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We uncover a hidden Gaussian ensemble inside each of the three circular ensembles of random matrices, which provide novel diagrammatic rules for the calculation of moments. The matrices involved are generic complex for $\beta=2$, complex symmetric for $\beta=1$ and complex self-dual for $\beta=4$, and their dimension must be set to $1-2/\beta$. As an application, we compute moments of traces of submatrices., Comment: 12 pages, 4 figures

We present a semiclassical approach for time delay statistics in quantum chaotic systems, in the presence of absorption, for broken time-reversal symmetry. We derive three kinds of expressions for Schur-moments of the time delay operator: as a power series in inverse channel number, $1/M$, whose coefficients are rational functions of absorption time, $\tau_a$; as a power series in $\tau_a$, tailored to strong absorption, whose coefficients are rational functions of $M$; as a power series in $1/\tau_a$, tailored to weak absorption, whose coefficients are rational functions of $M$.

We investigate the entanglement in the ground state of systems comprising two and three qubits with random interactions. Since the Hamiltonians also contain deterministic one-body terms, by varying the interaction strength, one can continuously interpolate between deterministic separable eigenstates and fully random entangled eigenstates, with non-trivial intermediate behavior. Entanglement strongly depends on the underlying topology of the interaction among the qubits. For a certain class of interactions GHZ entanglement is favoured by a non-separable collective interaction, while for fully separable pairwise interactions the ground states concentrate in the vicinity of W states.

This is a book for absolute beginners. If you have heard about random matrix theory, commonly denoted RMT, but you do not know what that is, then welcome!, this is the place for you. Our aim is to provide a truly accessible introductory account of RMT for physicists and mathematicians at the beginning of their research career. We tried to write the sort of text we would have loved to read when we were beginning Ph.D. students ourselves. Our book is structured with light and short chapters, and the style is informal. The calculations we found most instructive are spelt out in full. Particular attention is paid to the numerical verification of most analytical results. Our book covers standard material - classical ensembles, orthogonal polynomial techniques, spectral densities and spacings - but also more advanced and modern topics - replica approach and free probability - that are not normally included in elementary accounts on RMT. This book is dedicated to the fond memory of Oriol Bohigas.

Integration of polynomials over the classical groups of unitary, orthogonal and symplectic matrices can be reduced to basic building blocks known as Weingarten functions. We present an elementary derivation of these functions., Comment: v2 fixes some imprecisions

A basic introduction to the $su(1,1)$ algebra is presented, in which we discuss the relation with canonical transformations, the realization in terms of quantized radiation field modes and coherent states. Instead of going into details of these topics, we rather emphasize the existing connections between them. We discuss two parametrizations of the coherent states manifold $SU(1,1)/U(1)$: as the Poincar{��} disk in the complex plane and as the pseudosphere (a sphere in a Minkowskian space), and show that it is a natural phase space for quantum systems with SU(1,1) symmetry., 7 pages, no figures

Orientador: Marcus Aloizio Martinez de Aguiar Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin Resumo: Uma versão generalizada do método 'coupled coherent states' é desenvolvida para estados coerentes associados a grupos de Lie arbitrários. Em contraste com a abordagem original, restrita a funções de base gaussianas, o método estendido é adequado para propagação de estados quânticos de sistemas exibindo propriedades físicas destituídas de análogo clássico, tais como graus de liberdade de spin ou indistinguibilidade de partículas. A formulação para o caso de sistemas com um número fixo de partículas idênticas interagentes é examinada em detalhe, sendo este um caso relevante descrito em termos de estados coerentes do grupo especial unitário. A técnica é ilustrada com aplicações simples, envolvendo modelos de Hubbard bosônicos e fermiônicos. Diversos aspectos da implementação numérica são discutidos Abstract: A generalized version of the coupled coherent states method for coherent states of arbitrary Lie groups is developed. In contrast to the original approach, which is restricted to frozen-Gaussian basis sets, the extended method is suitable for propagating quantum states of systems featuring non-classical physical properties, such as spin degrees of freedom or particle interchange symmetry. The formulation for the relevant case of number-conserving systems of interacting identical particles, most adequately described in terms of coherent states of the special unitary group, is studied in detail. The technique is illustrated with applications to simple Hubbard-like models for both bosons and fermions. Several aspects of the numerical implementation are discussed Doutorado Física Doutor em Ciências FAPESP 2011/20065-4, 2013/00145-9 CNPQ 141338/2011-3 FAEPEX 2065/17

Orientador: Marcus Aloizio Martinez de Aguiar Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin Resumo: Representações no espaço de fase são ferramentas bastante difundidas no estudo e na simulação de sistemas quânticos, principalmente devido aos seus apelos clássicos. Tanto na mecânica quântica quanto na clássica, elementos similares, tal como densidades de probabilidade, podem ser definidos e usados para comparar ambos regimes. Neste trabalho construímos a partir de primeiros princípios uma corrente quântica no espaço de fase na representação de estados coerentes canônicos. Determinamos a corrente quântica para sistemas sob evolução de uma hamiltoniana genérica e mostramos que ela pode ser expandida numa série de potências em $hbar$ cujo termo de ordem mais baixa é a corrente clássica. Calculamos analiticamente a corrente para alguns sistemas uni-dimensionais simples. A corrente quântica apresenta propriedades não-clássicas, por exemplo, inversão de momento e surgimento de novos pontos de estagnação aos pares durante a dinâmica. Mostramos que estes pares são compostos por um ponto de sela, que é um zero da densidade de probabilidade e possui uma carga topológica de -1, e por um vórtice, que possui carga +1. Ambos pontos constituem o que denominamos dipolo topológico. Analisamos o papel destes dipolos no espalhamento de uma partícula por uma barreira gaussiana e mostramos que suas localizações em relação às superfícies de energia clássicas e em relação aos máximos da densidade de probabilidade são assinaturas de tunelamento Abstract: Phase space representations are widely used tools to study and simulate the quantum dynamics of systems, mainly due to its natural classical appeal. In both classical and quantum mechanics, corresponding but not equivalent structures, such as probability densities, can be defined and explored to compare both dynamical regimes. In this work, we constructed from first principles the quantum phase space current for a quantum system in the canonical coherent states representation. We determined the quantum current for systems evolving under a general Hamiltonian, and we showed that the current can be expanded as a power series in $hbar$, whose lowest order term is the classsical current. We also calculated analytically the quantum current for simple one-dimensional systems. The quantum current presents non-classical features, such as momentum inversion and emergence of new stagnation points which appear in pairs during the dynamics. We showed that the pairs are composed by a saddle point, which is a zero of the phase space probability density and bears a topological charge -1, and a vortex, with charge +1. Both points constitute what we named a topological dipole. We analysed the role the dipoles play in the scattering of a particle by a gaussian barrier, and we showed that the location of the dipoles in relation to the classical energy surfaces and the quantum probability density maxima is a fingerprint of quantum tunneling Doutorado Física Doutor em Ciências FAPESP 2013/02248-0 CNPQ 157615/2011-1