About Quantum Revivals, Quantum Fidelity, A semiclassical Approach

In this paper we develop the topics of Quantum Recurrences and of Quantum Fidelity which have attracted great interest in recent years. The return probability is given by the square modulus of the overlap between a given initial wavepacket and the corresponding evolved one; quantum recurrences in time can be observed if this overlap is unity. We provide some conditions under which this is semiclassically achieved taking as initial wavepacket a coherent state located on a closed orbit of the corresponding classical motion. The "quantum fidelity" (or Loschmidt Echo) is the square modulus of the overlap of an evoloved quantum state with the same evoloved by a slightly perturbed Hamiltonian. Its decrease in time measures the sensitivity of Quantum Evolution with respect to small perturbations. It is believed to have significantly different behavior in time when the underlying classical motion is chaotic or regular. Starting with suitable initial quantum states, we develop a semiclassical estimate of this quantum fidelity in the Linear Response framework (appropriate for the small perturbation regime), assuming some ergodicity conditions on the corresponding classical motion

Quadratic Quantum Hamiltonians revisited

Time dependent quadratic Hamiltonians are well known as well in classical mechanics and in quantum mechanics. In particular for them the correspondance between classical and quantum mechanics is exact. But explicit formulas are non trivial (like the Mehler formula). Moreover, a good knowlege of quadratic Hamiltonians is very useful in the study of more general quantum Hamiltonians and associated Schr\"{o}dinger equations in the semiclassical regime. Our goal here is to give our own presentation of this important subject. We put emphasis on computations with Gaussian coherent states. Our main motivation to do that is application concerning revivals and Loschmidt echo

La femme et la transmission dans le Libro Verde de Aragon

En el Libro Verde de Aragon, obra manuscrita del siglo XVI que describe las genealogias de familas aragonesas de origen judio, la mujer ocupa un sitio importante. Participa a la transmision de la memoria familiar y a veces del patrimonio. Pero es en el dominio de la transmision de la "mancha" donde desempeñar un papel primordial. El autor utiliza lo impreciso de su identidad para desacreditar los mas ardientes opositores al poder real.Dans le Libro Verde de Aragon, oeuvre manuscrite du XVIe siècle, décrivant les généalogies de familles aragonaises d'origine juive, la femme occupe une place importante. Elle participe activement à la transmission de la mémoire familiale et parfois du patrimoine. Mais c'est dans le domaine de la transmission de la "tache" qu'elle joue un rôle primordial. L'auteur utilise le flou de son identité pour falsifier certaines généalogies dans le but de discréditer les plus ardents opposants au pouvoir royal

A proof of the Gutzwiller Semiclassical Trace Formula using Coherent States Decomposition

The Gutzwiller semiclassical trace formula links the eigenvalues of the Scrodinger operator ^H with the closed orbits of the corresponding classical mechanical system, associated with the Hamiltonian H, when the Planck constant is small ("semiclassical regime"). Gutzwiller gave a heuristic proof, using the Feynman integral representation for the propagator of ^H. Later on mathematicians gave rigorous proofs of this trace formula, under different settings, using the theory of Fourier Integral Operators and Lagrangian manifolds. Here we want to show how the use of coherent states (or gaussian beams) allows us to give a simple and direct proof.Comment: 17 pages, LaTeX, available on http://qcd.th.u-psud.f

The Mutually Unbiased Bases Revisited

The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in $\mathbb C^d, B {and} B'$ are said mutually unbiased if $\forall b\in B, b'\in B'$ the scalar product $b\cdot b'$ has modulus $d^{-1/2}$. In particular this property has been introduced in order to allow an optimization of the measurement-driven quantum evolution process of any state $\psi \in \mathbb C^d$ when measured in the mutually unbiased bases $B\_{j} {of} \mathbb C^d$. At present it is an open problem to find the maximal umber of mutually Unbiased Bases when $d$ is not a power of a prime number. \noindent In this article, we revisit the problem of finding Mutually Unbiased Bases (MUB's) in any dimension $d$. The method is very elementary, using the simple unitary matrices introduced by Schwinger in 1960, together with their diagonalizations. The Vandermonde matrix based on the $d$-th roots of unity plays a major role. This allows us to show the existence of a set of 3 MUB's in any dimension, to give conditions for existence of more than 3 MUB's for $d$ even or odd number, and to recover the known result of existence of $d+1$ MUB's for $d$ a prime number. Furthermore the construction of these MUB's is very explicit. As a by-product, we recover results about Gauss Sums, known in number theory, but which have apparently not been previously derived from MUB properties.Comment: International Conference on Transport and Spectral Problems in Quantum Mechanics held in Honor of Jean-Michel Combes, Cergy Pontoise : France (2006

Genealogia y politica en el Siglo de Oro: El Libro Verde de Aragon

International audienceLe "Libro Verde de Aragon", recueil généalogique concernant les familles aragonaises les plus en vue de l'Espagne du Siècle d'Or, dénonce l'origine juive de ces lignages. Les trois manuscrits étudiés présentent des prolongations spécifiques à chaque exemplaires et conduisent, soit aux "fueristas" aragonais de 1591, soit aux résistants portugais à la domination hispanique après 1580. Le Livre Vert constituait, en fait, une arme aux mains du pouvoir royal contre ses opposants

A mathematical study of quantum revivals and quantum fidelity

Theoriepas de résum

Phase-space semiclassical analysis - Around semiclassical trace formulae

ThéorieInternational audienc

Cataluña en el Libro Verde de Aragon

Conférence présentée au "III Congrés per a l'estudi dels Jueus en Territoris de Llengua Catalana", Barcelone-Perpignan, 15-18 octobre 2007. Actes non publiésInternational audienceLe Libro Verde de Aragon, recueil généalogique d'importantes familles aragonaises comprenant toutes au moins un ancêtre juif, concerne peu la Catalogne. Toutefois, la présence discrète de cette province permet des manipulations qui ne sont pas sans conséquences. Ainsi, l'amalgame entre Maria Junquers, fille du châtelain de Rosas, et la juive Estenga Conejo permet d'attribuer faussement un sang "impur" à la descendance de don Alonso de Aragon, Grand Maître de l'Ordre de Calatrava

Les conversos aragonais et le Libro Verde de Aragon

On appelle Libro Verde de Aragon un ensemble de manuscrits qui décrivent les généalogies des familles les plus en vue de l'Aragon du Siècle d'Or. Une étude détaillée de ces documents, écrits en principe en 1507 par un inquisiteur voulant démontrer les origines juives de l'oligarchie aragonaise, décèle des intentions tout autres. En effet, l'un des se manuscrits se termine avec les indépendantistes portugais opposants au pouvoir royal au début du XVIIe siècle tandis que les autres mettent en évidence les fueristas aragonais soutien d'Antonio Perez en 1591