We derive a lower bound for the Wehrl entropy in the setting of SU(1,1). For asymptotically high values of the quantum number k, this bound coincides with the analogue of the Lieb-Wehrl conjecture for SU(1,1) coherent states. The bound on the entropy is proved via a sharp norm bound. The norm bound is deduced by using an interesting identity for Fisher information of SU(1,1) coherent state transforms on the hyperbolic plane and a new family of sharp Sobolev inequalities on the hyperbolic plane. To prove the sharpness of our Sobolev inequality, we need to first prove a uniqueness theorem for solutions of a semi-linear Poisson equation (which is actually the Euler-Lagrange equation for the variational problem associated with our sharp Sobolev inequality) on the hyperbolic plane. Uniqueness theorems proved for similar semi-linear equations in the past do not apply here and the new features of our proof are of independent interest, as are some of the consequences we derive from the new family of Sobolev inequalities. We also prove Fisher information identities for the groups SU(n,1) and SU(n,n).

O objetivo desta Tese foi a aplicação da teoria dos estados coerentes para sistemas quânticos não-triviais. A partir da definição de estados coerentes para grupos de Lie compactos em geral, nos dedicamos a uma investigação detalhada da construção de tais estados e de suas propriedades no caso do grupo simplético unitário Sp(4), que é extremamente importante tanto em mecânica quântica quanto em mecânica clássica. Esse grupo possui uma complexidade intermediária, que permite um tratamento analítico ainda que apresente propriedades não-triviais do ponto de vista de teoria de representação de álgebras de Lie. Os estados coerentes obtidos nos permitiram uma investigação do limite clássico para sistemas com simetria Sp(4) e uma conexão com a teoria do caos em mecânica quântica. Além disso, tratamos uma proposta recente de generalização do conceito de estados coerentes para sistemas de espectro discreto não-degenerado, os estados de Gazeau-Klauder. Esses estados foram aplicados a um problema de magnetização bidimensional e também ao potencial unidimensional de mínimos duplos, onde observamos o aparecimento dos estados chamados \"Gatos de Schrödinger\", que consistem na superposição de dois estados de mínima incerteza. / The subject of the Thesis was the aplication of the coherent states theory to non-trivial quantum systems. Starting from the general definition of coherent states for compact Lie groups, we made a detailed investigation of the construction of these states and its properties in the case of the unitary symplectic group Sp(4), which is extremely important in both quantum and classical mechanics. This group has an intermediate complexity, allowing an analytic treatment while presenting non-trivial properties from the point of view of represention theory of Lie algebras. The coherent states so obtained allowed us an investigation of the classical limit of systems with Sp(4) symmetry and a conection with the theory of chaos in quantum mechanics. Besides that, we have treated a recent generalization of the concept of coherent states for systems with discrete and nondegenerate spectrum, the Gazeau-Klauder states. These states were applied to a twodimensional magnetization problem and also to the onedimensional double-well potential, where we have observed the appearence of the so-called \"Schrödinger cats\", which consist in the superposition of two minimum-uncertainty states.

Nesta tese, obtemos estados quânticos que satisfazem a equação de Schrödinger, para Hamiltonianos quadráticos de forma geral e, ao mesmo tempo, permitem de maneira natural obter a correspondência com a descrição clássica. Usamos o método de integrais de movimento para construir operadores de criação e aniquilação, que satisfazem a álgebra de Weyl-Heisenberg. Dessa forma, construímos os estados de número generalizados (ENG) de maneira análoga ao que é feito para os estados de Fock. Obtemos diferentes famílias de estados coerentes (EC), através de uma superposição dos ENG, que chamamos de estados coerentes generalizados (ECG). Esses estados são rotulados pela constante complexa z escrita em termos do valor esperado inicial da coordenada e do momento. Escrevemos os ECG em função do desvio padrão inicial na coordenada, $\\sigma_q$, de modo a minimizar a relação de incerteza de Heisenberg no instante de tempo inicial. Obtemos, de forma pioneira, os ECG para partícula livre e discutimos em detalhes suas propriedades, tal como a relação de completeza, a minimização das relações de incerteza e a evolução da correspondente densidade de probabilidade. Mostramos que o valor esperado da coordenada e do momento segue ao longo da trajetória clássica no espaço de fase. Mostramos que, quando o comprimento de onda da partícula livre é muito menor que $\\sigma_q$, os EC se comportam como estados semiclássicos. Além da partícula livre, construímos pela primeira vez, os ECG para o oscilador invertido e discutimos em detalhes suas propriedades. Mostramos que os ECG de sistemas diferentes podem ser relacionados, impondo condições sobre os parâmetros do Hamiltoniano. Por fim, consideramos Hamiltonianos dependentes do tempo, em particular, construímos os ECG, de forma exata, para um oscilador harmônico cuja frequência varia explicitamente no tempo. Mostramos ainda modelos úteis para obter solução exata de sistemas dependentes do tempo, fazendo analogia com a equação de spin ou equação de Schrödinger unidimensional independente do tempo. Além disso, desenvolvemos um método próprio, que fixa a solução e em seguida determinamos a forma da frequência. / In this thesis we obtain quantum states that satisfy the Schrödinger equation for quadratic Hamiltonians in the general form and at the same time allow, naturally, to obtain the correspondence with the classical description. For this, we use the method of integrals of motion to construct creation and annihilation operators, which satisfy the algebra of Weyl-Heisenberg. Thus, we obtain the generalized number states (GNS) in the same way that is done for the Fock states. We obtain diferent families of coherent states (CS) that we call generalized CS (GCS), by a superposition of GNS. These states are labeled by a complex constant z which is written in terms of the initial expected values of the coordinate and momentum. We write the GCS in terms of the initial standard deviation of the coordinate, $\\sigma_q$, which provides the minimization of Heisenberg uncertainty relation at the initial instant time. In particular, we obtain for the first time the GCS for the free particle and discuss in detail their properties, such as the completeness relation, the minimization of uncertainty relations, and the evolution of the corresponding probability density. We show that the expected values of coordinated and momentum propagate along the classical trajectory in phas espace. When the Compton wavelength is much smaller than $\\sigma_q$, the CS can be considered a semiclassical state. In addition to the free particle, we obtain for the first time the GCS for the inverted oscillator and discuss in detail their properties. We show that the GCS of diferent systems can be related by imposing conditions on the parameters of the Hamiltonian. Finally, we consider the time-dependent Hamiltonian, especially to obtain the GCS for a harmonic oscillator whose frequency varies explicitly in time. We also show useful models to obtain exact solution for time-dependent systems, by analogy with the spin equation or one-dimensionaltime-independent Schrödinger equation, as well as a method which consists first to find the solution and then determine the shape of the frequency.

La propagation des états cohérents est un sujet actif de la recherche de ces dernières décennies. Un état cohérent est la modélisation quantique d'une particule classique. La trajectoire d'une particule chargée dans un champ magnétique fort, i.e. qui ne s'annule pas, est donnée par le mouvement centre guide : il s'agit d'oscillations rapides autour d'un centre virtuel dont la dynamique plus lente suit les lignes de champ magnétique. Le sujet de ma thèse est d'étudier le pendant magnétique de ce résultat. Il est connu que pour des temps courts, c'est à dire bornés en le paramètre semi-classique h, la dynamique d'un état cohérent est donnée par la dynamique classique de la particule associée. Le problème est alors l'étude de la dynamique en temps longs de tels états. Pour des états cohérents de basses énergies, en utilisant des résultats de forme normale et de régularité elliptique magnétique, on prouve le résultat suivant de propagation magnétique en temps longs. Un état cohérent sous un champs magnétique se propage pour des temps de l'ordre de l'inverse de h en une somme d'états cohérents évoluant sur une même trajectoire, mais à des vitesses différentes. / The propagation of coherent states was extensively studied over past decades. A coherent state is the quantum version of a classical particle. The trajectory of a charged particle under strong magnetic field, that is a non-vanishing field, is given by the center guide motion : it is high speeds oscillations around a virtual center that moves slowly along the field lines of the magnetic field. The aim of this thesis is the study of the quantum translation of this. It is well-known that for short times, i.e. bounded times with respect to the semiclassical parameter h, the dynamics of such a state is given by the classical trajectory of the associated particle. The thing is to study the dynamics of coherent states for long times, that are times going to infinite as h goes to 0. For low energy coherent states, using normal form and magnetic elliptical regularity results, we prove the following long time magnetic propagation result. A coherent state under strong magnetic field is propagated as a sum of coherent states following the same dynamics, but with different speeds.

Orientador: Marcus Aloizio Martinez de Aguiar / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-09-02T17:51:39Z (GMT). No. of bitstreams: 1 Grigolo_Adriano_D.pdf: 6040981 bytes, checksum: 874a21109be77dc3f306ac9ce74f90b5 (MD5) Previous issue date: 2017 / 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 / 2011/20065-4 / 141338/2011-3 / FAPESP / CNPQ

A eletrodinâmica clássica estocástica pode ser entendida como sendo a teoria clássica de Maxwell, onde se inclui um novo elemento da realidade física: As flutuações eletromagnéticas de ponto zero. Sob esse enfoque, estudamos a interação de um \"ensemble\", de osciladores harmônicos carregados com a radiação térmica e de ponto zero (atérmica). Incluímos os efeitos de dissipação através da força de reação da radiação. Além disso estudamos também a excitação do oscilador por uma força determinística com dependência temporal arbitrária. Nossa análise estatística do sistema físico é baseada na solução exata da equação de Fokker-Planck adequada ao problema. Obtém-se a evolução temporal, no espaço de fase, para uma dada distribuição inicial que caracteriza um \"ensemble\" de osciladores forçados que apresentam estados excitados na forma de estados coerentes e estados coerentes comprimidos e pulsantes. A comparação direta com a formulação quântica do mesmo problema nos faz reconhecer que é possível obter da física clássica alguns resultados antes só obtidos pela teoria quântica. Identificamos na radiação de ponto zero o ingrediente que torna possível entender a estabilidade do estado fundamental e o princípio de incerteza. / Classical stochastic electrodynamics may be understood as classical electrodynamics theory, when a new element of physical reality is included: The zero point electrodynamics fluctuations. Under this approach, we study the interaction of a charged harmonic oscillator with the thermal radiation and zero point radiation. We include the effect of dissipation by the radiation reaction force. We also study the excitation of this oscillator by a deterministic force with arbitrary temporal dependence. Our statistical analysis of the physical system is based on the exact solution of the appropriate Fokker-Planck equation. We get the temporal evolution in phase space for a given initial distribution that characterizes one \"ensemble\" of forced oscillators that presents excited states in the coherent and squeezed states form. A direct comparison with the quantum formulation of the same problem make us recognize that it is possible to get some results from classical physics which were accomplished previously only by quantum theory. We identify in the zero point radiation the ingredient that makes it possible to understand the stability of the fundamental state and the uncertainty principle.

An J=1→J=1 atomic system in Samarium with incident laser resonant radiation has been investigated. A linearly polarized laser at 686nm excites atoms from the level 4ƒ[superscript]66s[superscript]2 [superscript]7F[subscript]1 to a excited level 4ƒ[superscript]66s6p [superscript]9F[subscript]1 via the process of optical pumping. When an external magnetic field is applied to the atom-laser interaction and the decay fluorescence collected, a level-crossing profile appears. Theoretical predictions of the level-crossing profile can be made using spherically irreducible tensors to describe the density matrix which take advantage of the symmetry of the atomic system. By comparing theory with experimental data, a discussion is made of the various parameters and external factors that can affect this system, which show that Doppler broadening is the major influence. An additional investigation is made into the evolution of the J=1→J=1 atomic system with increasing laser exposure. Comparisons of the experimental data with theoretical predictions are made by analyzing the FWHM of the overall level-crossing profile, the FWHM of the dip about B = 0 and the relative depth of the dip. By charting the progress of these parameters with increasing laser exposure, it can be seen that the theory and experimental data agree qualitatively. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1320338 / Thesis (Ph.D.) -- University of Adelaide, School of Chemistry and Physics, 2008

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The study of coherent states (CS) for a quantum mechanical system has received a lot of attention. The definition, applications, generalizations of such states have been the subject of work by researchers. A common starting point of all these approaches is the observation of properties of the original CS for the harmonic oscillator. It is well-known that they are described equivalently as (a) eigenstates of the usual annihilation operator, (b) from a displacement operator acting on a fundamental state and (c) as minimum uncertainty states. What we observe in the different generalizations proposed is that the preceding definitions are no longer equivalent and only some of the properties of the harmonic oscillator CS are preserved. In this thesis we propose to study a new class of coherent states and its properties. We note that in one example our CS coincide with the ones proposed by Glauber where a set of three requirements for such states has been imposed. The set of our generalized coherent states remains invariant under the corresponding time evolution and this property is called temporal stability. Secondly, there is no state which is orthogonal to all coherent states (the coherent states form a total set). The third property is that we get all coherent states by acting on one of these states ['fiducial vector'] with operators. They are highly non-classical states, in the sense that in general, their Bargmann functions have zeros which are related to negative regions of their Wigner functions. Examples of these coherent states with Bargmann function that involve the Gamma and also the Riemann ⲝ functions are represented. The zeros of these Bargmann functions and the paths of the zeros during time evolution are also studied.

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (leaves [72]-80).