The aim of this work is to investigate projection operator method of deriva- tion of equations of motion for reduced density matrix and apply it to a model open quantum system. We gradually pass from quantum mechanical model of a molecule with one vibrational degree of freedom to an example of open quantum system relevant in the theory of nonlinear spectroscopy. In the thesis we present results of numerical simulations of the time evolution of the open quantum system performed with a program written for this purpose. We are specially concerned with simulations of the solution of the time-convolutionless generalized master equation up to the a second order of the perturbation expan- sion, and we show that under certain conditions it provides an exact solution of the problem. The text also contains derivation of the recurrence relations for the Franck-Condon factors for the most general case of two quantum harmonic oscillators in one space dimension, i. e. transformation matrix between two bases of the L2 (R) space determined by the solutions of the time-independent Schrödinger equation appropriate for these oscillators. 1

Dans cette thèse, on construit un grand nombre de solutions globales pour de nombreuses équations de Schrödinger sur-critiques. Le principe consiste à rendre la donnée initiale aléatoire, selon les mêmes méthodes que Nicolas Burq, Nikolay Tzvetkov et Laurent Thomann afin de gagner de la dérivabilité.On considère d'abord l'équation de Schrödinger cubique en dimension 3. En partant de variables aléatoires gaussiennes et de la base de L^2(R^3) formée des fonctions d'Hermite tensorielles, on construit des ensembles de solutions globales pour des données initiales qui sont moralement dans L^2(R^3). Les points clefs de la démonstration sont l'existence d'une estimée bilinéaire de type Bourgain pour l'oscillateur harmonique et la transformation de lentille qui permet de se ramener à prouver l'existence locale de solutions à l'équation de Schrödinger avec potentiel harmonique.On étudie ensuite l'effet régularisant pour prouver un théorème analogue où le gain de dérivée vaut 1/2-2/(p-1) où p correspond à la non linéarité de l'équation. Le gain est donc plus faible que précédemment mais la base de fonctions propres quelconques. De plus, la méthode s'appuyant sur des estimées linéaires, on établit le résultat pour des variables aléatoires dont la queue de distribution est à décroissance exponentielle.Enfin, on démontre des estimées multilinéaires en dimension 2 pour une base de fonctions propres quelconques ainsi que des inégalités de types chaos de Wiener pour une classe générale de variables aléatoires. Cela nous permet d'établir le théorème pour l'équation de Schrödinger quintique, avec un gain de dérivée égal à 1/3, dans le même cadre que la partie précédente. / In this thesis, we build a large number of global solutions for many supercritical Schrödinger equations. The method is to make the random initial data, using the same methods that Nicolas Burq, Nikolay Tzvetkov and Laurent Thomann in order to obtain differentiability. First, we consider the cubic Schrödinger equation in three dimensional. Using Gaussian random variables and the basis of L^2(R^3) consists of tensorial Hermite functions, we construct sets of solutions for initial data that are morally in L^2(R^3). The main ingredients of the proof are the existence of Bourgain type bilinear estimates for the harmonic oscillator and the lens transform which can be reduced to prove a local existence of solutions for the Schrödinger equation with harmonic potential. Next, we study the smoothing effect to prove an analogous theorem which the gain of differentiability is equalto 1/2-2/(p-1) which p is the nonlinearity of the equation. This gain is lower than previously but the basis of eigenfunctions are general. As the method uses only linear estimates, we establish the result for a general class of random variables.Finally, we prove multilinear estimates in two dimensional for a basis of ordinaries eigenfunctions and Wienerchaos type inequalities for classical random variables. This allows us to establish the theorem for the quinticSchrödinger equation, with a gain of differentiability equals to 1/3, in the same context as the previous chapter.

Made available in DSpace on 2017-07-21T19:25:58Z (GMT). No. of bitstreams: 1 Joao Frederico.pdf: 2218397 bytes, checksum: 781c58d12bce3113c45da4e65bd0e36d (MD5) Previous issue date: 2010-03-11 / Conselho Nacional de Desenvolvimento Científico e Tecnológico / In this dissertation, we studied entanglement in some fundamental systems of physics, such as an excited atom in free-space spontaneously decaying and coupled harmonic oscillators. In order to study entanglement in spontaneous emission in free-space, we employed theWeisskopf-Wigner theory which allowed us to obtain the time evolution of both the atom and field states. In the case of bipartite entanglement among field modes after spontaneous emission, we showed that the modes can become highly entangled and that the features of this entanglement strongly depend on the way the partitions are made. For the entanglement between atom and field during spontaneous emission, we were able to relate entanglement to a well known physical quantity namely the lifetime of an atom in a excited state. Keeping in mind the intention to study simple but relevant physical systems, we used in the second work a chain of coupled harmonic oscillators. It was well-known among researchers in the field of quantum information that a linear chain of coupled oscillators in the rotating wave approximation and prepared in classical states would never create entanglement. Then, we used two reference oscillators prepared in squeezed states to make creation of entanglement possible. We found results concerning the relationship between the phases in the reference oscillators’ state and dynamics of entanglement in the chain for some coupling configurations. We showed that it is not always true that squeezing can favor entanglement creation and that with the configuration used by us it is possible to localize entanglement. We proposed a possible implementation of our results in coupled microelectromechanical systems. / Nesta dissertação estudamos o emaranhamento em alguns sistemas fundamentais da Física, como um átomo no espaço livre realizando emissão espontânea e em osciladores harmônicos acoplados. Para o estudo do emaranhamento na emissão espontânea no espaço livre, utilizamos a teoria de Weisskopf-Wigner que nos permitiu obter a evolução temporal, tanto do estado do átomo, quanto do estado do campo. Para o caso de emaranhamento bipartido entre os modos do campo após a emissão espontânea, mostramos que os modos podem ficar altamente emaranhados e que as características desse emaranhamento dependem fortemente de como são realizadas as partições. Para o emaranhamento entre o átomo e o campo durante a emissão espontânea, pudemos relacionar o emaranhamento com uma quantidade Física bastante conhecida, o tempo de vida do átomo no seu estado excitado. Ainda com o intuito de estudar sistemas físicos simples, mas de relevância na Física, utilizamos, em um segundo trabalho, uma cadeia de osciladores harmônicos acoplados. Já era bem conhecido dos pesquisadores na área de informação quântica que uma cadeia linear de osciladores acoplados, na aproximação de onda girante e preparados em estados clássicos, não cria emaranhamento. Assim, utilizamos dois osciladores de referência em estados comprimidos para permitir a criação de emaranhamento. Encontramos resultados a respeito da relação das fases dos osciladores de referência e a dinâmica do emaranhamento na cadeia para algumas configurações de acoplamentos. Mostramos que nem sempre a compressão dos estados comprimidos favorece a criação de emaranhamento e que na configuração utilizada por nós é possível localizar o emaranhamento. Nós propusemos uma possível implementação de nossos estudos em sistemas microeletromecânicos acoplados.

Hamilton-Jacobi theory provides a powerful method for extracting the equations of motion out of some given systems in classical mechanics. On occasion it allows some systems to be solved by the method of separation of variables. If a system with n degrees of freedom has 2n - 1 constants of the motion that are polynomial in the momenta, then that system is called superintegrable. Such a system can usually be solved in multiple coordinate systems if the constants of the motion are quadratic in the momenta. All superintegrable two dimensional Hamiltonians of the form H = (p_x)sup2 + (p_y)sup2 + V(x,y), with constants that are quadratic in the momenta were classified by Kalnins et al [5], and the coordinate systems in which they separate were found. We discuss Hamilton-Jacobi theory and its development from a classical viewpoint, as well as superintegrability. We then proceed to use the theory to find equations of motion for some of the superintegrable Hamiltonians from Kalnins et al [5]. We also discuss some of the properties of the Poisson algebra of those systems, and examine the orbits.

Transmitting quantum information across quantum channels is an important task. However quantum information is delicate, and is easily corrupted. We address the task of protecting quantum information from an information theoretic perspective -- we encode some message qudits into a quantum code, send the encoded quantum information across the noisy quantum channel, then recover the message qudits by decoding. In this dissertation, we discuss the coding problem from several perspectives.} The noisy quantum channel is one of the central aspects of the quantum coding problem, and hence quantifying the noisy quantum channel from the physical model is an important problem. We work with an explicit physical model -- a pair of initially decoupled quantum harmonic oscillators interacting with a spring-like coupling, where the bath oscillator is initially in a thermal-like state. In particular, we treat the completely positive and trace preserving map on the system as a quantum channel, and study the truncation of the channel by truncating its Kraus set. We thereby derive the matrix elements of the Choi-Jamiolkowski operator of the corresponding truncated channel, which are truncated transition amplitudes. Finally, we give a computable approximation for these truncated transition amplitudes with explicit error bounds, and perform a case study of the oscillators in the off-resonant and weakly-coupled regime numerically. In the context of truncated noisy channels, we revisit the notion of approximate error correction of finite dimension codes. We derive a computationally simple lower bound on the worst case entanglement fidelity of a quantum code, when the truncated recovery map of Leung et. al. is rescaled. As an application, we apply our bound to construct a family of multi-error correcting amplitude damping codes that are permutation-invariant. This demonstrates an explicit example where the specific structure of the noisy channel allows code design out of the stabilizer formalism via purely algebraic means. We study lower bounds on the quantum capacity of adversarial channels, where we restrict the selection of quantum codes to the set of concatenated quantum codes. The adversarial channel is a quantum channel where an adversary corrupts a fixed fraction of qudits sent across a quantum channel in the most malicious way possible. The best known rates of communicating over adversarial channels are given by the quantum Gilbert-Varshamov (GV) bound, that is known to be attainable with random quantum codes. We generalize the classical result of Thommesen to the quantum case, thereby demonstrating the existence of concatenated quantum codes that can asymptotically attain the quantum GV bound. The outer codes are quantum generalized Reed-Solomon codes, and the inner codes are random independently chosen stabilizer codes, where the rates of the inner and outer codes lie in a specified feasible region. We next study upper bounds on the quantum capacity of some low dimension quantum channels. The quantum capacity of a quantum channel is the maximum rate at which quantum information can be transmitted reliably across it, given arbitrarily many uses of it. While it is known that random quantum codes can be used to attain the quantum capacity, the quantum capacity of many classes of channels is undetermined, even for channels of low input and output dimension. For example, depolarizing channels are important quantum channels, but do not have tight numerical bounds. We obtain upper bounds on the quantum capacity of some unital and non-unital channels -- two-qubit Pauli channels, two-qubit depolarizing channels, two-qubit locally symmetric channels, shifted qubit depolarizing channels, and shifted two-qubit Pauli channels -- using the coherent information of some degradable channels. We use the notion of twirling quantum channels, and Smith and Smolin's method of constructing degradable extensions of quantum channels extensively. The degradable channels we introduce, study and use are two-qubit amplitude damping channels. Exploiting the notion of covariant quantum channels, we give sufficient conditions for the quantum capacity of a degradable channel to be the optimal value of a concave program with linear constraints, and show that our two-qubit degradable amplitude damping channels have this property.

Transmitting quantum information across quantum channels is an important task. However quantum information is delicate, and is easily corrupted. We address the task of protecting quantum information from an information theoretic perspective -- we encode some message qudits into a quantum code, send the encoded quantum information across the noisy quantum channel, then recover the message qudits by decoding. In this dissertation, we discuss the coding problem from several perspectives.} The noisy quantum channel is one of the central aspects of the quantum coding problem, and hence quantifying the noisy quantum channel from the physical model is an important problem. We work with an explicit physical model -- a pair of initially decoupled quantum harmonic oscillators interacting with a spring-like coupling, where the bath oscillator is initially in a thermal-like state. In particular, we treat the completely positive and trace preserving map on the system as a quantum channel, and study the truncation of the channel by truncating its Kraus set. We thereby derive the matrix elements of the Choi-Jamiolkowski operator of the corresponding truncated channel, which are truncated transition amplitudes. Finally, we give a computable approximation for these truncated transition amplitudes with explicit error bounds, and perform a case study of the oscillators in the off-resonant and weakly-coupled regime numerically. In the context of truncated noisy channels, we revisit the notion of approximate error correction of finite dimension codes. We derive a computationally simple lower bound on the worst case entanglement fidelity of a quantum code, when the truncated recovery map of Leung et. al. is rescaled. As an application, we apply our bound to construct a family of multi-error correcting amplitude damping codes that are permutation-invariant. This demonstrates an explicit example where the specific structure of the noisy channel allows code design out of the stabilizer formalism via purely algebraic means. We study lower bounds on the quantum capacity of adversarial channels, where we restrict the selection of quantum codes to the set of concatenated quantum codes. The adversarial channel is a quantum channel where an adversary corrupts a fixed fraction of qudits sent across a quantum channel in the most malicious way possible. The best known rates of communicating over adversarial channels are given by the quantum Gilbert-Varshamov (GV) bound, that is known to be attainable with random quantum codes. We generalize the classical result of Thommesen to the quantum case, thereby demonstrating the existence of concatenated quantum codes that can asymptotically attain the quantum GV bound. The outer codes are quantum generalized Reed-Solomon codes, and the inner codes are random independently chosen stabilizer codes, where the rates of the inner and outer codes lie in a specified feasible region. We next study upper bounds on the quantum capacity of some low dimension quantum channels. The quantum capacity of a quantum channel is the maximum rate at which quantum information can be transmitted reliably across it, given arbitrarily many uses of it. While it is known that random quantum codes can be used to attain the quantum capacity, the quantum capacity of many classes of channels is undetermined, even for channels of low input and output dimension. For example, depolarizing channels are important quantum channels, but do not have tight numerical bounds. We obtain upper bounds on the quantum capacity of some unital and non-unital channels -- two-qubit Pauli channels, two-qubit depolarizing channels, two-qubit locally symmetric channels, shifted qubit depolarizing channels, and shifted two-qubit Pauli channels -- using the coherent information of some degradable channels. We use the notion of twirling quantum channels, and Smith and Smolin's method of constructing degradable extensions of quantum channels extensively. The degradable channels we introduce, study and use are two-qubit amplitude damping channels. Exploiting the notion of covariant quantum channels, we give sufficient conditions for the quantum capacity of a degradable channel to be the optimal value of a concave program with linear constraints, and show that our two-qubit degradable amplitude damping channels have this property.

Made available in DSpace on 2016-12-12T20:15:52Z (GMT). No. of bitstreams: 1 Graziele Bortolini.pdf: 3189488 bytes, checksum: 6e8f5c11b189fb8f0ee4624270ebcee3 (MD5) Previous issue date: 2015-03-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The Nosé-Hoover dynamics is deterministic and time-reversible and performs a statistical sampling of the canonical ensemble of a many-body system directly from the system dynamics. This system is coupled to one extra degree of freedom (extended system) acting as a heat reservoir. In this work, we study the dynamical of the prototype system: the harmonic oscillator thermalized in the Nosé-Hoover dynamics. We observe periodic and quasi periodic regular trajectories in the phase space for different initial conditions. However, the oscillator is non-ergodic and non-canonical in the Nosé-Hoover dynamics, when we add one reservoir degree of freedom (thermostats chain), the momentum distribution tends to the canonical one. The second system studied is an Argonne gas with 4000 particles which interact by means of a Lennard-Jones potential. For this system, we observe phase transitions, momentum distributions in agreement of canonical ensemble, as well, the thermostat efficiency to keep constant the average temperature. / A dinâmica de Nosé-Hoover é determinística e reversível no tempo e realiza uma amostragem estatística de um sistema de muitos corpos no ensemble canônico diretamente a partir da dinâmica do sistema. Esse sistema é acoplado a um grau de liberdade extra (sistema estendido) que desempenha o papel de um reservatório térmico. Neste trabalho, estudamos a dinâmica do sistema protótipo mínimo: o oscilador harmônico na dinâmica de Nosé-Hoover. Observamos trajetórias regulares periódicas e quase periódicas no espaço de fase para diferentes condições iniciais. Embora, o oscilador não seja ergódico e não reproduza as médias do ensemble canônico na dinâmica de Nosé-Hoover, ao adicionarmos mais um grau de liberdade ao reservatório (cadeia de termostatos), a distribuição de momentos tende a canônica. O segundo sistema estudado é um gás de Argônio constituído de 4000 partículas que interagem segundo um potencial de Lennard-Jones. Para esse sistema, observamos transições de fase, distribuições de momento em concordância com o ensemble canônico, assim como a eficiência do termostato de Nosé-Hoover em manter a temperatura média constante.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho analisamos as soluÃÃes para a equaÃÃo de movimento para os osciladores de Lane-Emden, onde a massa à dada por m(t)=t^α, onde α>0. Os osciladores de Lane-Emden sÃo osciladores harmÃnicos amortecidos, onde o fator de amortecimento depende do tempo, γ(t)=α/t. Obtivemos as expressÃes analÃticas de q(t), dq(t)/dt, and p(t)=m(t)(dq(t)/dt) para α=2 e α=4. Discutimos as diferenÃas entre as expressÃes da hamiltoniana e da energia para sistemas dependentes do tempo. TambÃm, comparamos nossos resultados com aqueles do oscilador de Caldirola-Kanai. Usamos o mÃtodo dos invariantes quÃnticos e uma transformaÃÃo unitÃria para obter a funÃÃo de onda exata de SchrÃdinger, ψn (q,t), e calcular para n=0 a entropia conjunta (entropia de Leipnik) dependente do tempo e as informaÃÃes Fisher para posiÃÃo (Fq) e para o momento (Fp) para duas classes de osciladores harmÃnicos quÃnticos amortecidos. Observamos que a entropia de Leipnik nÃo varia no tempo para o oscilador Caldirola-Kanai, enquanto diminui e tende a um valor constante (ln(e/2)) para tempos assintÃticos para o oscilador de Lane-Emden. Isto à devido ao fato de que, para este Ãltimo, o fator de amortecimento diminui à medida que o tempo aumenta. Os resultados mostram que a dependÃncia do tempo da entropia de Leipnik à bastante complexa e nÃo obedece a uma tendÃncia geral de aumento monotonicamente com o tempo e que Fq aumenta enquanto Fp diminui com o aumento do tempo. AlÃm disso, FqFp aumenta e tende a um valor constante (4/ℏ^2 ) no limite em que t->∞. NÃs comparamos os resultados com os do bem conhecido oscilador de Caldirola-Kanai. / In this work we analyze the solutions of the equations of motions for two Lane-Emden-type Caldirola-Kanai oscillators. For these oscillators the mass varies as m(t)=t^α, where α>0.We obtain the analytical expression of q(t), dq(t)/dt, and p(t)=m(t)(dq(t)/dt) for α=2 and α=4. These are damped-like harmonic oscillators with a time-dependent damping factor given by γ(t)=α/t. We discuss the differences between the expressions for the hamiltonian and the mechanical energy for time-dependent systems. We also compared our results to those of the well-known Caldirola-Kanai oscillators. We use the quantum invariant method and a unitary transformation to obtain the exact SchrÃdinger wave function, ψn (q,t), and calculate for n=0 the time-dependent joint entropy (LeipnikÂs entropy) and the position (Fq) and momentum (Fp) Fisher information for two classes of quantum damped harmonic oscillators. We observe that the joint entropy does not vary in time for the Caldirola-Kanai oscillator, while it decreases and tends to a constant value (ln(e/2)) for asymptotic times for the Lane-Emden ones. This is due to the fact that for the latter, the damping factor decreases as time increases. The results show that the time dependence of the joint entropy is quite complex and does not obey a general trend of monotonously increase with time and that F_q increases while F_p decreases with increasing time. Also, FqFp increases and tends to a constant value (4/ℏ^2 ) in the limit t->∞.We compare the results with those of the well-known Caldirola-Kanai oscillator.

Orientador: Edmundo Capelas de Oliveira / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T00:53:38Z (GMT). No. of bitstreams: 1 Oliveira_DanieladosSantosde_M.pdf: 3702602 bytes, checksum: c0b05792ff3ac3c5bdd5fad1b7586dd5 (MD5) Previous issue date: 2014 / Resumo: Neste trabalho apresentamos um estudo sobre as funções de Mittag-Leffler de um, dois e três parâmetros. Apresentamos a função de Mittag-Leffler como uma generalização da função exponencial bem como a relação que esta possui com outras funções especiais, tais como as funções beta, gama, gama incompleta e erro. Abordamos, também, a integração fracionária que se faz necessária para introduzir o conceito de derivação fracionária. Duas formulações para a derivada fracionária são estudadas, as formulações proposta por Riemann-Liouville e por Caputo. Investigamos quais regras clássicas de derivação são estendidas para estas formulações. Por fim, como uma aplicação, utilizamos a metodologia da transformada de Laplace para resolver a equação diferencial fracionária associada ao problema do oscilador harmônico fracionário / Abstract: This work presents a study about the one- two- and three-parameters Mittag-Leffler functions. We show that the Mittag-Leffler function is a generalization of the exponential function and present its relations to other special functions beta, gamma, incomplete gamma and error functions. We also approach fractional integration, which is necessary to introduce the concept of fractional derivatives. Two formulations for the fractional derivative are studied, the formulations proposed by Riemann-Liouville and by Caputo. We investigate which classical derivatives rules can be extended to these formulations. Finally, as an application, using the Laplace transform methodology, we discuss the fractional differential equation associated with the harmonic oscillator problem / Mestrado / Matematica Aplicada / Mestra em Matemática Aplicada

Ce travail est constitué de deux parties indépendantes traitant chacune d'un problème issu de la théorie du contrôle des équations aux dérivées partielles. La première partie est consacrée à l'étude d'un feedback explicite et déjà connu, s'appliquant à des systèmes linéaires, réversibles en temps et éventuellement munis d'un opérateur de contrôle non-borné. On justifie le caractère bien posé du problème en boucle fermée via la théorie des semi-groupes puis on étudie le taux de décroissance des solutions du système régulé. La seconde partie concerne un problème d'observation pour la corde vibrante : on détermine comment choisir des instants d'observation pour que la position de la corde à ces instants permette de retrouver les conditions initiales tout en préservant une certaine régularité. La méthode, qui repose sur des résultats d'approximation diophantienne, est ensuite étendue à d'autres systèmes. En utilisant une méthode de dualité on démontre aussi un résultat de contrôlabilité exacte. / This works contains two independent parts, each one dealing with the control of partial differential equations. In the first part, we study an explicit and already known feedback law that applies to linear, time-reversible systems, with a possibly unbounded control operator. We prove the well-posedness of the closed-loop problem in the semi-group framework and we study the decay rate of the solutions. In the second part, we give conditions on the choice of some time instants, such that the positions of a vibrating string (or beam) at these times enable to recover the initial data. The method relies on Diophantine approximation results. Using a duality method, we give a related exact controllability result.