials of the form exp(x (a*)^r a), r=1,2,..., under the composition of their exponential generating functions (egf). They turn out to be of Sheffer-type. We demonstrate that two key properties of these sequences remain preserved under substitutional composition: (a)the property of being the solution of the Stieltjes moment problem; and (b) the representation of these sequences through infinite series (Dobinski-type relations). We present a number of examples of such composition satisfying properties (a) and (b). We obtain new Dobinski-type formulas and solve the associated moment problem for several hierarchically defined combinatorial families of sequences.Comment: 14 pages, 31 reference

Dobinski-type relations: Some properties and physical applications

We introduce a generalization of the Dobinski relation through which we define a family of Bell-type numbers and polynomials. For all these sequences we find the weight function of the moment problem and give their generating functions. We provide a physical motivation of this extension in the context of the boson normal ordering problem and its relation to an extension of the Kerr Hamiltonian.Comment: 7 pages, 1 figur