Leibniz Central Extensions on Some Infinite-Dimensional Lie Algebras

Abstract

In this paper, all one-dimensional Leibniz central extensions on the algebras of differential operators over C[t, t ^?1] and C((t)), as well as on the quantum 2-torus, the Virasoro-like algebra and its q-analog are studied. We determine all nontrivial Leibniz 2-cocycles on these infinite dimensional Lie algebras.     

Dérivations des Algèbres de Weyl Quantiques Multiparamétrées et de Certaines de leurs Localisations Simples

ABSTRACT

On calcule une base explicite de l'espace des dérivations modulo les dérivations intérieures des algèbres de Weyl quantiques et de certaines de leurs localisations simples. On applique les résultats obtenus à des questions de séparation à équivalence de Morita près de ces algèbres.

We compute an explicit basis of the space of derivations modulo inner derivations for quantum Weyl algebras and some of their simple localizations. We apply these results to separate these algebras up to Morita equivalence.

     

Γ-X mixing in GaAs-Ga<Subscript>1-x</Subscript>Al<Subscript>x</Subscript>As quantum wells under hydrostatic pressure

The mixing between the Γ and X conduction-band valleys in GaAs-Ga_1-xAl_xAs quantum wells is investigated by using a phenomenological model which takes into account the effects of applied hydrostatic pressure. The dependencies of the variationally calculated photoluminescence peak-energy transitions on the applied hydrostatic pressure and quantum-well width are presented. A systematic study of the Γ-X mixing parameter is also reported. In particular, it is shown that the inclusion of the Γ-X mixing explains the non-linear behavior in the photoluminescence peak of confined exciton states that has been experimentally observed for pressures above 15 kbar in GaAs-Ga_1-xAl_xAs quantum wells.     

Implementation of controlled phase shift gates and Collins version of Deutsch-Jozsa algorithm on a quadrupolar spin-7/2 nucleus using non-adiabatic geometric phases

In this work controlled phase shift gates are implemented on a qaudrupolar system, by using non-adiabatic geometric phases. A general procedure is given, for implementing controlled phase shift gates in an ‘N’ level system. The utility of such controlled phase shift gates, is demonstrated here by implementing 3-qubit Deutsch–Jozsa algorithm on a spin-7/2 quadrupolar nucleus oriented in a liquid crystal matrix.     

The characteristics for H<Subscript> 2</Subscript><Superscript> +</Superscript>-like impurities confined by spherical quantum dots

The energy spectra of H₂ ⁺-like impurities confined in finite spherical quantum dots have been calculated as a function of the distance between nuclear with different sizes on the basis of effective-mass approximation by linear variational method. B-splines have been used as basis functions, which can easily construct the trial wavefunctions with appropriate boundary and cusp conditions. The quantitative analyses of the partial wave weights for ground state and some low lying states have been done.     

Entangled quantum heat engines based on two two-spin systems with Dzyaloshinski-Moriya anisotropic antisymmetric interaction

We construct an entangled quantum heat engine (EQHE) based on two two-spin systems with Dzyaloshinski-Moriya (DM) anisotropic antisymmetric interaction. By applying the explanations of heat transferred and work performed at the quantum level in Kieu’s work [Phys. Rev. Lett. 93, 140403 (2004)], the basic thermodynamic quantities, i.e., heat transferred, net work done in a cycle and efficiency of EQHE are investigated in terms of DM interaction and concurrence. The validity of the second law of thermodynamics is confirmed in the entangled system. It is found that there is a same efficiency for both antiferromagnetic and ferromagnetic cases, and the efficiency can be controlled in two manners: (1) only by spin-spin interaction J and DM interaction D; (2) only by the temperature T and concurrence C. In order to obtain a positive net work, we need not entangle all qubits in two two-spin systems and we only require the entanglement between qubits in a two-spin system not be zero. As the ratio of entanglement between qubits in two two-spin systems increases, the efficiency will approach infinitely the classical Carnot one. An interesting phenomenon is an abrupt transition of the efficiency when the entanglements between qubits in two two-spin systems are equal.     

Scattering effect of atoms through a Bose-Einstein condensate in an optical Lattice with a single defect

We study the transport of atoms across a localized Bose-Einstein condensate in an onedimensional optical lattice with a single defect. Our analytical and numerical results show that the defect as well as the nonlinear parameter can control the transmission of the atoms beam and the position of total reflection caused by Fano resonance. These interesting features may be a very useful basis for devising tunable atom filters or a button.     

Recurrence spectra and dynamical simulation of Rydberg hydrogen atom near a metal surface

By using the closed-orbit theory including the effect of Coulomb scattering together with an electrical image potential approach, the recurrence spectra and the dynamical behaviours of the Rydberg hydrogen atom near a metal surface are presented. Theoretical analysis and numerical simulation reveal that the impacts of the image potential contributing to the recurrence spectrum are qualitatively analogous to that of the parallel electrical and magnetic fields on the Rydberg atom. The recurrence spectra are computed for a few selected scaled energies and the results demonstrate that the scaled energy dominates the dynamical properties of system. With the increase of the scaled energy e from small to large, the whole trend of spectral structure is from simple to complex,and then simple.