Phase transitions abound in the physical world, from the subatomic length scales of quark condensation to the decoupling forces in the early universe. In the Bose-Einstein condensation phase transition, a gas of trapped bosonic atoms is cooled to a critical temperature. Below this temperature, a macroscopic number of atoms suddenly starts to occupy a single quantum state; these atoms comprise the Bose-Einstein condensate (BEC). The dynamics of the BEC phase transition are the focus of this dissertation and the experiments described here have provided new information on the details of BEC formation. New theoretical developments are proving to be valuable tools for describing BEC phase transition dynamics and interpreting new experimental results. With their amenability to optical manipulation and probing along with the advent of new microscopic theories, BECs provide an important new avenue for gaining insight into the universal dynamics of phase transitions in general.Spontaneous symmetry breaking in the system's order parameter may be one result of cooling through a phase transition. A potential consequence of this is the spontaneous formation of topological defects, which in a BEC appear as vortices. We experimentally observed and characterized the spontaneous formation of vortices during BEC growth. We attribute vortex creation to coherence length limitations during the initial stages of the phase transition. Parallel to these experimental observations, theory collaborators have used the Stochastic Gross-Pitaevski Equation formalism to simulate the growth of a condensate from a thermal cloud. The experimental and theoretical statistical results of the spontaneous formation of vortex cores during the growth of the condensate are in good quantitative agreement with one another, supporting our understanding of the dynamics of the phase transition. We believe that our results are also qualitatively consistent with the Kibble-Zurek mechanism, a universal model for topological defect formation.Ultimately, our understanding of the dynamics of the BEC phase transition may lead to a broader understanding of phase transitions in general, and provide new insight into the development of coherence in numerous systems.

À l'interface entre l'optique, le magnétisme et la supraconductivité, nous cherchons à dévelop-per de nouveaux concepts pour la manipulation optique, la génération et l'étude des vortex d'Abrikosov individuels dans des systèmes supraconducteurs. D'une part, nous démontrons l'efficacité d'une méthode optique de génération spontanée d'une paire de vortex/anti-vortex par effet Kibble Zurek basée sur l'utilisation d'une impulsion laser focalisée à la surface d'un film supraconducteur. C'est une technique en champ lointain, rapide qui permet de créer et piéger une paire dans le condensat supraconducteur à une position stable et reproductible. Cette expérience est également adaptée à l'étude du scénario Kibble-Zurek, relatif à la nucléation spontanée de défauts topologiques, tels que les vortex d'Abrikosov, lors de transitions de phase rapides du second ordre. D'autre part, nous présentons les résultats d'une expérience pompe-sonde visant à étudier l'effet Faraday inverse dans le grenat de BiLuIG servant à l'imagerie magnéto-optique des vortex. Nous montrons qu'il est possible de générer un champ magnétique femtoseconde de plusieurs Tesla localisé dans le grenat grâce à une impulsion laser ultra-courte de polarisation circulaire. Ces résultats nous permettent d'étudier les conditions expérimentales pour une génération de paires de vortex/anti-vortex par méthode magnéto-optique, basée sur l'application d'un fort champ magnétique à la surface du supraconducteur. La possibilité de manipuler et générer les vortex, véritables nano-objets de l'état supraconducteur, offre des perspectives prometteuses quant au développement du contrôle optique de micro-circuits supraconducteurs tels que les jonctions Josephson. / At the interface between optics, magnetism and superconductivity, we want to develop new concepts for the optical manipulation, the generation and the study of individual Abrikosov vortices in superconducting systems. On one hand, we demonstrate the efficiency of an optical method to perform spontaneous generation of a single vortex/anti-vortex pair by Kibble Zurek effect, based on a laser pulse focused at the superconductor film surface. It is a fast far field method to create and trap a pair into the superconducting condensate at a reproducible and stable position. This experiment is also adapted to the study of the Kibble Zurek mechanism, describing nucleation of topological defects such as Abrikosov vortices during a fast second order phase transition. On the other hand, we present the results of a pomp-probe experiment to the study the inverse Faraday effect into a BiLuIG garnet used for magneto-optical imaging of vortices. We show the possibility to produce a strong femtosecond magnetic field of a few Tesla localized into the garnet with a circularly polarized ultra-short laser pulse. Those results lead to the determination of the experimental conditions necessary to generate a vortex/anti-vortex pair with a magneto-optical method based on the application of a strong magnetic field close to the superconductor surface. Fast optical manipulation and generation of vortices, which are intrinsic nano-objects of the superconducting state, should enable the development of optically driven superconducting micro-circuits such as Josephson junctions.

The Kibble-Zurek mechanism (KZM) primarily characterizes scaling in the formation of topological defects when a system crosses a continuous phase transition. The KZM was first used to study the evolution of the early universe, describing the topology of cosmic domains and strings as the symmetry-breaking phase transitions acted on the vacuum fields during the initial cooling. A ferromagnetic spin-1 $^{87}$Rb Bose-Einstein condensate (BEC) exhibits a second-order gapless quantum phase transition due to a competition between the magnetic and collisional spin interaction energies. Unlike extended systems where the KZM is illustrated by topological defects, we focus our study on the temporal evolution of the spin populations and observe how the scaling of the spin dynamics depend on how fast the system is driven through the critical point. In our case, the excitations are manifest in the temporal evolution of the spin populations illustrating a Kibble-Zurek type scaling, where the dynamics of slow quenches through the critical point are predicted to exhibit universal scaling as a function of quench speed. The KZM has been studied theoretically and experimentally in a large variety of systems. There has also been a tremendous interest in the KZM in the cold atoms community in recent years. It has been observed not only in ion chains and in atomic gases in optical lattices, but also in Bose gases through the formation of vortices or solitons. The KZM in the context of crossing the quantum phase transition in a ferromagnetic BEC has been theoretically studied, but this thesis is the first experimental investigation of this phenomenon.

Quantum annealing represents an essential milestone towards the goal of adiabatic quantum computing. In quantum annealing, the computation involves finding the ground state of a classical Ising-like Hamiltonian realized as interactions between qubits. Quantum fluctuations are introduced to allow the wavefunction of the qubits to explore the energy landscape, the hope being that the wavefunction finds a minimum energy configuration and possibly giving the result of the computation. While quantum annealing likely may not be as powerful as adiabatic quantum computing, it is possible that it may be better at optimization compared to analogous classical algorithms. In physical realizations of quantum annealing, there are still questions as to the role of quantum fluctuations in the operation of a device given the short coherence times of the individual qubits. These questions have consistently posed a serious theoretical challenge making it difficult to verify experimental results. Here we simplify the problem by considering a system of qubits with ferromagnetic interactions, modeling the decoherence effects as classical noise in the transverse-field of each qubit. We compare the calculations to data collected from a system of manufactured qubits produced by D-wave Systems by performing a finite-size scaling analysis that captures the competition between quantum fluctuations of the transverse-field and bit-flip errors from the noise. We argue that on time-scales larger than the single-qubit decoherence time, the device produces the expected quantum fluctuations for the many-body system. Using this finite-size scaling, one can diagnose sources of noise in the system. Hopefully, in the near future, these devices will not only be realizing coherent quantum annealing but will likely be useful as another example of synthetic quantum matter.

Cette thèse analyse certains aspects de la dynamique hors équilibre de systèmes quantiques unidimensionnels lorsqu’ils sont soumis à des champs externes dépendant du temps. Nous considérons plus particulièrement le cas des forçages périodiques, et le cas d’une variation temporelle lente d’un paramètre de l’Hamiltonien qui permet de traverser une transition de phase quantique. La première partie contient une présentation des notions, des modèles et des outils nécessaires pour comprendre la suite de la thèse, avec notamment des rappels sur les modèles quantiques critiques (en particulier sur les chaines de spin et sur le modèle de Bose-Hubbard), le mécanisme de Kibble-Zurek, et la théorie de Floquet. Ensuite, nous étudions la dynamique hors équilibre des gaz de Tonks-Girardeau dans un potentiel harmonique dépendant du temps par différentes techniques : développements perturbatifs, diagonalisation numérique exacte et solutions analytiques exactes basées sur la théorie des invariants dynamiques d’Ermakov-Lewis. Enfin, nous analysons la dynamique hors équilibre des systèmes quantiques ouverts markoviens soumis à des variations périodiques des paramètres du système et de l’environnement. Nous formulons une théorie de Floquet afin d’obtenir des solutions exactes des équations de Lindblad périodiques. Ce formalisme de Lindblad-Floquet est utilisé pour obtenir une caractérisation exacte du fonctionnement en temps fini des machines thermiques quantiques. / This thesis analyzes some aspects regarding the dynamics of one-dimensional quantum systems which are driven out-of-equilibrium by the presence of time- dependent external fields. Among the possible kinds of driven systems, our focus is dedicated to the slow variation of a Hamiltonian’s parameter across a quantum phase transition and to the case of a time-periodic forcing. To begin with, we prepare the background and the tools needed in the following. This includes a brief introduction to quantum critical models (in particular to the xy spin chain and to the Bose-Hubbard model), the Kibble-Zurek mechanism and Floquet theory. Next, we consider the non-equilibrium dynamics of Tonks-Girardeau gases in time-dependent harmonic trap potentials. The analysis is made with different techniques: perturbative expansions, numerical exact diagonalization and exact methods based on the theory of Ermakov-Lewis dynamical invariants. The last part of the thesis deals instead with the non-equilibrium dynamics of markovian open quantum systems subject to time-periodic perturbations of the system parameters and of the environment. This has led to an exact formulation of Floquet theory for a Lindblad dynamics. Moreover, within the Lindblad-Floquet framework it is possible to have an exact characterization ofthe finite-time operation of quantum heat-engines.

La première partie de la thèse étudie le diagramme de phase d’une généralisation de la chaîne de Kitaev qui décrit un système fermionique avec un pairing p-wave à long rayon qui tombe avec la distance ℓ comme 1/ℓα. On a analysé les lignes critiques, les corrélations et le comportement de l’entropie d’entanglement avec la taille du système. Nous avons démontré l’existence de deux régimes massifs, (i) où les fonctions de corrélation tombent exponentiellement à de courtes distances et comme puissance à de longues distances (α > 1), (ii) où elles tombent à puissance seulement (α < 1). Dans la seconde région l’entropie d’intrication d’un sous-système diverge logarithmiquement. Remarquablement, sur les lignes critiques, le pairing à long rayon brise la symètrie conforme du modèle pour des α suffisamment petits. On a prouvé ça en calculant aussi l’évolution temporelle de l’entropie d’intrication après un quench. Dans la seconde partie de la thèse nous avons analysé la dynamique de l’entropie d’intrication du modèle d’Ising avec un champ magnétique qui dépend linéairement du temps avec de différentes vitesses. Nous avons un régime adiabatique (de basses vitesses) lorsque le système évolue selon son état fondamental instantané; un sudden quench (de hautes vitesses) lorsque le système est congelé dans son état initial; un régime intermédiaire où l’entropie croît linéairement et, ensuite, elle montre des oscillations du moment que le système se trouve dans une superposition des états excités de l’Hamiltonienne instantanée. Nous avons discuté aussi du mécanisme de Kibble-Zurek pour la transition entre la phase paramagnétique et antiferromagnétique. / In the first part of the thesis, we propose an exactly-solvable one-dimensional model for fermions with long-range p-wave pairing decaying with distance ℓ as a power law 1/ℓα. We studied the phase diagram by analyzing the critical lines, the decay of correlation functions and the scaling of the von Neumann entropy with the system size. We found two gapped regimes, where correlation functions decay (i) exponentially at short range and algebraically at long range (α > 1), (ii) purely algebraically (α < 1). In the latter the entanglement entropy is found to diverge logarithmically. Most interestingly, along the critical lines, long-range pairing breaks the conformal symmetry for sufficiently small α. This can be detected also via the dynamics of entanglement following a quench. In the second part of the thesis we studied the evolution in time of the entanglement entropy for the Ising model in a transverse field varying linearly in time with different velocities. We found different regimes: an adiabatic one (small velocities) when the system evolves according the instan- taneous ground state; a sudden quench (large velocities) when the system is essentially frozen to its initial state; and an intermediate one, where the entropy starts growing linearly but then displays oscillations (also as a function of the velocity). Finally, we discussed the Kibble-Zurek mechanism for the transition between the paramagnetic and the ordered phase

Les gaz quantiques atomiques constituent un outil de choix pour étudier la physique à N corps grâce à leurs nombreux paramètres de contrôle. Ils offrent la possibilité d’explorer la physique en basse dimension, modifiée par rapport au cas à trois dimensions (3D) à cause du rôle accru des fluctuations. Dans ce travail, nous étudions le gaz de Bose à deux dimensions (2D) avec un confine-ment original dans le plan atomique, uniforme et de motif arbitraire. Ces gaz2D et uniformes, développés sur un montage existant, ont été installés sur un nouveau montage grâce à des potentiels optiques polyvalents.Nous présentons une série d’expériences exploitant cette géométrie flexible.D’abord, nous étudions le comportement statique et dynamique d’un gaz uni-forme lors de la transition d’un état 3D normal vers un état 2D superfluide.Nous observons l’établissement de la cohérence de phase dans un gaz à l’équilibre puis nous montrons l’apparition après une trempe de défauts topologiques dont le nombre est comparé à la prédiction de Kibble-Zurek. Ensuite, nous étudions grâce au nouveau montage les effets collectifs dans l’interaction lumière-matière, où les propriétés de résonance d’un nuage d’atomes dense sont fortement modifiées par rapport à celles d’un atome unique. Enfin, nous proposons deux protocoles pour le nouveau montage. Le premier permet d’évaporer de manière uniforme un gaz 2D grâce au réseau incliné du confinement à 2D. Le second propose de produire des supercourants de manière déterministe dans des pièges en anneaux, soit par condensation dans un champ de jauge, soit en réalisant une pompe à vortex topologique. / Degenerate atomic gases are a versatile tool to study many-body physics. They offer the possibility to explore low-dimension physics, which strongly differs from the three dimensional (3D) case due to the enhanced role of fluctuations. In this work, we study degenerate 2D Bose gases whose original in-plane confinement is uniform and of arbitrary shape. These 2D uniform traps, which we first developed on an existing set-up, were subsequently implemented on a newset-up using versatile optical potentials. We present a series of experiments that take advantage of this flexible geometry. First, we study the static and dynamic behaviours of a uniform gas at the transition between a 3D normal and a 2D superfluid state. We observe the establishement of extended phase coherence, followed, as the gas is quench cooled, by the apparition of topological defects whose scaling is compared to the Kibble-Zurek prediction. Second, we present the first results of the new set-up : we investigate collective effects in light-matter interactions, where the resonance properties of a dense ensemble of atoms are strongly modified with respect to the single atom ones. Last, we develop two experimental proposals for the new set-up. The first one studies how a 2D gas can be uniformly evaporated using the tilted lattice providing the 2D confinement. In the second one, we propose to produce su-percurrents in a deterministic way in ring-shaped traps either by condensing inan artificial gauge field or by implementing a topological vortex pump.

No description available.

Dans cette thèse, nous avons répondu à certaines questions ouverts dans le domaine de la dynamique hors équilibre des systèmes quantiques unidimensionnels fermés. Durant ces dernières années, les avancées dans les techniques expérimentales ont revitalisé la recherche théorique en physique de la matière condensée et dans l'optique quantique. Nous avons traité trois sujets différents et en utilisant des techniques à la fois numériques et analytiques. Dans le cadre des techniques numériques, nous avons utilisé des méthodes de diagonalisation exacte, l'algorithme du groupe de renormalisation de la matrice densité en fonction du temps (t-DMRG) et l'algorithme de Lanczos. Au début, nous avons étudié la dynamique quantique adiabatique d'un système quantique près d'un point critique. Nous avons démontré que la présence d'un potentiel de confinement modifie fortement les propriétés d'échelle de la dynamique des observables en proximité du point critique quantique. La densité d'excitations moyenne et l'excès d'énergie, après le croisement du point critique, suivent une loi algébrique en fonction de la vitesse de la trempe avec un exposant qui dépend des propriétés spatio-temporelles du potentiel. Ensuite, nous avons étudié le comportement de bosons ultra-froids dans un réseau optique incliné. En commençant par l'hamiltonien de Bose-Hubbard, dans la limite de Hard-Core bosons, nous avons développé une théorie hydrodynamique qui reproduit exactement l'évolution temporelle d'une partie des observables du système. En particulier, nous avons observé qu'une partie de bosons reste piégée, et oscille avec une fréquence qui dépend de la pente du potentiel, au contraire, une autre partie est expulsée hors de la rampe. Nous avons également analysé la dynamique du modèle de Bose-Hubbard en utilisant l'algorithme t-DMRG et l'algorithme de Lanczos. De cette façon, nous avons mis en évidence le rôle de la non-intégrabilité du modèle dans son comportement dynamique. Enfin, nous avons abordé le problème de la thermalisation dans un système quantique étendu. À partir de considérations générales, nous avons introduit la notion de profil de température hors équilibre dans une chaîne des bosons à coeur dure. Nous avons analysé la dynamique du profil de temperature et, notamment, ses propriétés d'échelle / In this thesis we have addressed some open questions on the out-of-equilibrium dynamics of closed one-dimensional quantum systems. In recent years, advances in experimental techniques have revitalized the theoretical research in condensed matter physics and quantum optics. We have treated three different subjects using both numerical and analytical techniques. As far as the numerical techniques are concerned, we have used essentially exact diagonalization methods, the adaptive time-dependent density-matrix renormalization-group algorithm (t-DMRG) and the Lanczos algorithm. At first, we studied the adiabatic quantum dynamics of a quantum system close to a critical point. We have demonstrated that the presence of a confining potential strongly affects the scaling properties of the dynamical observables near the quantum critical point. The mean excitation density and the energy excess, after the crossing of the critical point, follow an algebraic law as a function of the sweeping rate with an exponent that depends on the space-time properties of the potential. After that, we have studied the behavior of ultra-cold bosons in a tilted optical lattice. Starting with the Bose-Hubbard Hamiltonian, in the limit of Hard-Core bosons, we have developed a hydrodynamic theory that exactly reproduces the temporal evolution of some of the observables of the system. In particular, it was observed that part of the boson density remains trapped, and oscillates with a frequency that depends on the slope of the potential, whereas the remaining packet part is expelled out of the ramp. We have also analyzed the dynamics of the Bose-Hubbard model using the tDMRG algorithm and the Lanczos algorithm. In this way we have highlighted the role of the non-integrability of the model on its dynamical behavior. Finally, we have addressed the issue of thermalization in an extended quantum system. Starting from quite general considerations, we have introduced the notion of out-of-equilibrium temperature profile in a chain of Hard-Core bosons. We have analyzed the dynamics of the temperature profile and especially its scaling properties

The physics of ultracold quantum gases has been the subject of a long-lasting and intense research activity, which started almost a century ago with purely theoretical studies and had a fluorishing experimental development after the implementation of laser and evaporative cooling techniques that led to the first realization of a Bose Einstein condensate (BEC) over 25 years ago. In recent years, a great interest in ultracold atoms has developed for their use as platforms for quantum technologies, given the high degree of control and tunability offered by ultracold atom systems. These features make ultracold atoms an ideal test bench for simulating and studying experimentally, in a controlled environment, physical phenomena analogous to those occurring in other, more complicated, or even inaccessible systems, which is the idea at the heart of quantum simulation. In the rapidly developing field of quantum technologies, it is highly important to acquire an in-depth understanding of the state of the quantum many-body system that is used, and of the processes needed to reach the desired state. The preparation of the system in a given target state often involves the crossing of second order phase transitions, bringing the system strongly out-of-equilibrium. A better understanding of the out-of-equilibrium processes occurring in the vicinity of the transition, and of the relaxation dynamics towards the final equilibrium condition, is crucial in order to produce well-controlled quantum states in an efficient way. In this thesis I present the results of the research activity that I performed during my PhD at the BEC1 laboratory of the BEC center, working on ultracold gases of 23Na atoms in an elongated harmonic trap. This work had two main goals: the accurate determination of the equilibrium properties of a Bose gas at finite temperature, by the measurement of its equation of state, and the investigation of the out-of-equilibrium dynamics occurring when a Bose Einstein condensate is prepared by cooling a thermal cloud at a finite rate across the BEC phase transition.To study the equilibrium physics of a trapped atomic cloud, it is crucial to be able to observe its density distribution in situ. This requires a high optical resolution to accurately obtain the density profile of the atomic distribution, from which thermodynamic quantities can then be extracted. In particular, in a partially condensed atomic cloud at finite temperature, it is challenging to resolve well also the boundaries of the BEC, where the condensate fraction rapidly drops in a narrow spatial region. This required an upgrade of the experimental apparatus in order to obtain a high enough resolution. I designed, tested and implemented in the experimental setup new imaging systems for all main directions of view. Particular attention was paid for the vertical imaging system, which was designed to image the condensates in trap with a resolution below 2 μm, with about a factor 4 improvement compared to the previous setup. The implementation of the new imaging systems involved a partial rebuilding of the experimental apparatus used for cooling the atoms. This created the occasion for an optimization of the whole system to obtain more stable working conditions. Concurrently I also realized and included in the experiment an optical setup for the use of a Digital Micromirror Device (DMD) to project time-dependent arbitrary light patterns on the atoms, creating optical potentials that can be controlled at will. The use of this device opens up exciting future scenarios where it will be possible to locally modify the trapping potential and to create well-controlled barriers moving through the atomic cloud. Another challenge in imaging the density distribution in situ is determined by the fact that the maximum optical density (OD) of the BEC, in the trap center, exceeds the low OD of the thermal tails by several orders of magnitude. In order to obtain an accurate image of the whole density profile, we developed a minimally destructive, multi-shot imaging technique, based on the partial transfer of a fraction of atoms to an auxiliary state, which is then probed. Taking multiple images at different extraction fractions, we are able to reconstruct the whole density profile of the atomic cloud avoiding saturation and maintaining a good signal to noise ratio. This technique, together with the improvements in the imaging resolution, has allowed us to accurately obtain the optical density profile of the Bose gas in trap, from which the 3D density profile was then calculated applying an inverse Abel transform, taking advantage of the symmetry of the trap. From images of the same cloud after a time-of-flight expansion, we measured the temperature of the gas. From these quantities we could find the pressure as a function of the density and temperature, determining the canonical equation of state of the weakly interacting Bose gas in equilibrium at finite temperature. These measurements also allowed us to clearly observe the non-monotonic temperature behavior of the chemical potential near the critical point for the phase transition, a feature that characterizes also other superfluid systems, but that had never been observed before in weakly interacting Bose gases. The second part of this thesis work is devoted to the study of the dynamical processes that occur during the formation of the BEC order parameter within a thermal cloud. The cooling at finite rate across the Bose-Einstein condensation transition brings the system in a strongly out-of-equilibrium state, which is worth investigating, together with the subsequent relaxation towards an equilibrium state. This is of interest also in view of achieving a better understanding of second order phase transitions in general, since such phenomena are ubiquitous in nature and relevant also in other platforms for quantum technologies. A milestone result in the study of second order phase transitions is given by the Kibble-Zurek mechanism, which provides a simple model capturing important aspects of the evolution of a system that crosses a second-order phase transition at finite rate. It is based on the principle that in an extended system the symmetry breaking associated with a continuous phase transition can take place only locally. This causes the formation of causally disconnected domains of the order parameter, at the boundaries of which topological defects can form, whose number and size scale with the rate at which the transition is crossed, following a universal power law. It was originally developed in the context of cosmology, but was later successfully tested in a variety of systems, including superfluid helium, superconductors, trapped ions and ultracold atoms. The BEC phase transition represents in this context a paradigmatic test-bench, given the high degree of control at which this second-order phase transition can be crossed by means of cooling ramps at different rates. Already early experiments investigated the formation of the BEC order parameter within a thermal cloud, after quasi-instantaneous temperature quenches or very slow evaporative cooling. In the framework of directly testing the Kibble-Zurek mechanism, further experiments were performed, both in 2D and 3D systems, focusing on the emergence of coherence and on the statistics of the spontaneously generated topological defects as a function of the cooling rate. The Kibble-Zurek mechanism, however, does not fully describe the out-of-equilibrium dynamics of the system at the transition, nor the post-quench interaction mechanisms between domains that lead to coarse-graining. Most theoretical models are based on a direct linear variation of a single control parameter, e.g. the temperature, across the transition. In real experiments, the cooling process is controlled by the tuning of other experimental parameters and a global temperature might not even be well defined, in a thermodynamic sense, during the whole process. Moreover, the temperature variation is usually accompanied by the variation of other quantities, such as the number of atoms and the collisional rate, making it difficult to accurately describe the system and predict the post-quench properties. Recent works included effects going beyond the Kibble-Zurek mechanism, such as the inhomogeneity introduced by the trapping potential, the role of atom number losses, and the saturation of the number of defects for high cooling rates. These works motivate further studies, in particular of the dynamics taking place at early times, close to the crossing of the critical point. The aim of the work presented in this thesis is to further investigate the timescales associated to the formation and evolution of the BEC order parameter and its spatial fluctuations, as a function of the rate at which the transition point is crossed. We performed experiments producing BECs by means of cooling protocols that are commonly used in cold-atom laboratories, involving evaporative cooling in a magnetic trap. We explored a wide range of cooling rates across the transition and found a universal scaling for the growth of the BEC order parameter with the cooling rate and a finite delay in its formation. The latter was already observed in earlier works, but for a much more limited range of cooling rates. The evolution of the fluctuations of the order parameter was also investigated, with an analysis of the timescale of their decay during the relaxation of the system, from an initial strongly out-of-equilibrium condition to a final equilibrium state. This thesis is structured as follows: The first chapter presents the theoretical background, starting with a brief introduction to the concept of Bose Einstein condensation and a presentation of different models describing the thermodynamics of an equilibrium Bose gas. The second part of this chapter then deals with the out-of-equilibrium dynamics that is inevitably involved in the crossing of a second-order phase transition such as the one for Bose-Einstein condensation. The Kibble-Zurek mechanism is briefly reviewed and beyond KZ effects are pointed out, motivating a more detailed investigation of the timescales involved in the BEC formation. In the second chapter, I describe the experimental apparatus that we use to cool and confine the atoms. Particular detail is dedicated to the parts that have been upgraded during my PhD, such as the imaging system. In the third chapter I show our experimental results on the measurement of the equation of state of the weakly interacting uniform Bose gas at finite temperature. In the fourth chapter I present our results on the out-of-equilibrium dynamics in the formation of the condensate order parameter and its spatial fluctuations, as a function of different cooling rates.