Made available in DSpace on 2014-06-11T19:32:10Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-07-02Bitstream added on 2014-06-13T20:23:08Z : No. of bitstreams: 1 oliveira_ja_dr_guara.pdf: 11550496 bytes, checksum: c15108557604781ca8b3f8a3107a2e3e (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / A busca por soluções exatas para sistemas quânticos vem despertando o interesse de muitos autores ao longo das décadas. Em particular para soluções que apresentam limite clássico. Nesta tese buscamos fazer um estudo sistemático da geração de soluções analíticas para uma classe de sistemas quânticos exatamente solúveis com massa dependente da posição. Analisamos o efeito da presença de campos magnéticos sobre alguns sistemas, discutimos o problema da ambiguidade de ordenamento quântico e apresentamos possíveis limite clássico para os sistemas em estudo. / The search for exact solutions of quantum systems has been raising the interest of many authors along the decades. Particularly for nding solutions that present classical limit. In this thesis we make a systematic study of the generation of analytic solutions for a class of quantum exactly solvable systems with position-dependent masses. We analyze the e ect of the presence of magnetic elds on some of those systems. We discuss the problem of the ordering quantum ambiguity and present possible classical limit for the systems considered.

Quantum systems usually suffer from unavoidable interactions with the environment. In most experiments, it is virtually impossible to isolate the system under study from the environment. It is thus imperative to study and understand how quantum systems interact with their surroundings. In the present study, I consider the evolution of quantum entanglement in a two-qubit system interacting with an environment in a regime where the non-Markovian effects are important. The present thesis is organized as follows: Chapter one is a general introduction to the thesis. In chapter two, some preliminary concepts that will be used in the subsequent chapters will be introduced and defined. These concepts include qubits, density matrix, Quantum entanglement and Entropic measures in quantum information. Ways to detect or quantify entanglement in a quantum system will also be discussed in the section on quantum entanglement. Chapter three will be about open quantum systems in general: concepts like master equations, Markovian quantum systems and non-Markovian quantum systems will be briefly reviewed. Chapter four will deal with time dependent entanglement features of two-qubit and multi-qubit systems interacting with an environment, basically all the results obtained in my study will be presented there. Some general conclusions will be drawn in chapter five. Copyright / Dissertation (MSc)--University of Pretoria, 2010. / Physics / unrestricted

Although a closed quantum system lacks clear signatures of classical chaos, it has been shown numerically that correspondence between an open quantum system and open classical system can be established. This phenomenon is explored for the case of an unconditioned evolution, where a system interacts with its environment, but the environment does not extract any information. This has been dubbed the “weak” transition and stands in contrast to the “strong” version where information is extracted by the environment. The transition is numerically mapped for the classically chaotic Duffing oscillator. Closed quantum and classically chaotic systems fail to agree due to the presence of fine scale structure in the classical evolution and the abundance of nonlocal interference in the quantum evolution. We show how noise mitigates both of these effects by suppressing the foliation of the classical unstable manifold while simultaneously acting as a passive filter of nonlocal quantum interference. The predicted transition parameter values are tested numerically for the Duffing oscillator. Finally, we explore whether these mechanisms are responsible for the emergence of classical chaos. While they do modify closed system spectral arguments against chaotic behavior, they do not provide a signature of chaotic dynamics. This stands in contrast to the trajectory level chaos observed in the strong transition. Keywords: nonlinear dynamics, quantum-classical transition, theoretical physics

Open Effective Field Theories are a class of Effective Field Theories (EFTs) built using ideas from open quantum systems in which a measured sector (the system) interacts with an unmeasured sector (the environment). It is argued that Open EFTs are useful tools for any situation in which a quantum system couples to a gravitational background with an event horizon. The main reason for this is that for any EFT of gravity one generically expects perturbation theory to breakdown at late times (when interactions with the background persist indefinitely). It is shown that the tools of Open EFTs allow one to resum late-time perturbative breakdowns in order to make reliable late time predictions (without resorting to solving the dynamics exactly). To build evidence of their usefulness to these types of gravitational problems, Open EFT approximation methods are applied to two toy models relating to black hole physics. / Thesis / Doctor of Philosophy (PhD) / Open Effective Field Theories are a class of quantum theories in which a measured sector (the system) is used to make physical predictions with, while interacting with an unmeasured sector (the environment). In this thesis arguments are made that Open EFTs are useful for studying gravitating quantum systems, especially when there is an event horizon present (for example in gravitational fields like that of a black hole). Open EFTs are applied to simple toy problems in such settings to illustrate their usefulness.

Quantum systems with a finite Hilbert space, where position x and momen- tum p take values in Z(d) (integers modulo d), are studied. An analytic representation of finite quantum systems is considered. Quantum states are represented by analytic functions on a torus. This function has exactly d zeros, which define uniquely the quantum state. The analytic function of a state can be constructed using its zeros. As the system evolves in time, the d zeros follow d paths on the torus. Examples of the paths ³n(t) of the zeros, for various Hamiltonians, are given. In addition, for given paths ³n(t) of the d zeros, the Hamiltonian is calculated. Furthermore, periodic finite quantum systems are considered. Special cases where M of the zeros follow the same path are also studied, and general ideas are demonstrated with several ex- amples. Examples of the path with multiplicity M = 1; 2; 3; 4; 5 are given. It is evidenced within the study that a small perturbation of the initial values of the zeros splits a path with multiplicity M into M different paths. / Libyan Cultural Affairs

Unitary transformations in an angular momentum Hilbert space H(2j + 1), are considered. They are expressed as a finite sum of the displacement operators (which play the role of SU(2j + 1) generators) with the Weyl function as coefficients. The Chinese remainder theorem is used to factorize large qudits in the Hilbert space H(2j + 1) in terms of smaller qudits in Hilbert spaces H(2ji + 1). All unitary transformations on large qudits can be performed through appropriate unitary transformations on the smaller qudits.

Quantum systems with variables in the ring Z(d) are considered, and the concepts of weak mutually unbiased bases and mutually unbiased projectors are discussed. The lines through the origin in the Z(d) x Z(d) phase space, are classified into maximal lines (sets of d points), and sublines (sets of d(i) points where d(i)vertical bar d). The sublines are intersections of maximal lines. It is shown that there exists a duality between the properties of lines (resp., sublines), and the properties of weak mutually unbiased bases (resp., mutually unbiased projectors).

Yes / Lorenz values and the Gini index are popular quantities in Mathematical Economics, and are used here in the context of quantum systems with finite-dimensional Hilbert space. They quantify the uncertainty in the probability distribution related to an orthonormal basis. It is shown that Lorenz values are superadditive functions and the Gini indices are subadditive functions. The supremum over all density matrices of the sum of the two Gini indices with respect to position and momentum states is used to define an uncertainty coefficient which quantifies the uncertainty in the quantum system. It is shown that the uncertainty coefficient is positive, and an upper bound for it is given. Various examples demonstrate these ideas.

yes / An expansion of row Markov matrices in terms of matrices related to permutations with repetitions, is introduced. It generalises the Birkhoff-von Neumann expansion of doubly stochastic matrices in terms of permutation matrices (without repetitions). An interpretation of the formalism in terms of sequences of integers that open random safes described by the Markov matrices, is presented. Various quantities that describe probabilities and correlations in this context, are discussed. The Gini index is used to quantify the sparsity (certainty) of various probability vectors. The formalism is used in the context of multipartite quantum systems with finite dimensional Hilbert space, which can be viewed as quantum permutations with repetitions or as quantum safes. The scalar product of row Markov matrices, the various Gini indices, etc, are novel probabilistic quantities that describe the statistics of multipartite quantum systems. Local and global Fourier transforms are used to de ne locally dual and also globally dual statistical quantities. The latter depend on off-diagonal elements that entangle (in general) the various components of the system. Examples which demonstrate these ideas are also presented.

Στην παρούσα έρευνα εξετάζουμε την αλληλεπίδραση ενός ανοικτού κβα- ντικού συστήματος ενός ή δύο αρμονικών ταλαντωτών με το περιβάλλον εκκι- νώντας από το μοντέλο κβαντικής κίνησης Brown. Υπολογίζουμε τις λύσεις της ομογενούς εξίσωσης κίνησης καθώς και τους πίνακες απωλειών και θορύβου, από τους οποίους κατασκευάζεται ο διαδότης του συστήματος. Στη βάση αυτών μας των υπολογισμών ελέγχουμε την αξιοπιστία της εξίσωσης Master δεύτερης τάξης, της οποίας τα αποτελέσματα τα βρίσκουμε αναξιόπιστα. Τα αποτελέσματα αυτά επιτρέπουν α). την ολοκλήρωση του θεωρητικού μοντέλου για την επικοι- νωνία δύο απομακρυσμένων κβαντικών συστημάτων και β).την εφαρμογή σε ζητήματα κβαντικής πληροφορίας. / In this research we examine the interaction of an open quantum system one or two harmonic oscillators with the environment starting from the quantum Brownian motion model. We compute the solutions of the homogeneous equation of motion and the dissipation and noise kernel, of which is constructed the propagator of the system. Based on these calculations we check the reliability of the Master equation of second order, whose the results are unreliable. These results allow a). the completion of the theoretical model for communication between two remote quantum systems and b). the application in issues quantum information.