Matter at the nanoscale exhibits peculiar properties, often not shown by the bulk counterpart, and strongly coupled to the specific size, shape and structure of the atomic aggregate. Particularly, the enormous surface-to-volume ratio implies boosted reactivity with respect to the environment, while the electronic confinement might cause quantum effects to dominate physical properties. Characterization techniques are of course essential to investigate properties at the atomic scale. Scattering techniques have tremendously evolved in the recent past benefiting from third and fourth generation light sources, producing beams with unprecedented spatial and temporal resolution. In a different realm, atomistic simulations have also greatly evolved deriving advantages from both recent theories and modern computing units. In this framework, a detailed description of the system in a spatial and temporal range compatible with lengths probed by scattering techniques is provided. In a single sentence, the subject of this Thesis is the effort of tying atomistic methods and scattering techniques so to increase the comprehension around size, shape and structure of nanostructured particles.

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.

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 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 AEgIS experiment pursues the ambitious goal of measuring for the first time the gravitational pull on neutral antimatter. The envisioned method consists in producing a beam of cold anti-hydrogen and measuring the deflection of its free fall by means of a Moiré deflectometer. To do so the pulsed production of abundant cold anti-hydrogen is paramount, therefore the charge exchange production mechanism has been elected as the most promising candidate production method. Performing the charge exchange anti-hydrogen production requires access to an abundant source of cold positronium which can be achieved by the employment of oxide-coated nanochanneled silica plates (NCPs). We spend chapter 1 formulating a classical model of positronium production and thermalisation in NCPs and validating it by testing it against the available experimental data. In chapter 2 we describe the measurement of the energy spectrum of positronium produced by nanochanneled plates using the beam produced by the SURF machine. We then compare the measured energy spectra with the model proposed in chapter 1 showing, in the comparison, the indication of a transition during thermalisation process to a regime where quantum phenomena become significant. We describe in detail in chapter 3 several positronium spectroscopy measurements that we performed during the course of the last three years by employing the positron beam line of the experiment AEgIS. We will the proceed to illustrate an improved version of the detrending technique commonly employed in signal analysis which, applied to the analysis of SSPALS spectra, improves the achievable precision on the experimental results. In chapter 4 we describe an innovative approach that we are currently pursuing to employ the detector FACT, part of the AEgIS apparatus, to confirm the successful production of anti-hydrogen.

Disorder is an inherent component of real materials, with significant implications for their application in functional devices. Despite this, the theoretical modelling of disorder remains restricted, primarily due to the large simulation cells required to adequately represent disordered systems, and the associated computational costs. This has been remedied in part by the increased availability of resources for high performance computing. In this thesis, using a combination of computational techniques, primarily density functional theory and ab initio as well as classical molecular dynamics, we investigate disorder in two broad categories – physical and chemical disorder, in three distinct classes of materials: palladium nanoparticles, the negative thermal expansion cuprite Ag2O and the complex quaternary chalcogenide Cu2ZnSnS4, known commonly as kesterite. The ‘physical’ disorder discussed in the thesis constitutes shape- and adsorption-induced mechanical softening on the surface of Palladium nanocrystals used for nanocatalysis. This includes one study on the the adsorption of organic capping agents, and another on the adsorption of oxygen molecules and the subsequent oxidation of Pd. In the former, it was observed that the strain effect due to adsorption-induced surface disorder is significantly greater than that due to variations in surface termination, i.e. nanoparticle shape. Moreover in the latter case, different crystallographic facets with different degrees of disorder were found to affect the spin-flip induced activation of oxygen atoms, relevant to the catalytic oxygen reduction reaction in hydrogen fuel cells. In each case, the computational results were combined with a sophisticated, phenomenological whole powder pattern modeling of X-ray diffraction data primarily from synchrotron radiation, leading to an accurate characterization of the Debye-Waller coefficient, which was established as a reliable metric for disorder in crystalline systems. In the case of Ag2O instead, we demonstrated that the large experimental Debye-Waller coefficient was due to thermal diffuse scattering arising from the strong distortion of the Ag4O coordination tetrahedra. The second form of disorder which was investigated is ‘chemical’ disorder, which refers to cation disorder in the quaternary chalcogenide Cu2ZnSnS4 studied for its performance as a thermoelectric material. Similar to the studies on palladium, the disorder was quantified through the Debye-Waller coefficient using molecular dynamics simulations, this time from ab initio methods, and compared with X-ray diffraction data from a synchrotron source. The ordered phase of CZTS is known to crystallize in a tetragonal phase, with alternating Cu-Zn and Cu-Sn cation layers sandwiched between sulfur layers. Two forms of cation disorder were studied: disorder only in the Cu-Zn layer, leading to a disordered tetragonal phase, and full cation site randomization, leading to a disordered cubic polymorph. In the former case, it was found that the higher symmetry of the disordered tetragonal structure led to an average symmetrization of the nearest neighborhood of each individual cation, as a result of which there was a convergence of bands at the valence band maximum, leading to an experimentally observed increase in p-type carrier concentration. In the case of CZTS with full cation disorder, inhomogenous bond led to favorable modifications of the electronic and phonon properties, allowing for a simultaneous improvement of the experimentally measured electrical and thermal conductivities as well as the Seebeck coefficients. Finally, by studying the atypical electronic band structure of this cubic polymorph, we were able to identify topologically non-trivial behavior evidence of bulk band inversion, robust surface states, and an adiabatically continuous connection to a known TI phase. As such, we were able predict disordered cubic CZTS to be the first disorder-induced topological Anderson insulator in a real material system.