To gain a deeper insight on the local origin of NTE in zincblende crystals, EXAFS measurements have been performed on CdTe, which has NTE properties intermediate between Ge and CuCl. In this work an accurate evaluation of the bond thermal expansion, parallel and perpendicular MSRDs and distribution asymmetry of the first shell of CdTe has been made, obtaining a good agreement between two different procedures of the data analysis (i) ratio method (ii) FEFF6-FEFFIT method. The values of the relevant parameters of CdTe were found intermediate between the corresponding values previously found for Ge and CuCl. The positive contribution to thermal expansion due to the bond stretching and the negative contribution due to the tension effects are disentangled and quatified in terms of the bond thermal expansion and the perpendicular MSRD, respectively, determined by the EXAFS analysis. A critical comparison of EXAFS and Bragg diffraction results; thermal expansion, thermal factors and correlation of atomic vibration, for the iso-structural crystals Ge, CdTe and CuCl has been performed. The correlation between several quantities measured by EXAFS and NTE properties is confirmed.

In the present thesis two main results are presented. The first is a study of the statistical properties of the finite-time maximum Lyapunov exponent determined out of a time series by using the divergent rate method. To reach this goal, we developed a new, completely automatic algorithm based on the method developed by Gao and Zheng. A main achievement of this part of the work is the interpretation of the uncertainty in the light of the work by Grassberger, Badii e Politi of 1988 on the theoretical distribution of maximum Lyapunov exponents. We showed that the analysis and identification of clusters in diagrams representing uncertainty vs. maximum Lyapunov exponent can provide useful information about the optimal choice of the embedding parameters. In addition, our results allow us to identify systems that can provide suitable benchmarks for the comparison and ranking of different embedding methods. The second main result concerns the development of a new method for the assessment of the optimal embedding parameters. Our method is based on two assumptions: a potential-like quantity is defined on the lattice of points that characterize the embedding; the optimal embedding choice coincides with local extrema (maxima or minima) of this potential. Throughout the work, we used "synthetic" time series generated by numerically integrating the difference and differential equations that describe the following dynamical systems: the Hénon map, the Lorenz attractor, the Rössler attractor and the Mackey-Glass attractor. These four systems are widely used as references in the scientific literature. In the last part of the work, we have started to examine EEG recordings by using the techniques developed in the main part of the work. The EEG recordings are sampled on healthy subjects in resting-state. These investigations are still at a starting phase.

When a second-order phase transition is crossed at finite speed, domains with independent order parameters can appear in the system, with the consequent formation of defects at the domain boundaries. The Kibble-Zurek theory provides a description for this universal phenomenon, which applies to many different systems in nature, and it predicts a power-law dependence of the defect density on the quench rate. This thesis reports on the results of the experimental study of the Kibble-Zurek mechanism in elongated Bose-Einstein condensates of atomic sodium gases, following the observations on the spontaneous formation of defects after temperature quenches across the BEC transition. The power-law scaling of the defect number with the quench speed was observed and characterized for the first time in ultracold gases. The characterization of the density and phase profiles of the defects allowed their identification as solitonic vortices, representing the first direct experimental evidence for this kind of long living excitation, which sets a link between solitons and vortices. The measurements reported in this thesis provide a novel approach to the study of the critical phenomena happening at phase transitions, and introduce to the possibility of exploring the turbulent dynamics of quenched systems through the spontaneous production of solitonic vortices.

The experimental achievement of Bose-Einstein condensation and Fermi degeneracy with ultracold gases boosted tremendous progresses both in theoretical methods and in the development of new experimental tools. Among them, intriguing possibilities have been opened by the implementation of optical lattices: periodic potentials for neutral atoms created by interfering laser beams. Degenerate gases in optical lattices can be forced in highly anisotropic traps, reducing the effective dimensionality of the system. From a fundamental point of view, the behavior of matter waves in reduced dimensions sheds light on the intimate properties of interparticle interactions. Furthermore, such reduced-dimensional systems can be engineered to quantum-simulate fasci- nating solid state systems, like bidimensional crystals, in a clean and controllable environment. Motivated by the exciting perspectives of this field, we devote this Thesis to the theoretical study of two systems where matter waves propagate in reduced dimensions. The long-range and anisotropic character of the dipole-dipole interaction critically affects the behavior of dipolar quantum gases. The continuous experimental progresses in this flourishing field might lead very soon to the creation of degenerate dipolar gases in optical potentials. In the first part of this Thesis, we investigate the emergence of a single dipolar-induced resonance in the two-body scattering process in quasi-one dimensional geometries. We develop a two-channel approach to describe such a resonance in a highly elongated cigar-shaped harmonic trap, which approximates the single site of a quasi-one-dimensional optical lattice. At this stage, we develop a novel atom-dimer extended Bose-Hubbard model for dipolar bosons in this quasi-one-dimensional optical lattice. Hence we investigate the T = 0 phase diagram of the model by exact diagonalization of a small-sized system, highlighting the effects of the dipolar-induced resonance on the many-body behavior in the lattice. In the second part of the Thesis, we present a general scheme to realize cold-atom quantum simulators of bidimensional atomic crystals, based on the possibility to independently trap two different atomic species. The first one constitutes a two-dimensional matter wave which interacts only with the atoms of the second species, deeply trapped around the nodes of a two-dimensional optical lattice. By introducing a general analytic approach, we investigate the matter-wave transport properties. We propose some illustrative applications to both Bravais (square, triangular) and non-Bravais (graphene, kagomeÌ ) lattices, studying both ideal periodic systems and experimental- sized, eventually disordered, ones. The features of the artificial atomic crystal critically depend on the two-body interspecies interaction strength, which is shown to be widely tunable via 0D-2D mixed-dimensional resonances. Keywords: matter waves, reduced dimensions, dipolar-induced resonances, mixed-dimensional resonances, extended Bose-Hubbard model, atomic artificial crystals.

The optomechanics field of research has been gathering a lot of momentum during the last couple of years. Recent experimental results show that the field is finally entering the quantum era. In this context, we have worked to develop new and competitive optomechanical devices. We have also worked towards the generation and observation of ponderomotive squeezing and we have identified, and experimentally demonstrated, an optomechanical effect that can ease the achievement of this goal. Finally, we have developed a stabilization technique that have been instrumental for the success of two experiments: the implementation of the Wiener-Kolmogorov data analysis and the squeezing of a mechanical thermal oscillator.

In this thesis we investigate the properties of mixtures of Bose gases in one dimensions at zero temperature using quantum Monte-Carlo methods. First we investigate the limiting case of an impurity interacting with an atomic bath. We characterize the impurity, by calculating its effective mass, binding energy as well as the contact parameter between the impurity and the bath. In particular, we find that the effective mass rapidly increases to very large values when the impurity gets strongly coupled to an otherwise weakly repulsive bath. Then we describe uniform balanced mixtures with repulsive interactions. We investigate the miscibility phase diagram of the two components and find that correlations do not alter the phase diagram predicted by mean-field theories. We investigate the Andreev-Bashkin effect , a non-dissipative drag between the the two components of the gas and find that the drag becomes very large in the strongly interacting regime. In non-homogeneous systems we also investigate the frequency of the spin-dipole mode. Finally we describe mixtures with attractive inter-species interactions, where one can obtain a liquid ground state because of the competition between the inter-species attraction and intra-species repulsion. We characterize the the liquid and we find that the liquid state can be formed if the ratio of coupling strengths between inter-species attractive and intra-species repulsive interactions exceeds a critical value.

General Relativity is known to suffer from singularities at short distances, which indicates the breakdown of its predictability, for instance at the center of black holes, and in the very early universe. This is one of the main reason to look for a Quantum Theory of Gravity, that would describe spacetime geometry as a quantum field, and possibly cure these classical singularities. However, no consensus on the topic has yet been reached, as many different approaches have been proposed, but none has yet received an experimental confirmation. This is in part due to the extraordinary small scale at which quantum gravitational effects are expected to become dominant, and to the technical difficulty to make unambiguous predictions. For this reason, many works have focused on the so-called effective approaches in which the possible high energy corrections to General Relativity are classified, and their theoretical and ob- servational predictions derived, with the idea that among these modifications, some could come as the semi-classical limits of quantum gravity theories. A way to discriminate between the different proposals is precisely the absence of singular geometries in their solutions. In the first two Chapters of this thesis, we will present such an effective approach, in which the action of General Relativity is modified at high energy by non-polynomial curvature invariants, which are constructed in such a way that the dynamical spherically symmetric sector of these theories (which contain both cosmological and non-rotating black hole spacetimes) yield second order field equations. These properties of the non-polynomial invariants follow from a peculiar algebraic identity satisfied by the Cotton tensor in this class of geometries. As we will see in the last two Chapters, having second order dynamical spherically symmetric field equations is necessary in order to recover some quantum corrected geometries that have been found from more fundamental approaches like Loop Quantum Cosmology and Asymptotic Safety, within its Einstein-Hilbert truncation. The existence of such gravitational models provides an interpretation of two-dimensional Horn- deski theory as describing the dynamical spherically symmetric sector of specific higher dimensional non-polynomial gravity theories. Therefore, it allows to have some concrete d-dimensional formu- lations of the two-dimensional Einstein-Dilaton and Lovelock Designer effective approaches that have been studied extensively, in particular to find and study the properties of non-singular black holes. This enables us to propose two four-dimensional effective-like actions, which are constructed in such a way that their dynamical spherically symmetric sectors decompose in the same way as those of General Relativity and Gauss-Bonnet gravity. In the remaining Chapters, we essentially investigate the solutions and properties of these theories. It is shown that the first one leads to regular (A)dS-core black hole solutions, with the correct quantum correction to their Newton potentials and logarithmic correction to their entropies. The charged generalization is considered, and a way to avoid the mass inflation instability of their inner horizons is found, provided that a bound between the mass and the charge is satisfied. In Chap. 4, we establish a reconstruction procedure able to find theories admitting as solutions the Modesto semi-polymeric black hole, as well as the D’Ambrosio-Rovelli and Visser-Hochberg geometries. All these black holes are regular and derived or inspired by quantum gravity results. They have many properties in common, as for example the fact that they automatically regularize the Coulomb singularity of a static electric field. Finally, the last Chapter is devoted to the theory whose dynamical spherically symmetric sector is a generalization of the one of Gauss-Bonnet gravity. It is shown that the Loop quantum cosmology bounce universe and some Asymptotic Safety black holes can be reconstructed from two members of these theories. In particular, the associated black hole solutions of the first are regular, and the associated cosmological solution of the second is as well, and describe a universe which is eternal in the past, and behaves as de Sitter spacetime in the limit of infinite past. Some generalizations of these results are provided, and the Mimetic gravity formulations of the cosmological solutions are found.

The behavior of surface tension within the random first-order theory (RFOT) of glass transition is studied in a glass-forming liquid model by means of ad-hoc numerical methods. The spinodal point for RFOT excitations turns out to be well defined as a function of the energy of inherent structures (IS), i.e. the minima of potential energy which underlie the equilibrium configurations. The corresponding spinodal temperature, although not sharply defined, lies definitely above the mode coupling one. The role played by surface tension within the context of dynamical heterogeneities is also studied by means of a dynamic algorithm in which the overlap with the initial configuration is constrained along equilibrium dynamics. Indications are found that, in the proximity of the mode coupling temperature, a phase-separation between high and low overlap regions occurs, driven by surface tension. The existence of a positive surface tension between amorphous excitations, in the proximity of the mode-coupling temperature, is therefore observed for both static and dynamic excitations.

In this work we discuss the role of topological degrees of freedom in very low-energy hadronic processes (vacuum polarization and vacuum birefringence). We also present an approach which enables to investigate the microscopic dynamics of non-perturbative processes: this is achieved by constructing an effective statistical theory for topological vacuum gauge configurations, by means of Lattice QCD simulations.

In this thesis I theoretically study the dynamic and ground state properties of ultracold dipolar Fermi gases. The mean-field approach based on the Thomas-Fermi energy functional is applied to consider the dynamic properties of bilayer harmonically trapped dipolar Fermi gases. The fixed-node Diffusion Monte Carlo method (FNDMC) is used instead to investigate the ground-state properties of two dimensional dipolar Fermi gases. This technique is also applied to the problem of one impurity in a bilayer configuration with dipolar fermions.