<p>The curvatures in Finsler geometry can be defined in similar ways as in Riemannian geometry. However, since there are fewer restrictions on the metrics, many geometric quantities arise in Finsler geometry which vanish in the Riemannian case. These quantities are generally known as non-Riemannian quantities and interact with the curvatures in controlling the global geometrical and topological properties of Finsler manifolds. In the present work, we study general weighted Ricci curvatures which combine the Ricci curvature and the S-curvature, and define a weighted flag curvature which combines the flag curvature and the T -curvature. We characterize Randers metrics of almost isotropic weighted Ricci curvatures and show the general weighted Ricci curvatures can be divided into three types. On the other hand, we show that a proper open forward complete Finsler manifold with positive weighted flag curvature is necessarily diffeomorphic to the Euclidean space, generalizing the Gromoll-Meyer theorem in Riemannian geometry.</p>

Bose-Einstein condensates (BECs) of ultra-cold atoms have been subjects of a large research effort, that started a century ago as a purely theoretical subject and is now, since the invention of evaporative cooling thirty years ago, a rich research topic with many experimental apparatuses around the world. A deep knowledge of its underlying physics has been now acquired, for example on the thermodynamics of the gas, superfluidity, topological excitations and many-body physics. However, many topics are still open for investigation, thanks to the flexibility and the high degree of control of these systems. During the course of my PhD, I developed and realized a new experimental apparatus for the realization of coherently-coupled mixtures of sodium BECs. The highly stable and low-noise magnetic environment of this apparatus enables the experimental investigation of a previously inaccessible regime, where the energy of the coupling becomes comparable to the energy of spin excitations of the mixture. With this apparatus, I concluded two experimental investigations: I produced and investigated non-dispersive spin-waves in an two-component BEC and I experimentally observed the quantum spin-torque effect on a elongated bosonic Josephson junction.The research activity in multi-component BECs of alkali atoms begun shortly after the first realization of a condensate, thanks to the low energy splitting between the internal sub-states of the electronic ground state. These internal states can be coherently coupled with an external electromagnetic field and can interact via mutual mean-field interaction, generating interestinc effects such as ground states with different magnetic ordering depending on their interaction constants, density as well as spin dynamics and internal Josephson effects. The research interest on mixtures of sodium atoms sparks from the peculiar characteristic of the system: in the $ket{F = 1, m_F = pm 1}$ states, the interaction constants are such that the ground state has anti-ferromagnetic ordering and the system is perfectly symmetric for exchanges of the two species. In these peculiar system, density- and spin-excitations have very different energetic cost, with the latter being much less energetic, and can be completely decoupled. Moreover, spin-excitations, that are connected to excitations in the relative-phase between the components, change drastically in nature when a coupling of comparable energy is added between the states. The presence of the coupling effectively locks the relative-phase in the bulk and spin excitations become localized. While extensive theoretical predictions on the spin dynamics of this system has been already performed, experimental confirmation was still lacking because of the high sensitivity to external forces (due to the very low energy of the spin excitations) and the impossibility of realizing a low-energy coupling between these states in the presence of environmental magnetic noise. During my PhD, I realized an experimental apparatus where magnetic noises are suppressed by five orders of magnitude using a multi-layer magnetic shield made of an high-permeability metal alloy (μ-metal), that encases the science chamber. In this apparatus, I developed a protocol, compatible with the technical limitations of the magnetic shield, to produce BECs in a spin-insensitive optical trapping potential. I then characterized the residual magnetic noise and found it compatible with the requirements for observing spin-dynamics effects. Finally, I realized a system and a set of protocols for the manipulation of the internal state of the sample allowing arbitrary preparation of the sample while maintaining the long coherence times necessary to observe the spin dynamics, that have been used in the subsequent experimental observations. The first experimental result discussed in this thesis, is the production of magnetic solitons and the observation of their dynamic in a trapped sample. Waves in general spread during their propagation in a medium, however this tendency can be counterbalanced by a self-focusing effect if dispersion of the wave is non-linear, generating non-dispersive and long-lived wavepackets commonly named solitons. These have been found in many fields of physics, such as fluid dynamics, plasma physics, non-linear optics and cold-atoms BECs, attracting interest because of their ability to transport information or energy unaltered over long distances, as they are robust against the interaction with in-homogeneities in the medium. Of these systems, cold-atoms can be widely manipulated to generated different kinds of solitons, both in single and in multi-components systems. A new kind of them, named magnetic solitons, has been predicted in a balanced mixture of BECs of sodium in $ket{F = 1, m_F = pm 1}$, however experimental observation was still lacking. I deterministically produced magnetic solitons via phase engineering of the condensate using a spin-sensitive optical potential. I then developed a tomographic imaging technique to semi-concurrently measure the densities of both components and the discontinuities in their relative phase, allowing for the reconstruction of all the relevant quantities of the spinor wavefunction. This allowed to observe the dispersionless dynamics of the solitons as they perform multiple oscillation in the trapped sample in a timescale of the order of the second. Moreover, I engineered collisions between different kinds of magnetic solitons and observed their robustness to mutual interaction. The second experimental results presented in this thesis is the observation of the breaking of magnetic hetero-structures in BECs due to the quantum spin torque effect, an effect with strong analogies with electronic spins traveling through magnetic devices. Spins in magnetic material precess around the axis of the effective magnetic field, and their dynamics must take into account the external field as well as non-linear magnetization and the inhomogeneity of the material. These effects are commonly described by the Landau-Lifshitz equation and have been mainly studied for electronic spins in magnetic hetero-structures, where the inhomogeneity in the material at the interfaces enhances the exchange effects between spins. For homogeneous materials, this description reduces to the Josephson system, a closely related effect that is more known in cold-atoms systems. The Josephson effect arises when a macroscopic number of interacting bosonic particles are distributed in two possible states, weakly tunnel-coupled together, with the average energy of particles occupying each of the states depending on the occupation number itself. In these conditions, the dynamics of the system depends on the difference in occupation numbers, the relative phase between the states and the self-interaction to tunneling ratio, giving raise to macroscopic quantum effects such as oscillating AC and DC Josephson currents and self-trapping. While these phenomena has been historically studied in junctions between superconducting systems, they can be also realized with cold-atoms systems, allowing the study of Josephson junctions with finite dimensions and in regimes that are hard to reach for superconducting systems. In this thesis, I realized a magnetic hetero-structure in a two-component elongated BECs thanks to the simultaneous presence of self-trapped (ferromagnetic) and oscillating (paramagnetic) regions in the sample. While the dynamics at short times is correctly described by the Josephson effects, at the interface between the regions the particle nature of the gas creates a strong exchange effect, named the quantum spin torque, that produces magnetic excitations that spread trough the sample and break the local Josephson behaviour. I experimentally studied the spread and nature of these magnetic excitations, while numerical simulations confirmed the dominant role played by the quantum spin torque effect. The structure of this thesis is the following: in the first chapter is given a review of theoretical concepts and existing literature. In the second chapter is described the experimental apparatus and the protocols developed to prepare the ultra-cold atoms sample. In the third chapter is presented the experimental observation of magnetic solitons. In the fourth chapter is presented the experimental investigation of the quantum spin torque effect in magnetic heterostructures. The last chapter is devoted to conclusions and outlook of this work.

Cosmology is an attempt to mathematically describe the behaviour of the universe, the most commonly used models are the Friedmann-Lemaître-Robertson-Walker solutions. These models seem to be accurate for an old universe, which is homogeneous with low anisotropy. However for an earlier universe these models might not be that accurate or even correct. The almost non-existent anisotropy observed today might have played a bigger role in the earlier universe. For this reason we will study another model known as Bianchi Type I, where the universe is not necessarily isotropic. We utilize a 1+3-covariant formalism to obtain the equations that determine the behaviour of the universe and then use a tetrad formalism to complement the 1+3-covariant equations. Using these equations we examine the geometry of space-time and its dynamical properties. Finally we briefly discuss the different singularities possible and examine some special cases of geodesic movement.

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<p>The subject of this work is the theoretical analysis of the mean-field and many-body properties of an impurity in a Bose-Einstein condensate in a double well potential. By investigating the stationary mean-field properties we show that a critical value of the boson-impurity interaction energy, W_c, corresponds to a pitchfork bifurcation in the number difference variable in the mean-field theory. Comparing W_c to the value of W where the many-body ground state wave function begins to split shows a direct correlation signaling a connection between the many-body and mean-field theories. Investigation of the mean-field dynamics shows that chaos emerges for W > W_c in the vicinity of an unstable equilibrium point generated by the pitchfork bifurcation. An entropy is defined to quantify the chaos and compared to the entanglement entropy between the BEC and the impurity. The mean-field entropy shows a large gradient at W_c whereas the entanglement entropy shows no apparent features around the same value of W. An increase in correlations between nearest neighbour many-body eigenvalues is seen as W is increased providing evidence for ``quantum chaos''.</p> / Master of Science (MSc)

Quantum degenerate molecular gases are of interest for the unique level of control they offer over chemical interactions and processes. To reach the quantum degenerate regime, these molecular gases must be cooled to ultracold temperatures, typically on the order of 100 nanoKelvins. Unlike atoms, with a few-level system that facilitates cooling, molecules represent a many-level system, which makes these temperatures experimentally difficult to achieve. As a result, experiments have turned to photoassociation and Feshbach resonance as shortcuts to form ultracold molecules from already ultracold atoms. Photoassociation and Feshbach resonance have been utilized to successfully create stable quantum degenerate molecules, but not on a routine basis, and only for a small range of molecular species. The primary focus of this thesis will be to study photoassociation and Feshbach resonance, and investigate possible routes to more efficient long-lived quantum degenerate molecule formation. We will also investigate realistic limiting conditions to open the possibility to more routine molecules, and to molecular species that are currently inaccessible. Overall, we find combined photoassociation and Feshbach resonance are viable schemes for efficiently creating quantum degenerate molecules, under realistic restrictions such as low laser intensity, narrow Feshbach resonance, and strong elastic collisions. As the techniques to create quantum degenerate molecules become more robust and experimentally available, the creation of colder, larger, and more long-lived samples will facilitate study of these molecules, and spur development into new applications. / Physics

No / The purpose of this paper is to highlight the importance of constraints in the theory of relativity and, in particular, what philosophical work they do for Einstein's views on the laws of physics. Einstein presents a view of local ``structure laws'' which he characterizes as the most appropriate form of physical laws. Einstein was committed to a view of science, which presents a synthesis between rational and empirical elements as its hallmark. If scientific constructs are free inventions of the human mind, as Einstein, held, the question arises how such rational constructs, including the symbolic formulation of the laws of physics, can represent physical reality. Representation in turn raises the question of realism. Einstein uses a number of constraints in the theory of relativity to show that by imposing constraints on the rational elements a certain ``fit'' between theory and reality can be achieved. Fit is to be understood as satisfaction of constraint. His emphasis on reference frames in the STR and more general coordinate systems in the GTR, as well as his emphasis on the symmetries of the theory of relativity suggests that Einstein's realism is akin to a certain form of structural realism. His version of structural realism follows from the theory of relativity and is independent of any current philosophical debates about structural realism.

Yes / The essay revisits the puzzle of the ‘passage’ of time in relation to EPR-type measurements and asks what philosophical consequences can be drawn from them. Some argue that the lack of invariance of temporal order in the measurement of a space-like related EPR pair, under relativistic motion, casts serious doubts on the ‘reality’ of the lapse of time. Others argue that certain features of quantum mechanics establish a tensed theory of time – understood here as Possibilism or the growing block universe. The paper analyzes the employment of frame-invariant entropic clocks in a relativistic setting and argues that tenselessness does not imply timelessness. But this conclusion does not support a tensed theory of time, which requires a preferred foliation. It is argued that the only reliable inference from the EPR example and the use of entropic clocks is an inference not just to a Leibnizian order of the succession of events but a frame-invariant order according to some selected clocks.