Low-dimensional Quantum Systems Research Articles

The distillability problem revisited

An important open problem in quantum information theory is the question of the existence of NPT bound entanglement. In the past years, little progress has been made, mainly because of the lack of mathematical tools to address the problem. (i) In an attempt to overcome this, we show how the distillability problem can be reformulated as a special instance of the separability problem, for which a large number of tools and techniques are available. (ii) Building up to this we also show how the problem can be formulated as a Schmidt number problem. (iii) A numerical method for detecting distillability is presented and strong evidence is given that all 1-copy undistillable Werner states are also 4-copy undistillable. (iv) The same method is used to estimate the volume of distillable states, and the results suggest that bound entanglement is primarily a phenomenon found in low dimensional quantum systems. (v) Finally, a set of one parameter states is presented which we conjecture to exhibit all forms of distillability.

Divergence of the entanglement range in low-dimensional quantum systems

We study the pairwise entanglement close to separable ground states of a class of one-dimensional quantum spin models. At $T=0$ we find that such ground states separate regions, in the space of the Hamiltonian parameters, which are characterized by qualitatively different types of entanglement, namely parallel and antiparallel entanglement; we further demonstrate that the range of the concurrence diverges while approaching separable ground states, therefore evidencing that such states, with uncorrelated fluctuations, are reached by a long range reshuffling of the entanglement. We generalize our results to the analysis of quantum phase transitions occurring in bosonic and fermionic systems. Finally, the effects of finite temperature are considered: At $T>0$ we evidence the existence of a region where no pairwise entanglement survives, so that entanglement, if present, is genuinely multipartite.

Method of intermediate problems in the theory of Gaussian quantum dots placed in a magnetic field

Applicability of the method of intermediate problems to the investigation of the energy eigenvalues and eigen- states of a quantum dot (QD) formed by a Gaussian confining potential in the presence of an external mag- netic field is discussed. Being smooth at the QD boundaries and of finite depth and range, this potential can only confine a finite number of excess electrons thus forming a realistic model of a QD with smooth inter- face between the QD and its embedding environment. It is argued that the method of intermediate problems, which provides convergent improvable lower bound estimates for eigenvalues of linear half-bound Hermitian operators in Hilbert space, can be fused with the classical Rayleigh-Ritz variational method and stochastic variational method thus resulting in an efficient tool for an alytical and numerical studies of the energy spec- trum and eigenstates of the Gaussian quantum dots, confining small-to-medium number of excess electrons, with controllable or prescribed precision. Various theoretical and experimental aspects of the physics of nano-sized and low-dimensional systems have been under steady uninterrupted development for decades, but only in the last ten- fifteen years of research activities in the field, considerable extra momentum has been gained which can be partly ascribed to the progress in nanofabrication of these systems, partly to the development of new experimental techniques but, most of all, to a general continuous trend and desire to diminish the size of components of integrated electronic circuits down to the so-called mesoscopic scale. Among low-dimensional systems of various kinds, quantum dots are potentially fit for many practical applications and are currently thought to be promising building blocks for novel electronic, spintronic and optoelectronic devices. Reliable estimation of the energy spectrum and eigenstates of a quantum dot is a typical purpose of almost every theoretical study because their properties crucially stipulate the relevant physical characteristics of the quantum dot standing alone as a part of electric circuits or interacting with the environment through its various interfaces. As a matter of fact, nearly all the mathematical methods, developed within the domain of quan- tum mechanics so far, have already been employed in the theory of quantum dots in various specific applications though on a varying scale. The most frequently applicable methods of approximate calculation of eigenvalues and eigenstates of realistic physical models of low-dimensional quantum systems, quantum dots among them, are various numerical methods which permit either direct calculations of the magnitudes in question without proper error estimates or, at best, provide the calculations with nonincreasing or even convergent upper bounds for the eigenvalues. The widely applicable Rayleigh-Ritz method does it but, again, without error estimates. To control the error of the approximations provided by the upper bounds for some quantity it would be enough to derive the corresponding lower bounds which are highly desirable to be

Fröhlich Electron-Interface Optical Phonon Interactions in n-Layer Low-dimensional Coupling Quantum Structures

Within the framework of the dielectric continuum (DC) approximation, by using the transfer matrix method, uniform descriptions for the interface optical (IO) phonon modes and the Frhlich electron-IO phonon interaction Hamiltonians in n-layer coupling low-dimensional quantum systems (including the coupling quantum well (CQW), coupling quantum-well wire (CQWW) and coupling quantum dot (CQD)) have been deduced and investigated. Numerical calculation on three-layer AlGaAs/GaAs systems reveals that, when the wave-vector or quantum number in the three types of quantum confined systems (CQW, CQWW and CQD) approach infinity, each frequency of the IO phonon approaches one of the frequencies in single planar heterostructure, and the math and physical reasons for this feature have been explained and specified. The limited behavior of IO phonon frequency when the wave-vector or quantum number in these systems approach zero have also been displayed and analyzed.

Charge dynamics in low-dimensional quantum systems

We compare our optical data collected for two families of low-dimensionalcorrelated electron systems: the ladders and the organic Bechgaard salts. Our datadisplay a remarkable optical anisotropy between the spectra along the threecrystallographic axes, indicative of the low dimensionality in the electronicband structure. Moreover, we optically identify a dimensionality crossover,induced by Ca doping in the ladders and by chemical pressure in theBechgaard salts. An outlook on the ongoing research on carbon and MoS2nanotubes, as novel one-dimensional quantum wires, is also addressed.

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.

Disorder and two-particle interaction in low-dimensional quantum systems

We review some of the recent results on two-interacting particles (TIP) in low-dimensional disordered quantum models. Special attention is given to the mapping of the problem onto random band matrices. In particular, we construct two simple, seemingly closely related examples for which an analogous mapping leads to incorrect results. We briefly discuss possible reasons for this discrepancy based on the physical differences between the TIP problem and our examples.

Electronic-level calculations for semiconductor quantum dots: Deterministic numerical method using Green’s functions

A numerical method to find the bound states of a low-dimensional quantum system is described. The method is used to calculate the energy levels and the wave functions of three-dimensional systems. In particular, the electronic level structure of self-assembled ${\mathrm{In}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{A}\mathrm{s}/\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}$ quantum dots is studied.

A short introduction to quantum symmetry

In quantum theory, symmetries more general than groups are possible. We give a general definition of a quantum symmetry, such that symmetry operations act on the Hilbert space H> of physical states and notions of unitarity, invariance and covariance are defined. Within this frame, weak quasi quantum groups are described as a natural generalization of group algebras. Consistency with locality distinguishes them from more general quantum symmetries. To find the new kinds of symmetry one should investigate low dimensional quantum systems such as two-dimensional layers.