Special functions with mod n symmetry and kaleidoscope of quantum coherent states (original) (raw)

Kaleidoscope of Classical Vortex Images and Quantum Coherent States

2018

The Schrodinger cat states, constructed from Glauber coherent states and applied for description of qubits are generalized to the kaleidoscope of coherent states, related with regular n-polygon symmetry and the roots of unity. This quantum kaleidoscope is motivated by our method of classical hydrodynamics images in a wedge domain, described by q-calculus of analytic functions with q as a primitive root of unity. First we treat in detail the trinity states and the quartet states as descriptive for qutrit and ququat units of quantum information. Normalization formula for these states requires introduction of specific combinations of exponential functions with mod 3 and mod 4 symmetry, which are known also as generalized hyperbolic functions. We show that these states can be generated for an arbitrary n by the Quantum Fourier transform and can provide in general, qudit unit of quantum information. Relations of our states with quantum groups and quantum calculus are discussed.

Quantum deformed algebras : Coherent states and special functions

arXiv (Cornell University), 2013

The Heisenberg algebra is first deformed with the set of parameters {q, l, λ} to generate a new family of generalized coherent states. The matrix elements of relevant operators are exactly computed. A proof on sub-Poissonian character of the statistics of the main deformed states is provided. This property is used to determine a generalized metric. A unified method of calculating structure functions from commutation relations of deformed single-mode oscillator algebras is then presented. A natural approach to building coherent states associated to deformed algebras is deduced. Known deformed algebras are given as illustration. Futhermore, we generalize a class of two-parameter deformed Heisenberg algebras related to meromorphic functions, called R(p, q)-deformed algebra. Relevant families of coherent states maps are probed and their corresponding hypergeometric series are computed. The latter generalizes known hypergeometric series and gives a generalization of the binomial theorem. We provide new noncommutative algebras and show that the involved notions of differentiation and integration generalize the usual q-and (p, q)differentiation and integration. A Hopf algebra structure compatible with the R(p, q)-algebra is deduced. Besides, we succeed in giving a new characterization of Rogers-Szegö polynomials, called R(p, q)-deformed Rogers-Szegö polynomials. Continuous R(p, q)-deformed Hermite polynomials and their recursion relation are also deduced. Novel algebraic relations are provided and discussed. The whole formalism is performed in a unified way, generalizing known relevant results which are straightforwardly derived as particular cases.

Special deformed exponential functions leading to more consistent Klauder's coherent states

Physics Letters A, 2002

We give a general approach for the construction of deformed oscillators. These ones could be seen as describing deformed bosons. Basing on new definitions of certain quantum series, we demonstrate that they are nothing but the ordinary exponential functions in the limit when the deformation parameters goes to one. We also prove that these series converge to a complex function, in a given convergence radius that we calculate. Klauder's Coherent States are explicitly found through these functions that we design by deformed exponential functions

Representations of two-qubit and ququart states via discrete Wigner functions

arXiv: Quantum Physics, 2019

By means of a well-grounded mapping scheme linking Schwinger unitary operators and generators of the special unitary group mathrmSU(N)\mathrm{SU(N)}mathrmSU(N), it is possible to establish a self-consistent theoretical framework for finite-dimensional discrete phase spaces which has the discrete mathrmSU(N)\mathrm{SU(N)}mathrmSU(N) Wigner function as a legitimate by-product. In this paper, we apply these results with the aim of putting forth a detailed study on the discrete mathrmSU(2)otimesmathrmSU(2)\mathrm{SU(2)} \otimes \mathrm{SU(2)}mathrmSU(2)otimesmathrmSU(2) and mathrmSU(4)\mathrm{SU(4)}mathrmSU(4) Wigner functions, in straight connection with experiments involving, among other things, the tomographic reconstruction of density matrices related to the two-qubit and ququart states. Next, we establish a formal correspondence between both the descriptions that allows us to visualize the quantum correlation effects of these states in finite-dimensional discrete phase spaces. Moreover, we perform a theoretical investigation on the two-qubit X-states, which combines discrete Wigner functions a...

A ] 2 9 D ec 2 00 3 Quantum symmetric functions

2003

We study quantum deformations of Poisson orbivarieties. Given a Poisson manifold (R m , α) we consider the Poisson orbivariety (R m) n /Sn. The Kontsevich star product on functions on (R m) n induces a star product on functions on (R m) n /Sn. We provide explicit formulae for the case h × h/W , where h is the Cartan subalgebra of a classical Lie algebra g and W is the Weyl group of h. We approach our problem from a fairly general point of view, introducing Polya functors for categories over non-symmetric Hopf operads.

A ] 2 1 Ju l 2 00 4 Quantum symmetric functions

2003

We study quantum deformations of Poisson orbivarieties. Given a Poisson manifold (R m , α) we consider the Poisson orbivariety (R m) n /Sn. The Kontsevich star product on functions on (R m) n induces a star product on functions on (R m) n /Sn. We provide explicit formulae for the case h × h/W, where h is the Cartan subalgebra of a classical Lie algebra g and W is the Weyl group of h. We approach our problem from a fairly general point of view, introducing Polya functors for categories over non-symmetric Hopf operads.

Orthogonal polynomials, associated special functions and coherent states

The Schrödinger equations which are exactly solvable in terms of associated special functions are directly related to some self-adjoint operators defined in the theory of hypergeometric type equations. The mathematical formulae occurring in the study of these quantum systems are consequences of some fundamental results from the theory of special functions. We present in a unified and explicit way the systems of orthogonal polynomials defined by hypergeometric-type equations, the associated special functions and corresponding systems of coherent states. This general formalism allows us to extend certain results known in the case of some particular potentials.

Associated special functions and coherent states

A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated special functions and some systems of coherent states. This general formalism allows us to extend some results known only in particular cases.

A note on multi-index polynomials of Dickson type and their applications in quantum optics

Journal of Computational and Applied Mathematics, 2002

We discuss the properties of a new family of multi-index Lucas type polynomials, which are often encountered in problems of intracavity photon statistics. We develop an approach based on the integral representation method and show that this class of polynomials can be derived from recently introduced multi-index Hermite like polynomials.