A Tutorial on Quantum Master Equations: Tips and tricks for quantum optics, quantum computing and beyond

Quantum master equations are an invaluable tool to model the dynamics of a plethora of microscopic systems, ranging from quantum optics and quantum information processing, to energy and charge transport, electronic and nuclear spin resonance, photochemistry, and more. This tutorial offers a concise and pedagogical introduction to quantum master equations, accessible to a broad, cross-disciplinary audience. The reader is guided through the basics of quantum dynamics with hands-on examples that build up in complexity. The tutorial covers essential methods like the Lindblad master equation, Redfield relaxation, and Floquet theory, as well as techniques like Suzuki-Trotter expansion and numerical approaches for sparse solvers. These methods are illustrated with code snippets implemented in python and other languages, which can be used as a starting point for generalisation and more sophisticated implementations.Comment: 57 pages, 12 figures, 34 code example

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developements are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, quantum mechanics, representation theory, and the theory of error-correcting codes

Efficient Computational Approaches for Treatment of Transformed Path Integrals

In this thesis, efficient computational approaches for treatment of transformed path integrals (TPIs) are proposed. The TPI-based method allows us to calculate the time evolution probability density functions (PDFs) using a short time propagator matrix that accounts for the transition probability in a transformed domain. A grid-based implementation of the TPI, in contrast to the conventional fixed-grid implementation of a path integral (PI), allows the propagation of the PDF to be performed on a dynamically adaptive grid parametrized by the mean and covariance of the PDF. TPI-based methods generate PDFs from all possible paths within the transformed space, and while these methods are found to be highly effective at capturing tail information in systems with large drifts, diffusions, and concentrations, they can become somewhat computationally expensive when applied to systems that must be represented by large numbers of data points. The purpose of this thesis is to develop computationally efficient TPI-based methods that largely preserve the accuracy and other desirable features of the original TPI method. The first proposed method, referred to as the bandlimited TPI (BL-TPI) method, takes advantage of the fact that the transition probability is often concentrated around a set of peaks, with one natural peak occurring for each source state. This allows us to consider sparse matrix representations of the transition probability matrix operator and consider a region of importance about the peak transition probability curve for consideration in PDF propagation while neglecting all values outside of this region. With the use of sparse matrix tools, the BL-TPI enables us to perform PDF propagation using far fewer operations than the standard implementation. In the second proposed method, a TPI implementation based on the Symmetric Fast Gauss Transform (SFGT) is proposed. This method utilizes a Taylor series expansion of the Gaussian kernel in the propagator matrix to reduce the convolution operation for the PDF to an infinite sum of moments. This allows us to perform calculations involving source and target terms separately, eliminating their convolution and in the process potentially reducing the associated computational complexity. In order to demonstrate the effectiveness of the proposed approaches, comparisons with the standard TPI implementation are performed for canonical problems in onedimensional and multi-dimensional state spaces. The results from the BL-TPI method appear promising and indicate that the method is applicable to a wide range of cases. In contrast, the effectiveness of the SFGT approach is found to be inherently conditional, and the computational cost of this method can exceed that of the standard TPI method in many cases

Dynamiques stochastiques sur réseaux complexes

Tableau d’honneur de la Faculté des études supérieures et postdoctorales, 2012-2013.Cette thèse a pour but d'élaborer et d'étudier des modèles mathématiques reproduisant le comportement de systèmes composés de plusieurs éléments dont les interactions forment un réseau complexe. Le corps du document est découpé en trois parties ; un chapitre introductif et une conclusion récapitulative complétent la thèse. La partie I s'intéresse à une dynamique spécifique (propagation de type susceptibleinfectieux- retiré, SIR) sur une classe de réseaux également spécifique (modèle de configuration). Ce problème a entre autres déjà été étudié comme un processus de branchement dans la limite où la taille du système est infinie, fournissant une solution probabiliste pour l'état final de ce processus stochastique. La principale contribution originale de la partie I consiste à modifier ce modèle afin d'introduire des éffets dûs à la taille finie du système et de permettre l'étude de son évolution temporelle (temps discret) tout en préservant la nature probabiliste de la solution. La partie II, contenant les principales contributions originales de cette thèse, s'intéresse aux processus stochastiques sur réseaux complexes en général. L'état du système (incluant la structure d'interaction) est partiellement représenté à l'aide de motifs, et l'évolution temporelle (temps continu) est étudiée à l'aide d'un processus de Markov. Malgré que l'état ne soit que partiellement représenté, des résultats satisfaisants sont souvent possibles. Dans le cas particulier du problème étudié en partie I, les résultats sont exacts. L'approche se révèle très générale, et de simples méthodes d'approximation permettent d'obtenir une solution pour des cas d'une complexité appréciable. La partie III cherche une solution analytique exacte sous forme fermée au modèle développé en partie II pour le problème initialement étudié en partie I. Le système est réexprimé en terme d'opérateurs et différentes relations sont utilisées afinn de tenter de le résoudre. Malgré l'échec de cette entreprise, certains résultats méritent mention, notamment une généralisation de la relation de Sack, un cas particulier de la relation de Zassenhaus.The goal of this thesis is to develop and study mathematical models reproducing the behaviour of systems composed of numerous elements whose interactions make a complex network structure. The body of the document is divided in three parts; an introductory chapter and a recapitulative conclusion complete the thesis. Part I pertains to a specific dynamics (susceptible-infectious-removed propagation, SIR) on a class of networks that is also specific (configuration model). This problem has already been studied, among other ways, as a branching process in the infinite system size limit, providing a probabilistic solution for the final state of this stochastic process. The principal original contribution of part I consists of modifying this model in order to introduce finite-size effects and to allow the study of its (discrete) time evolution while preserving the probabilistic nature of the solution. Part II, containing the principal contributions of this thesis, is interested in the general problem of stochastic processes on complex networks. The state of the system (including the interaction structure) is partially represented through motifs, then the (continuous) time evolution is studied with a Markov process. Although the state is only partially represented, satisfactory results are often possible. In the particular case of the problem studied in part I, the results are exact. The approach turns out to be very general, and simple approximation methods allow one to obtain a solution for cases of considerable complexity. Part III searches for a closed form exact analytical solution to the the model developed in part II for the problem initially studied in part I. The system is re-expressed in terms of operators and different relations are used in an attempt to solve it. Despite the failure of this enterprise, some results deserve mention, notably a generalization of Sack's relationship, a special case of the Zassenhaus relationship

Mueller matrix polarimetry of anisotropic chiral media

[eng] Esta tesis se centra en el estudio de medios quirales mediante polarimetría de matriz de Mueller. Ópticamente los medios quirales se caracterizan por tener actividad óptica, que se manifiesta en procesos dispersivos con la birefringencia circular o en procesos de absorción mediante el cicroismo circular. Nuestro ámbito de estudio han estado los anisótropos no es posible aplicar los métodos convencionales de determinación de la actividad óptica ya que la descripción de la propagación de la luz polarizada se vuelve mucho más compleja ya que el dicroismo y la birefringencia lineales también están presentes. Una parte importante del trabajo han sido el desarrollo teórico necesario para poder obtener los parámetros de dicroismo circular o birefringencia circular a partir de las medidas de la Matriz de Mueller de una muestra anisótropa arbitraria. Otra parte importante del trabajo ha sido la y construcción de un polarímetro de matriz de Mueller de alta resolución basado en el uso de dos moduladores fotoelásticos y es capaz de trabajar en dos modos de funcionamiento: espectroscópico y con resolución espacial. Los desarrollos instrumentales teórico nos han llevado a poder realizar caracterizar muestras de diversa índole. En el ámbito cristalográfico hemos medido espectroscópicamente el tensor de girotropía del cuarzo y hemos mostrado la posibilidad de distinguir dominios quirales en capas delgadas policristalinas. Otro apartado experimental fundamental ha sido la caracterización de procesos de inducción de quiralidad supramolecular mediante efectos hidrodinámicos en soluciones agitadas de nanopartículas orgánicas de formas alargadas

Probabilistic Inference in Piecewise Graphical Models

In many applications of probabilistic inference the models contain piecewise densities that are differentiable except at partition boundaries. For instance, (1) some models may intrinsically have finite support, being constrained to some regions; (2) arbitrary density functions may be approximated by mixtures of piecewise functions such as piecewise polynomials or piecewise exponentials; (3) distributions derived from other distributions (via random variable transformations) may be highly piecewise; (4) in applications of Bayesian inference such as Bayesian discrete classification and preference learning, the likelihood functions may be piecewise; (5) context-specific conditional probability density functions (tree-CPDs) are intrinsically piecewise; (6) influence diagrams (generalizations of Bayesian networks in which along with probabilistic inference, decision making problems are modeled) are in many applications piecewise; (7) in probabilistic programming, conditional statements lead to piecewise models. As we will show, exact inference on piecewise models is not often scalable (if applicable) and the performance of the existing approximate inference techniques on such models is usually quite poor. This thesis fills this gap by presenting scalable and accurate algorithms for inference in piecewise probabilistic graphical models. Our first contribution is to present a variation of Gibbs sampling algorithm that achieves an exponential sampling speedup on a large class of models (including Bayesian models with piecewise likelihood functions). As a second contribution, we show that for a large range of models, the time-consuming Gibbs sampling computations that are traditionally carried out per sample, can be computed symbolically, once and prior to the sampling process. Among many potential applications, the resulting symbolic Gibbs sampler can be used for fully automated reasoning in the presence of deterministic constraints among random variables. As a third contribution, we are motivated by the behavior of Hamiltonian dynamics in optics —in particular, the reflection and refraction of light on the refractive surfaces— to present a new Hamiltonian Monte Carlo method that demonstrates a significantly improved performance on piecewise models. Hopefully, the present work represents a step towards scalable and accurate inference in an important class of probabilistic models that has largely been overlooked in the literature

Developments in the Centroid Phase Space Formulation of Quantum Statistical Mechanics

The centroid formalism provides a phase space representation of quantum statistical mechanics based on the Feynman path integral. Real time quantum correlation functions can be exactly calculated using the centroid formalism, though this requires diagonalizing the system Hamiltonian which is intractable for large collections of molecules. A computational method for computing real time correlation functions called centroid molecular dynamics (CMD) has been formulated to circumvent this issue though the results are approximations. The centroid formalism had previously only been able to treat systems moving in Euclidean space. This is insufficient to capture rotational motion and intramolecular torsions, which may be viewed as motion in a constrained subspace of the Euclidean space. Herein we present a method for incorporating this type of motion into the centroid formalism and test the validity by examining the motion of a particle on a ring. Past work has also seen the centroid formalism extended to pairs of particles obeying Bose-Einstein and Fermi-Dirac statistics by way of a projection operator. In this work we examine the case where this projection operator projects onto an individual quantum state. This will allow the centroid formalism, and hence CMD, to be extended to microcanonical ensembles. Results are shown for the quantum harmonic oscillator, quartic well system and double well system