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Holographic aspects of 2D Fresnel type memory patterns with 3D couplings and mutual diffractions are considered. Encoding topological charges and spin, the nonlinear interaction potentials are assigned to a shared holographic memory phase space with limited information storage density. Neutral intermediate structures arise in the overlap region. The neutralizing process is regarded as a generator of anti-charge around the topological monopole singularity and maps Coulomb/Kepler to oscillator potentials. The twistorial overlap pattern projects a quasiparticle into the orthogonal holographic extra dimension while providing for a massless and stereographic coupling (Gudermannian function). Basic interactions are: local strong between overlapping topological charges, weak by neutral intermediate quasiparticle overlap patterns, topological charge stereographic long-range interaction in 2D, and a weak long-range higher-dimensional interaction. It is conjectured that the mediating neutral patterns responsible for interaction and energy shifts behave like neutrons, neutrinos, photons, and quasiparticles in 2D crystals (graphene). A cluster of Z = 4 partially overlapping topological charge memory patterns could explain the exceptional stability of Beryllium with 5 neutrons. Forcing special phase conditions this principle could act as a prototype for holographic energy releases/drives.

In this paper we show that the Berry phase can always be cancelled by a unitary transformation if the space of external parameters is appropriately restricted, but it simply reappears as part of the dynamical phase. Moreover the overall extra geometrical phase measured in experiments is unambiguous and retains its geometrical character.

In this paper we introduce some mathematical tools to further study the classical adiabatic holo- nomy effect known as Hannay's angle. In particular, we prove with purely classical methods that the area (or angle) two-form associated with this effect can be seen as a modification of the symplectic structure of the slow-variable dynamics. We also show that, as in the quantum case, degeneracies cause singularities in this two-form. We conclude with some considerations concerning the triviality or nontriviality of the phase-space bundle associated with this phenomenon.

The ``geometrical phase factor,'' found by Berry in his study of the quantum adiabatic theorem, has a classical analogue discovered recently by Hannay and Berry himself. This classical analogue consists of an extra angle swept by the system in its classical evolution during an adiabatic excursion. As in Berry's phase, Hannay's angle reflects a nontrivial holonomy of the system and it has a purely geometric character. In this paper we calculate Hannay's angles for the classical Grassmannian system associated to a quantum spin in a magnetic field. We also show their close relation to the corresponding quantum-mechanical phase of Berry for the same system.

In this paper, we show that it is possible to manipulate the many-body wave function of an isolated dot with a few electrons by locally applying magnetic and electric fields. We polarize the dot at a level crossing, where the sensitivity is at its maximum. Time-dependent fields produce a superposition of the states involved in the avoided crossing. In the case of N = 2 and N =3 electrons, the results of exact di-agonalisation give information about the nature of these states and allow us to construct an effective Hamiltonian describing the coupling. The formalism for evaluating the Berry phase arises naturally. We argue that a quantum dot, capacitively coupled to a quantum point contact, can influence its conductance. The quantum superposition of the states produced by cycling the fields on the dot can be measured this way.

The first part of this article attempts to highlight in chronological order some prominent papers that have direct connexion with the Born-Oppenheimer approximation. This timeline successively points to the role of level crossing, the notions of gauge field and geometrical or Berry phase, early experiments that were accounted for with that concept, and its major role in understanding anomalous transport in solids. The second part analyzes the time evolution of states under the adiabatic approximation when the states are spin degenerate. The geometrical phase thus found entails an electron spin evolution. Then, we examine the case of moleclar dynamics. Starting from the Born-Oppenheimer approximation, we obtain a spin evolution by replacing the classical nuclear positions by their quantum mechanical counterparts in the terms that describe electrostatic interactions between nuclei and electrons.

This thesis deals with the consequences of periodic structures in quantum mechanics in different semiclassical regimes. The first chapter introduces the Floquet theory. It is applied on the Hill equation. As a special case, the solutions to the Mathieu equation are studied exemplifying the properties of the spectrum of periodic Schrödinger operators. Special attention is brought to the properties of the spectrum and the corresponding Floquet exponent of solutions. We present new results concerning the relation between the eigenvalues and the Floquet exponent. We introduce the Weyl quantization in the second chapter. In particular, we introduce semiclassical propagation results which, up to some known errors, describes the propagation of an initial Gaußian under quantum evolution (Combescure, Robert 1997). This approximation breaks down at what is commonly known as the Ehrenfest time. We use these results to investigate the localization properties of an initial Gaußian. We can show localization of the state up to the Ehrenfest time under the assumption that the flow differential of the corresponding classical motion admits Floquet solutions with purely imaginary Floquet exponents. In this case, we also show the existence of what we call classical revivals. The last part is a study of the spectral properties of perturbed periodic Schrödinger operators. This is done in the limit of weak perturbation epsilon . As the structure of the band spectrum of the unperturbed operator remains unchanged under small enough perturbations, the effective dynamics can be described adiabatically under certain prerequisites. We consider the construction of WKB ansätze (Littlejohn, Flynn, 1991, Emmrich, Weinstein 1996). Our results include Bohr-Sommerfeld quantization conditions modulo O(epsilon²). This is made possible by including geometric terms. The Wannier-Stark system is studied as an example.