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.

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.

The recent realization of synthetic spin-orbit coupling represents an outstanding achievement in the physics of ultracold quantum gases. In this thesis we explore the properties of spin-orbit-coupled Bose-Einstein condensates with equal Rashba and Dresselhaus strengths. These systems present a rich phase diagram, which exhibits a tricritical point separating a single-minimum phase, a spin-polarized plane-wave phase, and a stripe phase. In the stripe phase translational invariance is spontaneously broken, in analogy with supersolids. Spin-orbit coupling also strongly affects the dynamics of the system. In particular, the excitation spectrum exhibits intriguing features, including the suppression of the sound velocity, the emergence of a roton minimum in the plane-wave phase, and the appearance of a double gapless band structure in the stripe phase. Finally, we discuss a combined procedure to make the stripes visible and stable, thus allowing for a direct experimental detection.

The thesis is devoted to my Ph.D. research activities during last three years within the CMS collaboration. My primary field of interest was the investigation of the Higgs sector of the Standard Model and in connection with Beyond-Standard-Model New Physics searches.

This work is mainly divided in two parts and deals with some aspects of quantum field theory in curved space-time and some open problems related with the cosmological history of our Universe.

In this thesis, we used CMOS-like technologies to produce improved, hierarchical multifunctional bioinspired surfaces. Different natural surfaces have been surveyed including well-known lotus leaf, sharkskin, back of the Namib Desert beetle, butterfly wings, and legs of water-walking insects. The lotus leaf features superhydrophobicity, which leads to low adhesion and self-cleaning. Sharkskin is composed of ripples that manage to reduce skin-friction and thus drag resistance. The Namib Desert beetle, harvests water from the heterogeneous pattern having hydrophilic/hydrophobic bumps on his back. Butterfly wings have re-entrant structures that manage to reach superhydrophobicity from a hydrophilic substrate. Hairy legs of water-walking insects are superhydrophobic with low adhesion that allows them to fight and jump on water. In chapter 1, we have undertaken a review of bioinspired surfaces that emulate the abilities of such natural surfaces. Then, in chapter 2 we have described the innovative CMOS-like techniques used for generating several hierarchical and re-entrant microstructures. Chapter 3 depicts the analysis of surfaces with hierarchical structures generated with a fast and easy process; this latter forms a second hierarchical level composed of random pyramidal elements using wet etching. Surfaces realized with this process manage to reach remarkably high contact angle and low contact angle hysteresis. Additionally, in this chapter we have introduced an analytical model to study the stability of Cassie-Baxter state over Wenzel state for these hierarchical surfaces. In chapter 4 the fabrication and analysis of surfaces composed of controlled hierarchical levels, which combine sharkskin with single-level lotus leaf-inspired pillared structures are reported. These particular hierarchical surfaces are demonstrated to hold high superhydrophobic properties along with low skin-friction. The superhydrophobicity of these surfaces has been characterized in a series of tests on an inclined plane. The data extrapolated from this measurement was used to evaluate the total dissipated energy of the sliding drop. Combining the data collected during this experiment with contact angle and contact angle hysteresis measurements we propose a global parameter that evaluates the superhydrophobic “level†of a surface. Furthermore, in chapter 5 similar hierarchical surfaces have also been tested for water harvesting together with single-level pillared surfaces that feature heterogeneous chemistry with hydrophilic/hydrophobic spot on every single pillar. In chapter 6 a series of tests have also been performed on butterfly-inspired surfaces. Although the substrate of such surfaces is hydrophilic, thanks to the re-entrant structures the surfaces reach high level of hydrophobicity. An implemented mathematical model and experimental test confirm the stability of this hydrophobic state. In chapter 7, we describe two sets of surfaces inspired by the hairy legs of water walking insect the first is composed of stretchable pyramidal-pillars and the second of truncated-conical silicon pillars. The ability of sharp structures to easily detach from water surfaces is exploited to change the contact angle value of a water drop deposed on this fast type of stretchable micropatterned surface. A mathematical model has been implemented and experimental tests have been carried out to evaluate the stability of the water-air composite interface on both types of microstructured surfaces. In particular, in the polymeric surfaces elasto-capillarity seams to influence the metastability of the Cassie-Baxter state.