Abstract:
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.

Abstract:
We consider a quantum system of non-interacting fermions at temperature T, in the framework of linear response theory. We show that semiclassical theory is an appropriate framework to describe some of their thermodynamic properties, in particular through asymptotic expansions in $\hbar$ (Planck constant) of the dynamical susceptibilities. We show how the closed orbits of the classical motion in phase space manifest themselves in these expansions, in the regime where T is of the order of $\hbar$.

Abstract:
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.

Abstract:
We study the behavior of energy levels in two dimensions for exotic atoms, i.e., when a long-range attractive potential is supplemented by a short-range interaction, and compare the results with these of the one- and three-dimensional cases. The energy shifts are well reproduced by a scattering length formula $ \delta{E}= A_0^2/\ln (a/R)$, where $a$ is the scattering length in the short-range potential, $A_0^2/(2\,\pi)$ the square of the wave function at the origin in the external potential, and $R$ is related to the derivative with respect to the energy of the solution that is regular at large distances.

Abstract:
A presentation and a generalisation are given of the phenomenon of level rearrangement, which occurs when an attractive long-range potential is supplemented by a short-range attractive potential of increasing strength. This problem has been discovered in condensate-matter physics and has also been studied in the physics of exotic atoms. A similar phenomenon occurs in a situation inspired by quantum dots, where a short-range interaction is added to an harmonic confinement.

Abstract:
Let us consider the quantum/versus classical dynamics for Hamiltonians of the form \beq \label{0.1} H\_{g}^{\epsilon} := \frac{P^2}{2}+ \epsilon \frac{Q^2}{2}+ \frac{g^2}{Q^2} \edq where $\epsilon = \pm 1$, $g$ is a real constant. We shall in particular study the Quantum Fidelity between $H\_{g}^{\epsilon}$ and $H\_{0}^{\epsilon}$ defined as \beq \label{0.2} F\_{Q}^{\epsilon}(t,g):= < \exp(-it H\_{0}^{\epsilon})\psi, exp(-itH\_{g}^ {\epsilon})\psi > \edq for some reference state $\psi$ in the domain of the relevant operators. We shall also propose a definition of the Classical Fidelity, already present in the literature (\cite{becave1}, \cite{becave2}, \cite{ec}, \cite{prozni}, \cite{vepro}) and compare it with the behaviour of the Quantum Fidelity, as time evolves, and as the coupling constant $g$ is varied.

Abstract:
In our previous paper \cite{co1} we have shown that the theory of circulant matrices allows to recover the result that there exists $p+1$ Mutually Unbiased Bases in dimension $p$, $p$ being an arbitrary prime number. Two orthonormal bases $\mathcal B, \mathcal B'$ of $\mathbb C^d$ are said mutually unbiased if $\forall b\in \mathcal B, \forall b' \in \mathcal B'$ one has that $$| b\cdot b'| = \frac{1}{\sqrt d}$$ ($b\cdot b'$ hermitian scalar product in $\mathbb C^d$). In this paper we show that the theory of block-circulant matrices with circulant blocks allows to show very simply the known result that if $d=p^n$ ($p$ a prime number, $n$ any integer) there exists $d+1$ mutually Unbiased Bases in $\mathbb C^d$. Our result relies heavily on an idea of Klimov, Munoz, Romero \cite{klimuro}. As a subproduct we recover properties of quadratic Weil sums for $p\ge 3$, which generalizes the fact that in the prime case the quadratic Gauss sums properties follow from our results.

Abstract:
In this paper, we consider the problem of Mutually Unbiased Bases in prime dimension $d$. It is known to provide exactly $d+1$ mutually unbiased bases. We revisit this problem using a class of circulant $d \times d$ matrices. The constructive proof of a set of $d+1$ mutually unbiased bases follows, together with a set of properties of Gauss sums, and of bi-unimodular sequences.

Abstract:
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.

Abstract:
In this paper we perform an exact study of ``Quantum Fidelity'' (also called Loschmidt Echo) for the time-periodic quantum Harmonic Oscillator of Hamiltonian : $$ \hat H\_{g}(t):=\frac{P^2}{2}+ f(t)\frac{Q^2}{2}+\frac{g^2}{Q^2} $$ when compared with the quantum evolution induced by $\hat H\_{0}(t)$ ($g=0$), in the case where $f$ is a $T$-periodic function and $g$ a real constant. The reference (initial) state is taken to be an arbitrary ``generalized coherent state'' in the sense of Perelomov. We show that, starting with a quadratic decrease in time in the neighborhood of $t=0$, this quantum fidelity may recur to its initial value 1 at an infinite sequence of times {$t\_{k}$}. We discuss the result when the classical motion induced by Hamiltonian $\hat H\_{0}(t)$ is assumed to be stable versus unstable. A beautiful relationship between the quantum and the classical fidelity is also demonstrated.