A pesquisa na area de redes complexas tem evoluido bastante, e e nesta linha que o presente trabalho visa aportar, dando enfase especial no processo epidemico sobre redes. Desse modo, foi feito uma analise geral das redes complexas em conjunto com suas propriedades. Apos isso, desenvolveu-se o processo de contagio da epidemia do tipo suscetivel-infectado sobre uma rede aleatoria uniforme e sobre uma rede aleatoria com ligacoes preferenciais. Ambas abordagens foram desenvolvidas usando equacoes mestras para finalmente fazer sua analise com metodos analiticos e computacionais. / Research in the area of complex networks has evolved greatly, and over this line that this present work aims to contribute, with particular emphasis on the epidemic process over networks. Along these lines, a general review about complex networks is made with their main properties. After that, a susceptible-infected contagion process is developed over a uniform random network and a preferential attachment network. Both approaches were developed using master equations to finally analyze them with analytical and computatio- nal methods.

A pesquisa na area de redes complexas tem evoluido bastante, e e nesta linha que o presente trabalho visa aportar, dando enfase especial no processo epidemico sobre redes. Desse modo, foi feito uma analise geral das redes complexas em conjunto com suas propriedades. Apos isso, desenvolveu-se o processo de contagio da epidemia do tipo suscetivel-infectado sobre uma rede aleatoria uniforme e sobre uma rede aleatoria com ligacoes preferenciais. Ambas abordagens foram desenvolvidas usando equacoes mestras para finalmente fazer sua analise com metodos analiticos e computacionais. / Research in the area of complex networks has evolved greatly, and over this line that this present work aims to contribute, with particular emphasis on the epidemic process over networks. Along these lines, a general review about complex networks is made with their main properties. After that, a susceptible-infected contagion process is developed over a uniform random network and a preferential attachment network. Both approaches were developed using master equations to finally analyze them with analytical and computatio- nal methods.

Computational and mathematical models of cellular processes promise great benets in important elds such as molecular biology and medicine. Increasingly, researchers are incorporating the fundamentally discrete and stochastic nature of biochemical processes into the mathematical models that are intended to represent them. This has led to the formulation of models for genetic networks as continuous-time, discrete state, Markov processes, giving rise to the so-called Chemical Master Equation (CME), which is a discrete, partial dierential equation, that governs the evolution of the associated probability distribution function (PDF). While promising many insights, the CME is computationally challenging, especially as the dimension of the model grows. In this thesis, novel methods are developed for computing the PDF of the Master Equation. The problems associated with the high-dimensional nature of the Chemical Master Equation are addressed by adapting Krylov methods, in combination with Finite State Projection methods, to derive algorithms well-suited to the Master Equation. Variations of the approach that incorporate the Strang splitting and a stochastic analogue of the total quasi-steady-state approximation are also derived for chemical systems with disparate rates. Monte Carlo approaches, such as the Stochastic Simulation Algorithm, that simulate trajectories of the process governed by the CME have been a very popular approach and we compare these with the PDF approaches developed in this thesis. The thesis concludes with a discussion of various implementation issues along with numerical results for important applications in systems biology, including the gene toggle, the Goldbeter-Koshland switch and the Mitogen-Activated Protein Kinase Cascade.

Computational and mathematical models of cellular processes promise great benets in important elds such as molecular biology and medicine. Increasingly, researchers are incorporating the fundamentally discrete and stochastic nature of biochemical processes into the mathematical models that are intended to represent them. This has led to the formulation of models for genetic networks as continuous-time, discrete state, Markov processes, giving rise to the so-called Chemical Master Equation (CME), which is a discrete, partial dierential equation, that governs the evolution of the associated probability distribution function (PDF). While promising many insights, the CME is computationally challenging, especially as the dimension of the model grows. In this thesis, novel methods are developed for computing the PDF of the Master Equation. The problems associated with the high-dimensional nature of the Chemical Master Equation are addressed by adapting Krylov methods, in combination with Finite State Projection methods, to derive algorithms well-suited to the Master Equation. Variations of the approach that incorporate the Strang splitting and a stochastic analogue of the total quasi-steady-state approximation are also derived for chemical systems with disparate rates. Monte Carlo approaches, such as the Stochastic Simulation Algorithm, that simulate trajectories of the process governed by the CME have been a very popular approach and we compare these with the PDF approaches developed in this thesis. The thesis concludes with a discussion of various implementation issues along with numerical results for important applications in systems biology, including the gene toggle, the Goldbeter-Koshland switch and the Mitogen-Activated Protein Kinase Cascade.

Decoherence is a major barrier to the implementation of quantum technologies. Theoretical techniques for understanding decoherence in composite systems have traditionally been focused on systems with distinguishable emission spectra, where measuring the frequency of an emitted photon allows one to determine which process took place. Here the photon contains information about the state of the system. On the other hand, systems with indistinguishable spectra do not necessarily completely reveal information about the state of the system when a photon is emitted. It can be impossible to say for certain which of two nearly degenerate transitions has occurred just by measuring the photon's frequency. It is then possible to preserve information within the system throughout the decay process. In this Thesis we show that indistinguishable spectra can lead to protected coherences within one part of a coupled quantum system, even as another part decays. We develop a zero-temperature exact approach for modelling such systems, and compare it to the microscopically derived Born-Markov master equation. This comparison helps us to understand the range of validity of the Markovian approximation. We use this understanding to extend the master equation approach to finite temperature within the Markovian regime, and we compare its high temperature results to a semiclassical model. We examine the physical conditions required for coherence protection, and remarkably we find that heating the system can improve coherence protection. Similarly, increasing the decay rate of the unprotected part of the coupled system can also enhance the coherence of the protected part. These effects are the results of linewidth broadening and thus greater spectral indistinguishability. The findings in this Thesis are of interest to both those seeking to engineer hybrid quantum systems and those seeking to develop theoretical techniques for dealing with the decoherence of composite quantum systems.

Understanding energy transfer at the molecular scale is both essential for the design of novel molecular level devices and vital for uncovering the fundamental properties of non-equilibrium open quantum systems. In this thesis, we first establish the connection between molecular scale devices -- molecular electronics and phononics -- and open quantum system models. We then develop theoretical tools to study various properties of these models. We extend the standard master equation method to calculate the steady state thermal current and conductance coefficients. We then study the scaling laws of the thermal current with molecular chain size and energy, and apply this tool to investigate the onset of nonlinear thermal current - temperature characteristics, thermal rectification and negative differential conductance. Our master equation technique is valid in the ``on-resonance" regime, referring to the situation in which bath modes in resonance with the subsystem modes are thermally populated. In the opposite ``off-resonance" limit, we develop the Energy Transfer Born-Oppenheimer method to obtain the thermal current scaling without the need to solve for the subsystem dynamics. Finally, we develop a mapping scheme that allows the dynamics of a class of open quantum systems containing coupled subsystems to be treated by considering the separate dynamics in different subsections of the Hilbert space. We combine this mapping scheme with path integral numerical simulations to explore the rich phenomenon of entanglement dynamics within a dissipative two-qubit model. The formalisms developed in this thesis could be applied for the study of energy transfer in different realizations, including molecular electronic junctions, donor-acceptor molecules, artificial solid state qubits and cold-atom lattices.

Understanding energy transfer at the molecular scale is both essential for the design of novel molecular level devices and vital for uncovering the fundamental properties of non-equilibrium open quantum systems. In this thesis, we first establish the connection between molecular scale devices -- molecular electronics and phononics -- and open quantum system models. We then develop theoretical tools to study various properties of these models. We extend the standard master equation method to calculate the steady state thermal current and conductance coefficients. We then study the scaling laws of the thermal current with molecular chain size and energy, and apply this tool to investigate the onset of nonlinear thermal current - temperature characteristics, thermal rectification and negative differential conductance. Our master equation technique is valid in the ``on-resonance" regime, referring to the situation in which bath modes in resonance with the subsystem modes are thermally populated. In the opposite ``off-resonance" limit, we develop the Energy Transfer Born-Oppenheimer method to obtain the thermal current scaling without the need to solve for the subsystem dynamics. Finally, we develop a mapping scheme that allows the dynamics of a class of open quantum systems containing coupled subsystems to be treated by considering the separate dynamics in different subsections of the Hilbert space. We combine this mapping scheme with path integral numerical simulations to explore the rich phenomenon of entanglement dynamics within a dissipative two-qubit model. The formalisms developed in this thesis could be applied for the study of energy transfer in different realizations, including molecular electronic junctions, donor-acceptor molecules, artificial solid state qubits and cold-atom lattices.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23450号 / 理博第4744号 / 新制||理||1680(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 早川 尚男, 教授 佐々 真一, 教授 川上 則雄 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Realizamos nesse trabalho um tratamento abrangente da interação entre um sistema quântico e o meio ambiente modelado como um conjunto de osciladores harmônicos. Partimos para isso de um tratamento prévio de redes de osciladores harmônicos quânticos dissipativos. Utilizando a função característica, transformamos a equação de von Neumann em uma equação diferencial, e explorando a sua linearidade, essa é transformada em uma equação vetorial, cuja resolução é computacionalmente eficiente. Nosso formalismo, que parte de uma rede de osciladores harmônicos, não necessariamente dividida entre sistema e meio ambiente, permite que se contorne a necessidade da hipótese de acoplamento súbito sistema-reservatório para o tratamento exato da evolução do sistema. Em seguida, mostramos que essa evolução pode ser sempre descrita por uma equação mestra na forma usual de Lindblad, embora os coeficientes que a definem possam ser dependentes do tempo. Isso abre novas possibilidades para a dinâmica do sistema, e leva a efeitos que podem ser classificados de não-Markovianos, embora sejam descritos por uma equação mestra completamente local no tempo. Ressaltamos que, por ser baseado em uma solução exata, o método pode ser aplicado para qualquer intensidade de acoplamento, e é consideravelmente mais simples do que outros métodos disponíveis para esse fim, como os baseados em integrais de trajetória. Por fim, utilizamos simulações computacionais para explorar a validade das aproximações de ondas girantes e de Born-Markov, e os fenômenos que podem ser observados nos regimes em que elas deixam de ser válidas. / We present a comprehensive treatment of the interaction of a quantum system with an environment modeled as a set of harmonic oscillators. We start from a previous treatment of a network of quantum dissipative harmonic oscillators. Using the characteristic function, we transform the von Neumann equation in a differential equation, and exploring its linearity, this is transformed in a vector equation, whose solution is computationally efficient. Our method, whose origin lies on a network not necessarily divided into system and reservoir, allows us to circumvent the necessity of the sudden-coupling approximation for the exact treatment of the system evolution. After this, we show that this dynamics can always be described by a master equation in standard Lindblad form, although its coefficients may be functions of time. This opens new possibilities for the system dynamics, and lead to effects that may be called non-Markovian, even if they are described by a completely local-in-time master equation. It should be emphasized that, as it is based on an exact solution, the method may be applied for any strength of the system-reservoir interaction, and it is considerably simpler than other available methods, such as those based on path integrals. Finally, we employ computer simulations to investigate the validity of the rotating-wave and Born-Markov approximations, and the phenomena that migth be observed in regimes in which they fail to be valid.

Realizamos nesse trabalho um tratamento abrangente da interação entre um sistema quântico e o meio ambiente modelado como um conjunto de osciladores harmônicos. Partimos para isso de um tratamento prévio de redes de osciladores harmônicos quânticos dissipativos. Utilizando a função característica, transformamos a equação de von Neumann em uma equação diferencial, e explorando a sua linearidade, essa é transformada em uma equação vetorial, cuja resolução é computacionalmente eficiente. Nosso formalismo, que parte de uma rede de osciladores harmônicos, não necessariamente dividida entre sistema e meio ambiente, permite que se contorne a necessidade da hipótese de acoplamento súbito sistema-reservatório para o tratamento exato da evolução do sistema. Em seguida, mostramos que essa evolução pode ser sempre descrita por uma equação mestra na forma usual de Lindblad, embora os coeficientes que a definem possam ser dependentes do tempo. Isso abre novas possibilidades para a dinâmica do sistema, e leva a efeitos que podem ser classificados de não-Markovianos, embora sejam descritos por uma equação mestra completamente local no tempo. Ressaltamos que, por ser baseado em uma solução exata, o método pode ser aplicado para qualquer intensidade de acoplamento, e é consideravelmente mais simples do que outros métodos disponíveis para esse fim, como os baseados em integrais de trajetória. Por fim, utilizamos simulações computacionais para explorar a validade das aproximações de ondas girantes e de Born-Markov, e os fenômenos que podem ser observados nos regimes em que elas deixam de ser válidas. / We present a comprehensive treatment of the interaction of a quantum system with an environment modeled as a set of harmonic oscillators. We start from a previous treatment of a network of quantum dissipative harmonic oscillators. Using the characteristic function, we transform the von Neumann equation in a differential equation, and exploring its linearity, this is transformed in a vector equation, whose solution is computationally efficient. Our method, whose origin lies on a network not necessarily divided into system and reservoir, allows us to circumvent the necessity of the sudden-coupling approximation for the exact treatment of the system evolution. After this, we show that this dynamics can always be described by a master equation in standard Lindblad form, although its coefficients may be functions of time. This opens new possibilities for the system dynamics, and lead to effects that may be called non-Markovian, even if they are described by a completely local-in-time master equation. It should be emphasized that, as it is based on an exact solution, the method may be applied for any strength of the system-reservoir interaction, and it is considerably simpler than other available methods, such as those based on path integrals. Finally, we employ computer simulations to investigate the validity of the rotating-wave and Born-Markov approximations, and the phenomena that migth be observed in regimes in which they fail to be valid.