QUASIDISTRIBUTIONS and COHERENT STATES FOR FINITE-DIMENSIONAL QUANTUM SYSTEMS

In recent years, an approach to discrete quantum phase spaces which comprehends all the main quasiprobability distributions known has been developed. It is the research that started with the pioneering work of Galetti and Piza, where the idea of operator bases constructed of discrete Fourier transforms of unitary displacement operators was first introduced. Subsequently, the discrete coherent states were introduced, and finally, the s-parametrized distributions, that include the Wigner, Husimi, and Glauber-Sudarshan distribution functions as particular cases. In the present work, we adapt its formulation to encompass some additional discrete symmetries, achieving an elegant yet physically sound formalism.

An introduction to numerical methods in low-dimensional quantum systems

This is an introductory course to the Lanczos Method and Density Matrix Renormalization Group Algorithms (DMRG), two among the leading numerical techniques applied in studies of low-dimensional quantum models. The idea of studying the models on clusters of a finite size in order to extract their physical properties is briefly discussed. The important role played by the model symmetries is also examined. Special emphasis is given to the DMRG.

Remarks on supersymmetry of quantum systems with position-dependent effective masses

We apply the supersymmetry approach to one-dimensional quantum systems with spatially dependent mass, by including their ordering ambiguities dependence. In this way we extend the results recently reported in the literature. Furthermore, we point out a connection between these systems and others with constant masses. This is done through convenient transformations in the coordinates and wave functions.

Extended Cahill-Glauber formalism for finite-dimensional spaces. II. Applications in quantum tomography and quantum teleportation

By means of a mod(N)-invariant operator basis, s-parametrized phase-space functions associated with bounded operators in a finite-dimensional Hilbert space are introduced in the context of the extended Cahill-Glauber formalism, and their properties are discussed in details. The discrete Glauber-Sudarshan, Wigner, and Husimi functions emerge from this formalism as specific cases of s-parametrized phase-space functions where, in particular, a hierarchical process among them is promptly established. In addition, a phase-space description of quantum tomography and quantum teleportation is presented and new results are obtained.

Quasiprobability distribution functions for finite-dimensional discrete phase spaces: Spin-tunneling effects in a toy model

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Spin squeezing and entanglement via finite-dimensional discrete phase-space description

We show how mapping techniques inherent to N2-dimensional discrete phase spaces can be used to treat a wide family of spin systems which exhibits squeezing and entanglement effects. This algebraic framework is then applied to the modified Lipkin-Meshkov-Glick (LMG) model in order to obtain the time evolution of certain special parameters related to the Robertson- Schrödinger (RS) uncertainty principle and some particular proposals of entanglement measure based on collective angular-momentum generators. Our results reinforce the connection between both the squeezing and entanglement effects, as well as allow to investigate the basic role of spin correlations through the discrete representatives of quasiprobability distribution functions. Entropy functionals are also discussed in this context. The main sequence correlations → entanglement → squeezing of quantum effects embraces a new set of insights and interpretations in this framework, which represents an effective gain for future researches in different spin systems. © 2013 World Scientific Publishing Company.

Discrete squeezed states for finite-dimensional spaces

We show how discrete squeezed states in an N-2-dimensional phase space can be properly constructed out of the finite-dimensional context. Such discrete extensions are then applied to the framework of quantum tomography and quantum information theory with the aim of establishing an initial study on the interference effects between discrete variables in a finite phase space. Moreover, the interpretation of the squeezing effects is seen to be direct in the present approach, and has some potential applications in different branches of physics.

Extended Cahill-Glauber formalism for finite-dimensional spaces: I. Fundamentals

The Cahill-Glauber approach for quantum mechanics on phase space is extended to the finite-dimensional case through the use of discrete coherent states. All properties and features of the continuous formalism are appropriately generalized. The continuum results are promptly recovered as a limiting case. The Jacobi theta functions are shown to have a prominent role in the context.

A discrete finite-dimensional phase space approach for the description of Fe8 magnetic clusters: Wigner and Husimi functions

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Time-reversal aspect of the point interactions in one-dimensional quantum mechanics

It is known that there is a four-parameter family of point interactions in one-dimensional quantum mechanics. We point out that, as far as physics is concerned, it is sufficient to use three of the four parameters. The fourth parameter is redundant. The apparent violation of time-reversal invariance in the presence of the fourth parameter is an artifact.

Finite dimensional vertex

The spectrum of linearized excitations of the Type IIB SUGRA on AdS(5) x S-5 contains both unitary and non-unitary representations. Among the non-unitary, some are finite-dimensional. We explicitly construct the pure spinor vertex operators for a family of such finite-dimensional representations. The construction can also be applied to in finite-dimensional representations, including unitary, although it becomes in this case somewhat less explicit.

Crossover between the extended and localized regimes in stochastic partially self-avoiding walks in one-dimensional disordered systems

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

2-DIMENSIONAL QUANTUM GAS

Identical impenetrable particles in a 2-dimensional configuration space obey braid statistics, intermediate between bosons and fermions. This statistics, based on braid groups, is introduced as a generalization of the usual statistics founded on the symmetric groups. The main properties of an ideal gas of such particles are presented. They do interpolate the properties of bosons and fermions but include classical particles as a special case. Restriction to 2 dimensions precludes lambda points but originates a peculiar symmetry, responsible in particular for the identity of boson and fermion specific heats.

Generalized point interactions in one-dimensional quantum mechanics

There is a four-parameter family of point interactions in one-dimensional quantum mechanics. They represent all possible self-adjoint extensions of the kinetic energy operator. If time-reversal invariance is imposed, the number of parameters is reduced to three. One of these point interactions is the familiar delta function potential but the other generalized ones do not seem to be widely known. We present a pedestrian approach to this subject and comment on a recent controversy in the literature concerning the so-called delta' interaction. We emphasize that there is little resemblance between the delta' interaction and what its name suggests.

Discrete coherent states and probability distributions in finite-dimensional spaces

Operator bases are discussed in connection with the construction of phase space representatives of operators in finite-dimensional spaces, and their properties are presented. It is also shown how these operator bases allow for the construction of a finite harmonic oscillator-like coherent state. Creation and annihilation operators for the Fock finite-dimensional space are discussed and their expressions in terms of the operator bases are explicitly written. The relevant finite-dimensional probability distributions are obtained and their limiting behavior for an infinite-dimensional space are calculated which agree with the well known results. (C) 1996 Academic Press, Inc.