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This thesis covers most of my work in the field of ultracold atoms loaded in optical lattices. This thesis can be divided into five different parts. In Chapter 1, after a brief introduction to the field of optical lattices I review the fundamental aspects pertaining to the physics of systems in periodic potentials and a short overview of the experiments on ultracold atoms in an optical lattice. In Chapter 2 we develop an inhomogeneous mean-field theory for the extended Bose-Hubbard model with a quadratic, confining potential. In the absence of this poten¬tial, our mean-field theory yields the phase diagram of the homogeneous extended Bose-Hubbard model. This phase diagram shows a superfluid (SF) phase and lobes of Mott-insulator(MI), density-wave(DW), and supersolid (SS) phases in the plane of the chemical potential and on-site repulsion ; we present phase diagrams for representative values of , the repulsive energy for bosons on nearest-neighbor sites. We demonstrate that, when the confining potential is present, superfluid and density-wave order parameters are nonuniform; in particular, we obtain, for a few representative values of parameters, spherical shells of SF, MI ,DW ,and SSphases. We explore the implications of our study for experiments on cold-atom dipolar con¬densates in optical lattices in a confining potential. In Chapter3 we present an extensive study of Mottinsulator( MI) and superfluid (SF) shells in Bose-Hubbard (BH) models for bosons in optical lattices with har¬monic traps. For this we develop an inhomogeneous mean-field theory. Our results for the BH model with one type of spinless bosons agrees quantitatively with quan¬tum Monte Carlo(QMC) simulations. Our approach is numerically less intensive than such simulations, so we are able to perform calculations on experimentally realistic, large three-dimensional(3D) systems, explore a wide range of parameter values, and make direct contact with a variety of experimental measurements. We also generalize our inhomogeneous mean-field theory to study BH models with har¬monic traps and(a) two species of bosons or(b) spin-1bosons. With two species of bosons we obtain rich phase diagrams with a variety of SF and MI phases and as¬sociated shells, when we include a quadratic confining potential. For the spin-1BH model we show, in a representative case, that the system can display alternating shells of polar SF and MI phases; and we make interesting predictions for experi¬ments in such systems. . In Chapter 4 we carry out an extensive study of the phase diagrams of the ex-tended Bose Hubbard model, with a mean filling of one boson per site, in one dimension by using the density matrix renormalization group and show that it contains Superfluid (SF), Mott-insulator (MI), density-wave (DW) and Haldane ¬insulator(HI) phases. We show that the critical exponents and central charge for the HI-DW,MI-HI and SF-MI transitions are consistent with those for models in the two-dimensional Ising, Gaussian, and Berezinskii-Kosterlitz-Thouless (BKT) uni¬versality classes, respectively; and we suggest that the SF-HI transition may be more exotic than a simple BKT transition. We show explicitly that different bound¬ary conditions lead to different phase diagrams.. In Chapter 5 we obtain the excitation spectra of the following three generalized of Bose-Hubbard(BH) models:(1) a two-species generalization of the spinless BH model, (2) a single-species, spin-1 BH model, and (3) the extended Bose-Hubbard model (EBH) for spinless interacting bosons of one species. In all the phases of these models we show how to obtain excitation spectra by using the random phase approximation (RPA). We compare the results of our work with earlier studies of related models and discuss implications for experiments.

When bulk helium-4 is cooled below T = 2.18 K, it undergoes a phase transition to a superfluid, characterized by a complex wave function with a macroscopic phase and exhibits inviscid, quantized flow. The macroscopic phase coherence can be probed in a container filled with helium-4, by reducing one or more of its dimensions until they are smaller than the coherence length, the spatial distance over which order propagates. As this dimensional reduction occurs, enhanced thermal and quantum fluctuations push the transition to the superfluid state to lower temperatures. However, this trend can be countered via the proximity effect, where a bulk 3-dimensional (3d) superfluid is coupled to a low (2d) dimensional superfluid via a weak link producing superfluid correlations in the film at temperatures above the Kosterlitz-Thouless temperature. Recent experiments probing the coupling between 3d and 2d superfluid helium-4 have uncovered an anomalously large proximity effect, leading to an enhanced superfluid density that cannot be explained using the correlation length alone. In this work, we have determined the origin of this enhanced proximity effect via large scale quantum Monte Carlo simulations of helium-4 in a topologically non-trivial geometry that incorporates the important aspects of the experiments. We find that due to the bosonic symmetry of helium-4, identical particle permutations lead to correlations between contiguous spatial regions at a length scale greater than the coherence length. We show that quantum exchange plays a large role in explaining the anomalous experimental results while simultaneously showing how classical arguments fall short of this task.

Nesta dissertação estudaremos sistemas de bósons ultrafrios armadilhados em uma rede ótica quadrada bidimensional sem levar em consideração o confinamento harmônico. A dinâmica desses sistemas é bem descrita pelo modelo de Bose-Hubbard, que prevê uma transição de fase quântica de um superfluido para um isolante de Mott a temperaturas baixas, e pode ser induzida variando a profundidade do potencial da rede ótica. Apresentaremos o diagrama de fases dessa transição construído a partir de uma aproximação de campo médio e também com um cálculo numérico usando um algoritmo de Monte Carlo Quântico, denominado algoritmo Worm. Encontramos o ponto crítico para o primeiro lobo de Mott em ambos os casos, concordando com trabalhos anteriores. / This work study the two-dimensional ultracold bosonic atoms loaded in a square optical lattice, without harmonic confinement. The dynamics of this system is described by the Bose-Hubbard model, which predicts a quantum phase transition from a superfluid to a Mott-insulator at low temperatures that can be induced by varying the depth of the optical potential. We present here the phase diagram of this transition built from a mean field approach and from a numerical calculation using a Quantum Monte Carlo algorithm, namely the Worm algorithm. We found the critical transition point for the first Mott lobe in both cases, in agreement with the standard literature.

Dans cette thèse, nous étudions deux aspects de la dynamique d'impuretés atomiques interagissant avec des nanogouttes d'hélium superfluide (He) : la photo-excitation d'alcalins sur une nanogoutte et le dopage de nanogouttes contenant des tourbillons (vortex) quantiques avec des atomes de gaz rares. Nous utilisons la théorie de la fonctionnelle de la densité d'hélium ainsi que sa version dépendante du temps pour en faire la description théorique. Le premier aspect a été effectué dans le cadre d'une collaboration avec des expérimentateurs sur la photo-excitation du rubidium (Rb). Les alcalins sont une sonde très intéressante des gouttelettes d'hélium car ils résident dans leur zone de surface, où il a été prédit qu'un taux de condensation de Bose-Einstein de 100% était possible en raison d'une densité inférieure à celle de l'hélium superfluide. Nos simulations montrent que les états excités 5p et 6p désorbent à des échelles de temps très différentes, séparées par 2 ordres de grandeur (~100 ps et ~1 ps pour 5p et 6p respectivement). Ces résultats sont en accord avec ceux de l'expérience pompe-sonde à l'échelle femtoseconde qui a étudié la photodesorption d'atomes de Rb. Cependant, dans nos simulations, l'excitation vers 5pPi_{3/2} aboutit à un exciplexe RbHe lié à la surface, contrairement à l'expérience où RbHe est éjecté. L'introduction de la relaxation de spin de Pi_{3/2} à Pi_{1/2} nous a permis de résoudre ce désaccord, l'exciplexe RbHe ayant alors assez d'énergie pour désorber. Le deuxième aspect concerne une investigation purement théorique inspirée par les travaux récents de Gomez et Vilesov et al., où les tourbillons quantiques étaient visualisés en dopant les nanogouttes d'hélium avec des atomes d'argent, puis en les faisant atterrir en douceur (soft landing) sur un écran de carbone. Les images au microscope électronique montrent de longs filaments d'agrégats d'atomes d'argent qui s'étaient accumulés le long des coeurs des vortex. La formation de réseaux de tourbillons quantiques à l'intérieur de nanogoutelettes dopées par du xénon est également mise en évidence par diffraction de rayons X qui montrent des pics de Bragg caractéristiques d'agrégats de xénon piégés dans les coeurs des vortex. Nous avons d'abord étudié des collisions frontales entre un atome de xénon, héliophile, et une nanogoutte de 1000 héliums, et comparé les résultats à ceux d'une étude précédente sur le même processus avec le césium (Cs), qui est héliophobe. Dans le cas de Xe une «boule de neige» se forme autour de lui quand il entre dans la nanogoutte, et il lui faut beaucoup plus d'énergie qu'au Cs pour qu'il puisse en ressortir. Quand il le fait, il emporte des héliums avec lui, contrairement au Cs. Nous avons ensuite simulé des collisions entre Ar/Xe et des nanogouttes d'hélium superfluides pour différentes vitesses initiales et paramètres d'impact afin de déterminer leur section efficace de capture. Ces simulations ont ensuite été répétées pour des gouttelettes hébergeant un vortex quantique. On observe que l'impact des impuretés induit de grandes déformations de flexion et de torsion de la ligne de vortex, allant jusqu'à la génération d'ondes de Kelvin hélicoïdales qui se propagent le long du coeur du vortex. Ar/Xe est bien finalement capturé par le vortex, mais pas dans son coeur. Nous avons également découvert que l'existence d'un réseau de 6 lignes de vortex dont les noyaux sont remplis d'atomes d'Ar donne une rigidité accrue à la nanogoutte qui permet de stabiliser le système nano-goutte + vortex même à de faibles vitesses angulaires. Nos simulations impliquant des nanogouttes d'hélium comportant des tourbillons quantiques ouvrent la voie à d'autres investigations sur des nanogouttes hébergeant un ensemble de vortex, en collision avec de multiples impuretés. / In this thesis we investigate two aspects of the dynamics of atomic impurities interacting with superfluid helium (He) nanodroplets, namely the photo-excitation of alkalis on a nanodroplet and the doping process of nanodroplets hosting quantised vortices with noble gas atoms. For the theoretical investigations we use He density functional theory and its time-dependent version. The first aspect involves a joint experimental and theoretical collaboration that focusses on the photo-excitation of the alkali rubidium (Rb). Alkalis are a very interesting probe of He droplets since they reside in their surface region, where it has been argued that almost 100% Bose-Einstein condensation could be achieved due to a density that is lower than in bulk superfluid He. In our simulations we find that states excited to the 5p and 6p manifold desorb at very different timescales, separated by 2 orders of magnitude (~100 ps and ~1 ps for 5p and 6p respectively). This is in good agreement with experimental results where the desorption behaviour of photo-excited Rb atoms is determined using a femtosecond pump-probe scheme. However, in our simulations excitation to the 5pPi_{3/2}-state results in a surface-bound RbHe exciplex, contrary to the experimental case where the RbHe exciplex desorbs from the droplets surface. Introducing spin-relaxation from Pi_{3/2} to Pi_{1/2} into the simulations, the RbHe exciplex is able to desorb from the droplet's surface, which resolves this contradiction. The second aspect concerns a purely theoretical investigation that is inspired by recent work of Gomez and Vilesov et al., where quantised vortices were visualised by doping He nanodroplets with silver atoms, subsequently "soft landing" them on a carbon screen. Electron-microscope images show long filaments of silver atom clusters that accumulated along the vortex cores. Also the formation of quantum-vortex lattices inside nanodroplets is evidenced by using X-ray diffractive imaging to visualise the characteristic Bragg patterns from xenon (Xe) clusters trapped inside the vortex cores. First, head-on collisions between heliophilic Xe and a He nanodroplet made of 1000 He atoms are studied. The results are then compared with the results of a previous study of an equivalent kinematic case with cesium (Cs), which is heliophobic. Xe acquires a "snowball" of He around itself when it traverses the droplet and much more kinetic energy is required before Xe is able to pierce the droplet completely. When it does, it takes away some He with it, contrary to the Cs case. Next, collisions between argon (Ar)/Xe and pristine superfluid He nanodroplets are performed for various initial velocities and impact parameters to determine the effective cross-section for capture. Finally, the simulations are then repeated for droplets hosting a single quantised vortex line. It is observed that the impact of the impurities induces large bending and twisting excitations of the vortex line, including the generation of helical Kelvin waves propagating along the vortex core. We conclude that Ar/Xe is captured by the quantised vortex line, although not in its core. Also we find that a He droplet, hosting a 6-vortex line array whose cores are filled with Ar atoms, results in added rigidity to the system which stabilises the droplets at low angular velocities. Our simulations involving droplets hosting quantum vortices open the way to further investigations on droplets hosting an array of vortices, involving multiple impurities.

In this thesis, we investigate strongly correlated phenomena in quantum spin systems. In the first part of this work, we study geometrically frustrated antiferromagnets (AFMs). Generalizing the SU(2) Heisenberg Hamiltonian to Sp(N) symmetry, we obtain, in the large-N limit, the mean-field phase diagrams for the planar pyrochlore and cubic AFMs. We then use gauge theories to consider fluctuation effects about their respective mean-field configurations. We find, in addition to conventional Neel states, a plethora of novel magnetically disordered phases: two kinds of spin liquids, Z2 in 2+1D and U(1)in 3+1D, and several valence bond solids such as two and three-dimensional plaquette and columnar singlet states. We use the same approach to study the diamond lattice AFM which possesses extended classical ground state degeneracy. We demonstrate that quantum and entropic fluctuations lift this degeneracy in different ways. In the second part of the thesis, we study ultracold spinor atoms confined in optical lattices. We first demonstrate the feasibility of experimental realization of rotor models using ultracold spin-one Bose atoms in a spin-dependent and disordered optical lattice. We show that the ground state of such disordered rotor models with quadrupolar interactions can exhibit biaxial nematic ordering in the disorder-averaged sense, and suggest an imaging experiment to detect the biaxial nematicity in such systems. Finally, using variational wavefunction methods, we study the Mott phases and superfluid-insulator transition of spin-three bosons in an optical lattice with an anisotropic two dimensional optical trap. We chart out the phase diagrams for Mott states with n = 1 and n = 2 atoms per lattice site. We show that the long-range dipolar interaction stabilizes a state characterized by antiferromagnetic chains made of ferromagnetically aligned spins. We also obtain the mean-field phase boundary for the superfluid-insulator transition, and show that inside the superfluid phase and near the superfluid-insulator phase boundary, the system undergoes a first order antiferromagnetic-ferromagnetic spin ordering transition.

In this thesis, we investigate strongly correlated phenomena in quantum spin systems. In the first part of this work, we study geometrically frustrated antiferromagnets (AFMs). Generalizing the SU(2) Heisenberg Hamiltonian to Sp(N) symmetry, we obtain, in the large-N limit, the mean-field phase diagrams for the planar pyrochlore and cubic AFMs. We then use gauge theories to consider fluctuation effects about their respective mean-field configurations. We find, in addition to conventional Neel states, a plethora of novel magnetically disordered phases: two kinds of spin liquids, Z2 in 2+1D and U(1)in 3+1D, and several valence bond solids such as two and three-dimensional plaquette and columnar singlet states. We use the same approach to study the diamond lattice AFM which possesses extended classical ground state degeneracy. We demonstrate that quantum and entropic fluctuations lift this degeneracy in different ways. In the second part of the thesis, we study ultracold spinor atoms confined in optical lattices. We first demonstrate the feasibility of experimental realization of rotor models using ultracold spin-one Bose atoms in a spin-dependent and disordered optical lattice. We show that the ground state of such disordered rotor models with quadrupolar interactions can exhibit biaxial nematic ordering in the disorder-averaged sense, and suggest an imaging experiment to detect the biaxial nematicity in such systems. Finally, using variational wavefunction methods, we study the Mott phases and superfluid-insulator transition of spin-three bosons in an optical lattice with an anisotropic two dimensional optical trap. We chart out the phase diagrams for Mott states with n = 1 and n = 2 atoms per lattice site. We show that the long-range dipolar interaction stabilizes a state characterized by antiferromagnetic chains made of ferromagnetically aligned spins. We also obtain the mean-field phase boundary for the superfluid-insulator transition, and show that inside the superfluid phase and near the superfluid-insulator phase boundary, the system undergoes a first order antiferromagnetic-ferromagnetic spin ordering transition.

Nesta dissertação estudaremos sistemas de bósons ultrafrios armadilhados em uma rede ótica quadrada bidimensional sem levar em consideração o confinamento harmônico. A dinâmica desses sistemas é bem descrita pelo modelo de Bose-Hubbard, que prevê uma transição de fase quântica de um superfluido para um isolante de Mott a temperaturas baixas, e pode ser induzida variando a profundidade do potencial da rede ótica. Apresentaremos o diagrama de fases dessa transição construído a partir de uma aproximação de campo médio e também com um cálculo numérico usando um algoritmo de Monte Carlo Quântico, denominado algoritmo Worm. Encontramos o ponto crítico para o primeiro lobo de Mott em ambos os casos, concordando com trabalhos anteriores. / This work study the two-dimensional ultracold bosonic atoms loaded in a square optical lattice, without harmonic confinement. The dynamics of this system is described by the Bose-Hubbard model, which predicts a quantum phase transition from a superfluid to a Mott-insulator at low temperatures that can be induced by varying the depth of the optical potential. We present here the phase diagram of this transition built from a mean field approach and from a numerical calculation using a Quantum Monte Carlo algorithm, namely the Worm algorithm. We found the critical transition point for the first Mott lobe in both cases, in agreement with the standard literature.

L’étude des vortex trouve sa justification dans le rôle que ces derniers jouent dans la turbulence quantique. L’équation de Gross-Pitaevskii ne peut pas nous permettre de modéliser convenablement l’Hélium superfluide, mais on peut l’utiliser pour obtenir le paramètre d'ordre d’un superfluide modèle, ayant le maximum de propriétés en commun avec l’Hélium, notamment une courbe de dispersion identique, par la modification du terme d’interactions.En supposant que le minimum roton influence l’essentiel de la physique, on détermine la forme du paramètre d’ordre loin de la perturbation créée par le vortex rectilinéaire axisymétrique par deux approches différentes - il apparaît alors que seuls deux paramètres sont nécessaires pour caractériser entièrement le profil.Le modèle proposé par Pomeau-Rica, qui offre la possibilité d’étudier le superfluide près de la cristallisation, met en lumière l’impact de la profondeur du minimum roton sur l’amplitude des oscillations. Par comparaison avec les résultats obtenus ab initio par Reatto, les résultats donnés par le modèle de Berloff-Roberts exhibent un déphasage marqué, qui semble être une conséquence non-physique de la forme du spectre d’excitation. Les calculs énergétiques laissent à penser que les oscillations portent une faible fraction de l’énergie du vortex, l'énergie cinétique dominant.Le calcul du paramètre d’ordre est effectué pour un anneau de grande taille par rapport à la distance interatomique, à vitesse nulle et à vitesse non-nulle. La détermination des énergies potentielle et cinétique permet d’accéder à la vitesse maximale atteinte par l’anneau en fonction de son rayon et de la comparer à la vitesse critique de Landau. / Vortices study's justification lays in the fact that those former play an important part in quantum turbulence. The Gross-Pitaevskii equation can't be a proper model for superfluid helium, but we can still use it to determine the order parameter of a theoretical superfluid, which has then the maximum amount of properties in common with liquid helium, and in particular, the same dispersion relation, thus gained by modifying the interaction terms.We then make the assumption that all the physical properties of the superfluid are triggered by the existence of the roton minimum, which allows us to calculate the order parameter far from the perturbation created by an axisymmetric rectilinear vortex, using two different methods. At that point, it appears that only two parameters are needed to fully characterize vortex profil.Pomeau-Rica's model offers the possibility to study the superfluid near crystallization and reveals the influence of the roton minimum's shape and depth on oscillations' amplitude. Results are subsequently compared to those given by Reatto's ab initio calculations. In Berloff-Roberts' model, profil displays a strong phase shift, which seems to be a non-physical consequence of the dispersion relation's shape at high frequencies. Energies reckoning leads us to think that oscillations carry a small fraction of the total vortex' energy, meaning that the kinetic energy is dominant.The order parameter for a vortex ring, whose radius is much larger than the interatomic distance, is calculated at zero and nonzero speed. Potential and kinetic energies are estimated and help us obtain the maximal speed reached by such a ring, depending on its radius and finally discussed this speed in regard to the Landau critical speed.

This Thesis is devoted to the investigation of superfluid helium flows due to torsional oscillators. In its first part, flow due to a torsionally oscillating disc suspended on a tungsten filament is studied (building upon the work of A. C. Hollis Hallett from 1952). Measurements of the motion of the torsionally oscillating disc were performed in superfluid helium at temperatures between 1.265 K and 2.157 K at saturated vapour pressure. Time traces of the disc angular deflection were obtained, and critical parameters related to the turbulent flow stability were determined. In laminar flow, scaling of drag forces with the dimensionless Donnelly number was verified. Based on these results, and comparison with the original work, a scenario of the decay of turbulent flow was suggested. The second part of this work is focused on the development and construction of a similar experiment for mK temperatures. A new type of oscillator was designed, a so-called "pillbox", and a series of testing measurements was performed both at room and mK temperatures.