Estudamos as restrições impostas pelo princípio da covariância sobre o procedimento de quantização em espaços planos e curvos. Mostramos que o conjunto de todas as quantizações covariantes em espaços planos em coordenadas retangulares é composto de ordenamentos de operadores de posição e momento e exibimos uma parametrização funcional deste conjunto. Deduzimos regras para a quantização covariante em espaços planos em coordenadas gerais. Generalizamos estas quantizações para espaços curvos e mostramos que nestes espaços, além da ambiguidade de ordenamento, surge uma nova ambiguidade relacionada à curvatura. Este novo tipo de ambiguidade explica o surgimento de uma classe grande de potenciais quânticos no problema da quantização de uma partícula não relativística em um espaço curvo. / We study the restrictions imposed by the covariance principle on the quantization procedure in flat and curved spaces. We show that the set of all covariant quantizations in flat spaces in rectangular coordinates is composed of position and momentum operator orderings and exhibit a functional parametrization of this set. We deduce rules for the covariant quantization in flat spaces in general coordinates. We generalize these quantizations for curved spaces and show that in such spaces, besides the ordering ambiguity, it appears a new ambiguity related to the curvature. This new kind of ambiguity explains the appearence of a wide class of quantum potentials in the problem of quantization of a non-relativistic particle in curved space.

In the present thesis the dynamics of the correlations between an open quantum system and its environment is investigated. This becomes feasible by means of a very useful representation of the total system-environment state. General conditions for separability and entanglement of the latter are derived, and investigated in the framework of an open quantum two-level system, which is coupled to a dissipative and a dephasing environment.

The problem under consideration in the thesis is a two level atom in a photonic crystal and a pumping laser. The photonic crystal provides an environment for the atom, that modifies the decay of the exited state, especially if the atom frequency is close to the band gap. The population inversion is investigated als well as the emission spectrum. The dynamics is analysed in the context of open quantum systems. Due to the multiple reflections in the photonic crystal, the system has a finite memory that inhibits the Markovian approximation. In the Heisenberg picture the equations of motion for the system variables form a infinite hierarchy of integro-differential equations. To get a closed system, approximations like a weak coupling approximation are needed. The thesis starts with a simple photonic crystal that is amenable to analytic calculations: a one-dimensional photonic crystal, that consists of alternating layers. The Bloch modes inside and the vacuum modes outside a finite crystal are linked with a transformation matrix that is interpreted as a transfer matrix. Formulas for the band structure, the reflection from a semi-infinite crystal, and the local density of states in absorbing crystals are found; defect modes and negative refraction are discussed. The quantum optics section of the work starts with the discussion of three problems, that are related to the full resonance fluorescence problem: a pure dephasing model, the driven atom and resonance fluorescence in free space. In the lowest order of the system-environment coupling, the one-time expectation values for the full problem are calculated analytically and the stationary states are discussed for certain cases. For the calculation of the two time correlation functions and spectra, the additional problem of correlations between the two times appears. In the Markovian case, the quantum regression theorem is valid. In the general case, the fluctuation dissipation theorem can be used instead. The two-time correlation functions are calculated by the two different methods. Within the chosen approximations, both methods deliver the same result. Several plots show the dependence of the spectrum on the parameters. Some examples for squeezing spectra are shown with different approximations. A projection operator method is used to establish two kinds of Markovian expansion with and without time convolution. The lowest order is identical with the lowest order of system environment coupling, but higher orders give different results. / Die Arbeit befasst sich mit der Emission eines 2-Niveau-Atoms in einem photonischen Kristall mit einem treibenden Laser. Der photonische Kristall stellt für das Atom eine Umgebung dar, die seinen Zerfall verändert, insbesondere wenn die Übergangsfrequenz des Atoms nahe an der Bandkante ist. Es werden sowohl die Besetzungen als auch das Emissionsspektrum untersucht. Die Dynamik wird im Kontext offener Quantensysteme analysiert. Durch die vielfachen Reflexionen im photonischen Kristall hat das System ein endliches Gedächtnis, das die Markov-Näherung verhindert. Im Heisenberg-Bild stellen die Bewegungsgleichungen für die Systemvariablen eineunendliche Hierachie von Integro-Differentialgleichungen dar. Um ein geschlossenes System zu erhalten, sind Näherungen wie eine schwache Kopplung nötig. Zunächst wird ein einfacher photonischer Kristall betrachtet.: Der eindimensionale photonische Kristall, der aus wechselnden Lagen besteht. Die Blochmoden innerhalb und die Vakuummoden außerhalb des endlichen photonischen Kristalls sind durch eine Transformationsmatrix, die als Transfermatrix interpretiert werden kann, miteinander verbunden. Einfache Formeln für die Bandstruktur, Reflexion eines halb-unendlichen Kristalls, die lokale Zustandsdichte im absorbierenden Kristall werden gefunden; außerdem werden Defektmoden und negative Brechung diskutiert. Im quantenoptischen Teil der Arbeit werden zu Anfang drei Probleme diskutiert, die im Zusammenhang zum Problem der Resonanzfluoreszenz stehen und die analytisch berechnet werden können: Ein Dephasierungsmodell, das getriebenen Atom und Resonanzfluoreszenz im freien Raum. In der niedrigsten Ordnung der System-Bad-Kopplung werden die Erwartungswerte analytisch berechnet und die stationären Zustände werden für bestimmte Fälle diskutiert. Bei der Berechnung der Zweizeitkorrelationsfunktion und der Spektren taucht das zusätzliche Problem der Korrelationen zwischen den beiden Zeiten auf. Im Markov-Fall gilt das Quantenregressionstheorem. Im allgemeinen Fall kann stattdessen das Fluktuations-Dissipations-Theorem benutzt werden. Die Korrelationsfunktionen werden mit zwei verschiedenen Methoden berechnet. Innerhalb der gewählten Näherungen liefern beide Methoden dasselbe Resultat. Einige Plots zeigen die Abhängigkeit des Spektrums von den verschiedenen Parametern. Mehrere Beispiele für Squeezing-Spektren werden mit den verschiedenen Näherungen gezeigt. Eine Projektions-Operator-Methode wird benutzt, um zwei Arten einer Markov-Entwicklung zu implementieren, mit und ohne Faltungsintegral. Die niedrigste Ordnung ist identisch mit der niedrigsten Ordnung der System-Bad-Kopplung, wohingegen höhere Ordnungen andere Resultate ergeben.

L'informatique quantique, brièvement introduite au chapitre 1, exploite les corrélations quantiques et en particulier l'intrication. Ces corrélations sont difficiles à maintenir car un système quantique n'est habituellement pas fermé, mais en interaction avec son environnement. Le traitement formel d'un système quantique ouvert requiert des outils spécifiques, introduits au chapitre 2. En utilisant ces notions, nous montrerons au chapitre 3 que l'interaction entre le système et son environnement aura pour effet de privilégier certains états, qualifiés de quasi-classiques, suggérant ainsi l'émergence d'un monde classique à partir d'un monde quantique. De plus, l'intrication qui se crée entre le système et son environnement détruira la cohérence d'une superposition d'états quasi-classiques. Il s'agit du phénomène de décohérence dont les mécanismes seront mis en évidence dans notre étude originale d'un gyroscope quantique au chapitre 4. Nous montrerons qu'une particule de grand spin servant à mesurer le moment angulaire d'électrons perd sa cohérence en un temps très court par rapport au temps caractéristique de relaxation. Afin de protéger la cohérence d'un système, essentielle pour l'informatique quantique, plusieurs techniques de protection ont été développées. Nous les rappelerons brièvement en début de chapitre 5, avant d'introduire une approche originale qui consiste à préparer l'environnement. Notre étude nous permet de caractériser l'existence d'états initiaux de l'environnement permettant une évolution sans décohérence du système dans une gamme de modèles où le système interagit avec un environnement présentant une dynamique propre. / Quantum information processing, briefly introduced in Chapter 1, relies on quantum correlations, namely on entanglement. Those correlations are difficult to maintain since a typical quantum system is not closed, but interacting with its environment. The analysis of an open quantum system requires specific tools which we introduce in Chapter 2. Using these concepts, we show in Chapter 3 that the interaction between the system and its environment will distinguish certain quasi-classical states, suggesting the emergence of a classical world from a quantum one. Furthermore, the entanglement created between the system and its environment will destroy the coherence of a superposition of such quasi-classical states. This phenomenon of decoherence exhibits mechanisms which we highlight in our original study of a quantum gyroscope in chapter 4. We demonstrate that a particle with large spin, used to measure the angular momentum of electrons, loses its coherence on a timescale much shorter than the characteristic timescale of relaxation. To protect the coherence of a system, essential to quantum information processing, several techniques have been developed. We briefly review them at the beginning of Chapter 5, before introducing a novel approach based on the preparation of the environment. Our analysis characterizes the existence of initial states of the environment allowing for decoherence-free evolution of the system in a large class of models in which the system interacts with a dynamical environment. / Ce travail de maîtrise a mené à la rédaction d'un article (Physical Review A 80, 062319 (2009)).

Estudamos o amortecimento radiativo e o ruído de spins de um material magnético acoplado a um circuito ressonante. O amortecimento radiativo em ressonância magnética é um fenômeno de dissipação, na qual a magnetização preparada após um pulso de Rabi sofre um decaimento até seu estado de equilíbrio. O material magnético perde energia através do seu acoplamento com o circuito ressonante, que deve estar sintonizado na freqüência de Larmor dos spins do material. Apesar deste fenômeno ter sido estudado há vários anos, nenhuma descrição quântica completa lhe foi dada. Apresentamos um modelo hamiltoniano quântico que descreve o amortecimento radiativo. Para isto usamos o método de equações de Langevin quânticas. Mostramos que além do amortecimento radiativo do material magnético, se o circuito está em um estado inicial coerente, a magnetização adquire um movimento complicado não-trivial. Usando as mesmas equações de Langevin, estudamos a influência da amostra no ruído do circuito ressonante. Calculamos a densidade espectral da corrente no caso em que todo o sistema está em equilíbrio térmico. Pudemos verifcar a efcácia do método comparando-o com estudos anteriores. Além disso, estudamos as alterações do ruído do circuito quando uma tensão oscilante externa é aplicada. Nesta situação surgem dois outros picos laterais ao pico central do espectro de absorção da amostra magnética. Isso leva a três depressões no espectro da corrente do circuito. Este efeito deve-se à separação dupla dos estados de energia dos spins. Comentamos sobre a analogia deste fenômeno com a fluorescência ressonante observada na Óptica Quântica. / We study the radiation damping and the spin noise of a magnetic material coupled with a resonant circuit. Radiation damping in magnetic resonance is a dissipation phenomenon, where magnetization prepared after a Rabi pulse decays toward its equilibrium state. The magnetic sample loses its energy by the coupling with resonant circuit, that must be tuned in Larmor frequency of the sample spins. Even though this phenomenon had been studied many years ago, no full quantum description was done. We present a quantum Hamiltonian model, that explains the radiation damping. We use quantum Langevin equation method for this task. Beyond radiation damping, we show the magnetization acquires an unusual intrincate motion, if the circuit initial state is coherent. Using the same Langevin equation, we study the sample influence on the resonant circuit noise. We calculate the current spectral density in the case of thermal equilibrium of whole system. We can verify the method efectiveness, comparing former papers. Moreover we study modifcations in the circuit noise, if an external oscillating tension is applied. In this situation, other two peaks emerge in the central peak sidebands of the sample absorption spectrum. It leads to appear three dips in circuit current spectrum. This efect is due to the splitting of the spin energy states. We comment about the analogy between this phenomenon and the resonance fluorescence in Quantum Optics.

Estudamos as restrições impostas pelo princípio da covariância sobre o procedimento de quantização em espaços planos e curvos. Mostramos que o conjunto de todas as quantizações covariantes em espaços planos em coordenadas retangulares é composto de ordenamentos de operadores de posição e momento e exibimos uma parametrização funcional deste conjunto. Deduzimos regras para a quantização covariante em espaços planos em coordenadas gerais. Generalizamos estas quantizações para espaços curvos e mostramos que nestes espaços, além da ambiguidade de ordenamento, surge uma nova ambiguidade relacionada à curvatura. Este novo tipo de ambiguidade explica o surgimento de uma classe grande de potenciais quânticos no problema da quantização de uma partícula não relativística em um espaço curvo. / We study the restrictions imposed by the covariance principle on the quantization procedure in flat and curved spaces. We show that the set of all covariant quantizations in flat spaces in rectangular coordinates is composed of position and momentum operator orderings and exhibit a functional parametrization of this set. We deduce rules for the covariant quantization in flat spaces in general coordinates. We generalize these quantizations for curved spaces and show that in such spaces, besides the ordering ambiguity, it appears a new ambiguity related to the curvature. This new kind of ambiguity explains the appearence of a wide class of quantum potentials in the problem of quantization of a non-relativistic particle in curved space.

La chromodynamique quantique (QCD) prédit l'existence d'un nouvel état de la matière : le plasma de quarks et de gluons (PQG). Celui-ci aurait existé dans les premiers instants suivant le Big Bang et peut en principe être produit sous les conditions extrêmes de température et de densité atteintes lors de collisions d'ions lourds à haute énergie (au LHC par exemple). Un des marqueurs de sa présence est la suppression des quarkonia (états liés de quark/antiquark lourds), caractérisée par une production inférieure de ces états dans les collisions d'ions lourds relativement aux collisions proton-proton où le PQG ne pourrait être créé. Cette suppression a bien été observée expérimentalement, mais l'évolution de ses tendances aux énergies du RHIC et du LHC est un véritable défi qui requiert une meilleure compréhension théorique. La présente thèse a pour but d’étudier l’évolution en temps réel de paires corrélées de quark/antiquark lourds considérées comme des systèmes quantiques ouverts en interaction permanente avec un PQG en refroidissement. Explicitement, l'interaction continue entre le milieu et les degrés de liberté internes de la paire est obtenue par 1) un écrantage de couleur dit « de Debye » dû à la présence de charges de couleur dans leur voisinage et 2) des mécanismes de fluctuation/dissipation qui reflètent les collisions permanentes. Cela mène à une image dynamique et continue de la dissociation des quarkonia, de leur recombinaison et des transitions entre états liés. L'étude est transversale à différents cadres théoriques : semi-classique, quantique et quantique des champs. Les prédictions du modèle sont comparées aux résultats expérimentaux et aux résultats d'autres modèles théoriques. / The theory of quantum chromodynamics (QCD) predicts the existence of a new state of matter: the Quark-GluonPlasma (QGP). The latter may have existed at the first moments of the Universe following the Big Bang and can be, in theory, re-produced under the extreme conditions of temperature and density reached in high energy heavy ion collisions (at the LHC for instance). One of the QGP observables is the suppression of the quarkonia (heavy quark/antiquark bound states), characterised by a smaller production of these states in heavy ion collisions in comparison to proton-protoncollisions, in which no QGP production would be possible. This suppression has indeed been observed experimentally, but the puzzling evolution of its trend from RHIC to LHC energies requires a better theoretical understanding. The present thesis aims at studying the real-time evolution of correlated heavy quark/antiquark pairs described as open quantum systems which permanently interact with a cooling QGP. More explicitly, the continuous interaction between the medium and the pair internal degrees of freedom is obtained through 1) a temperature dependent color screening (“Debye” like) due to color charges in the irvicinity and 2) some fluctuation/dissipation mechanisms reflecting the continuous collisions. It leads to a dynamical and continuous picture of the dissociation, recombination and possible transitions to other bound states. This investigation is at the crossroads of different theoretical frameworks: semi-classic, quantum and quantum fields. The deduced predictions are compared to experimental data and to the results of other theoretical models.

In the present thesis the dynamics of the correlations between an open quantum system and its environment is investigated. This becomes feasible by means of a very useful representation of the total system-environment state. General conditions for separability and entanglement of the latter are derived, and investigated in the framework of an open quantum two-level system, which is coupled to a dissipative and a dephasing environment.

The famous Moore's law states: Since the invention of the integrated circuit, the number of transistors that can be placed on an integrated circuit has increased exponentially, doubling approximately every two years. As a result of the downscaling of the size of the transistor, quantum effects have become increasingly important while affecting significantly the device performances. Nowadays, at the nanometer scale, inter-atomic interactions and quantum mechanical properties need to be studied extensively. Device and material simulations are important to achieve these goals because they are flexible and less expensive than experiments. They are also important for designing and characterizing new generation of electronic device such as silicon nanowire or carbon nanotube (CNT) transistors. Several modeling methods have been developed and applied to electronic structure calculations, such as: Hartree-Fock, density functional theory (DFT), empirical tight-binding, etc. For transport simulations, most of the device community focuses on studying the stationary problem for obtaining characteristics such as I-V curves. The non-equilibrium transport problem is then often addressed by solving a multitude of time-independent Schrodinger-type equation for all possible energies. On the other hand, for many other electronic applications including high-frequency electronics response (e.g. when a time-dependent potential is applied to the system), the description of the system behavior necessitate insights on the time dependent electron dynamics. To address this problem, it is then necessary to solve a time-dependent Schrodinger-type equation. In this thesis, we will focus on solving time-dependent problems with application to CNTs. We will be identifying all the numerical difficulties and propose new effective modeling and numerical schemes to address the current limitations in time-dependent quantum simulations. we will point out that two numerical errors may occur: an integration error and the anti-commutation issue error; the direct computation above being mathematically equivalent to performing the integration of the time dependent Hamiltonian using a rectangle numerical quadrature formula along the total simulation times. After careful study and many numerical experiments, we found that the Gaussian quadrature scheme provides a good trade off between computational consumption and numerically accuracy, meanwhile unitary, stability and time reversal properties are well preserved. The new Gaussian quadrature integration scheme uses (i) much fewer points in time to approximate the integral of the Hamiltonian, (ii) ordered exponential to factorize the time evolution operator, (iii) FEM discretize techniques (iv) and at last, the FEAST eigenvalue solver to diagonalize and solve each exponential.

The goal of optimal transport is to map a source probability measure to a destination one with the minimum possible cost. However, the optimal mapping might not be feasible under some practical constraints. One such example is to realize a transport mapping through an information bottleneck. As the optimal mapping may induce infinite mutual information between the source and the destination, the existence of an information bottleneck forces one to resort to some suboptimal mappings. Investigating this type of constrained optimal transport problems is clearly of both theoretical significance and practical interest. In this work, we substantiate a particular form of constrained optimal transport in the context of quantum-to-classical systems by establishing an Output-Constrained Rate-Distortion Theorem similar to the classical case introduced by Yuksel et al. This theorem develops a noiseless communication channel and finds the least required transmission rate R and common randomness Rc to transport a sufficiently large block of n i.i.d. source quantum states, to samples forming a perfectly i.i.d. classical destination distribution, while maintaining the distortion between them. The coding theorem provides operational meanings to the problem of Rate-Limited Optimal Transport, which finds the optimal transportation from source to destination subject to the rate constraints on transmission and common randomness. We further provide an analytical evaluation of the quantum-to-classical rate-limited optimal transportation cost for the case of qubit source state and Bernoulli output distributions with unlimited common randomness. The evaluation results in a transcendental system of equations whose solution provides the rate-distortion curve of the transportation protocol. We further extend this theorem to continuous-variable quantum systems by employing a clipping and quantization argument and using our discrete coding theorem. Moreover, we derive an analytical solution for rate-limited Wasserstein distance of 2nd order for Gaussian quantum systems with Gaussian output distribution. We also provide a Gaussian optimality theorem for the case of unlimited common randomness, showing that Gaussian measurement optimizes the rate in a system with Gaussian source and destination. / Thesis / Doctor of Philosophy (PhD) / We establish a coding theorem for rate-limited quantum-classical optimal transport systems with limited classical common randomness. The coding theorem, referred to as the output-constrained rate-distortion theorem, characterizes the rate region of measurement protocols on a product quantum source state for faithful construction of a given classical destination distribution while maintaining the source-destination distortion below a prescribed threshold with respect to a general distortion observable. This theorem provides a solution to the problem of rate-limited optimal transport, which aims to find the optimal cost of transforming a source quantum state to a destination distribution via a measurement channel with a limited classical communication rate. The coding theorem is further extended to cover Bosonic continuous-variable quantum systems. The analytical evaluation is provided for the case of a qubit measurement system with unlimited common randomness, as well as the case of Gaussian quantum systems.