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.

Ultracold atoms have been successfully used to study numerous systems, previously unaccessible, but a precise control over the atom number of the sample still remains a challenge. This dissertation describes our progress towards achieving Fock states of atoms. The first three chapters cover the basic physics necessary to understand the techniques we use in our lab to manipulate atoms. We then summarize our experimental results from an earlier setup where we did two experiments. In the first experiment we compare the transport of cold atoms and a Bose Einstein Condensate (BEC) in a periodic potential. We find a critical potential height beyond which the condensate behavior deviates significantly from that of thermal atoms. In the second experiment we study the effect of periodic temporal kicks by a spatially periodic potential on a BEC in a quasi one dimensional trap. We observe a limit on the energy that the system can absorb from the kicks, which we conclude is due to the finite height of the trap rather than quantum effects. The majority of the dissertation discusses our experimental setup designed to produce Fock states. The setup is designed to use the method of laser culling to produce Fock states. We are able to create a BEC and transport it into a glass cell 25 cm away. We tried different innovative methods to reduce vibrations during transport before finally settling to a commercial air bearing translation stage. We create a high confinement one dimensional optical trap using the Hermite Gaussian TEM₀₁ mode of a laser beam. Such a trap gives trapping frequencies comparable to an optical lattice and allows us to create a single one dimensional trap. We creating the TEM₀₁ mode using an appropriate phase object (phase plate) in the path of a TEM₀₀ mode beam. The method for producing the phase plate was very well controlled to obtain a good quality mode. Once the atoms are loaded into this one dimensional trap we can proceed to do laser culling to observe Sub-Poissonian number statistics and eventually create Fock states of few atoms. Finally, we describe a novel method to create a real time tunable optical lattice which would provide us with the ability of spatially resolved single atom detection. The majority of the dissertation discusses our experimental setup designed to produce Fock states. The setup is designed to use the method of laser culling to produce Fock states. We are able to create a BEC and transport it into a glass cell 25 cm away. We tried different innovative methods to reduce vibrations during tr₀ansport before finally settling to a commercial air bearing translation stage. We create a high confinement one dimensional optical trap using the Hermite Gaussian TEM₀₁ mode of a laser beam. Such a trap gives trapping frequencies comparable to an optical lattice and allows us to create a single one dimensional trap. We creating the TEM₀₁ mode using an appropriate phase object (phase plate) in the path of a TEM₀₀ mode beam. The method for producing the phase plate was very well controlled to obtain a good quality mode. Once the atoms are loaded into this one dimensional trap we can proceed to do laser culling to observe Sub-Poissonian number statistics and eventually create Fock states of few atoms. Finally, we describe a novel method to create a real time tunable optical lattice which would provide us with the ability of spatially resolved single atom detection. The majority of the dissertation discusses our experimental setup designed to produce Fock states. The setup is designed to use the method of laser culling to produce Fock states. We are able to create a BEC and transport it into a glass cell 25 cm away. We tried different innovative methods to reduce vibrations during transport before finally settling to a commercial air bearing translation stage. We create a high confinement one dimensional optical trap using the Hermite Gaussian TEM₀₁ mode of a laser beam. Such a trap gives trapping frequencies comparable to an optical lattice and allows us to create a single one dimensional trap. We creating the TEM₀₁ mode using an appropriate phase object (phase plate) in the path of a TEM₀₀ mode beam. The method for producing the phase plate was very well controlled to obtain a good quality mode. Once the atoms are loaded into this one dimensional trap we can proceed to do laser culling to observe Sub-Poissonian number statistics and eventually create Fock states of few atoms. Finally, we describe a novel method to create a real time tunable optical lattice which would provide us with the ability of spatially resolved single atom detection. The majority of the dissertation discusses our experimental setup designed to produce Fock states. The setup is designed to use the method of laser culling to produce Fock states. We are able to create a BEC and transport it into a glass cell 25 cm away. We tried different innovative methods to reduce vibrations during transport before finally settling to a commercial air bearing translation stage. We create a high confinement one dimensional optical trap using the Hermite Gaussian TEM₀₁ mode of a laser beam. Such a trap gives trapping frequencies comparable to an optical lattice and allows us to create a single one dimensional trap. We creating the TEM₀₁ mode using an appropriate phase object (phase plate) in the path of a TEM₀₀ mode beam. The method for producing the phase plate was very well controlled to obtain a good quality mode. Once the atoms are loaded into this one dimensional trap we can proceed to do laser culling to observe Sub-Poissonian number statistics and eventually create Fock states of few atoms. Finally, we describe a novel method to create a real time tunable optical lattice which would provide us with the ability of spatially resolved single atom detection. / text

This dissertation presents several experimental methods that were devised to generate or manipulate quantized vortices in highly oblate dilute-gas Bose-Einstein condensates (BECs). Studies that involve single vortex dynamics, vortex-vortex interactions, and vortex-impurity interactions are essential in developing a deeper understanding of the nature of superfluidity and in particular, superfluid turbulence. In highly oblate systems, vortex dynamics have a two-dimensional (2D) nature and the resulting superfluid characteristics may be substantially different from those in three-dimensional (3D) superfluids. However, there have been remarkably few experimental studies of 2D vortex dynamics in superfluids. Therefore, to study 2D vortex dynamics and interactions, it is necessary to first develop experimental methods that can generate vortices and vortex distributions in nominally 2D systems, such as highly oblate BECs. Four main experiments are discussed in this dissertation. Two of these experiments generate multiple singly quantized vortices in a relatively stochastic manner leading to disordered vortex distributions. From these two vortex methods, the physics of high vorticity and highly disordered systems may be observed and studied in a highly oblate system. These methods may prove useful in studies of 2D quantum turbulence. The other two experiments involve newly developed techniques for controlled generation and manipulation of vortices. One of these methods creates multiply quantized pinned vortices with a control in the generated vorticity. The other method reliably creates a pair of singly quantized vortices of opposite circulation, whose positions can be easily manipulated after creation, such that they can be placed in any location within the BEC. The two techniques may be scalable to higher number of vortices and may prove useful in superfluid dynamics and vortex interactions that require repeatable vortex distributions. Taken together, these tools and methods may be applicable to many further studies of vortex physics in highly oblate BECs.

Nesta tese utilizamos o Modelo de Bose-Hubbard (MBH) generalizado para duas espécies bosonicas para investigar a estabilidade dinâmica da fase superfluida de uma mistura binaria de átomos bosonicos ultra-frios confinados em uma rede optica periódica anelar com M sítios. Na primeira parte consideramos a Hamiltoniana do MBH sem a presença do tunelamento inter-especies. Deduzimos e resolvemos as equações de Gross-Pitaevskii para os estados de equilíbrio do MBH e mostramos que são misturas binarias de condensados nos quais os átomos de cada espécie ocupam um estado de quase-momento q bem definido. As excitações elementares foram determinadas resolvendo as equações de Bogoliubov-de Gennes o que foi possível graças a estrutura de acoplamento dos quase-momentos que reduziu a Hamiltoniana Efetiva a uma soma direta de um dubleto e quadrupletos. Através da analise do comportamento das energias de excitação como função dos parâmetros de controle do sistema, investigamos a estabilidade dinâmica de dois casos de misturas de condensados onde, em um caso, os átomos de cada espécie ocupam o mesmo estado de quase-momento, qA = qB e, no outro, quase-momentos opostos, qA = qB. Em ambos os casos as condições de estabilidade dependem do quase-momento q estar nos quartos centrais ou laterais da primeira zona de Brillouin. No caso qA = qB vemos que a forma do diagrama de estabilidade independe do quase-momento do condensado. Por outro lado, o mesmo não ocorre nos condensados contra-propagantes qA = qB. Esta diferença fica mais acentuada no limite termodinâmico onde os diagramas de estabilidade no centro e nas extremidades da primeira zona de Brillouin ficam idênticos nos dois casos. Já nas bordas que separam os quartos centrais e laterais o comportamento ´e diferente pois a presença de uma interação interespécies por menor que seja desestabiliza completamente a mistura com qA = qB. Em todos estes casos ficou evidente o papel desestabilizador da interação interespécies. Na segunda parte consideramos o efeito de um termo de tunelamento inter-especies. As soluções das equações de Gross-Pitaevskii revelam uma estrutura biestável de estados de equilíbrio essencial para a ocorrência de bifurcação no sistema e, portanto, a presença de catástrofe. Investigamos se a catástrofe e acessível a uma observação experimental. De acordo com nosso critério, esta observação e impossível se o plano de bifurcação for a fronteira de um domínio de instabilidade dinâmica. Através da analise da estabilidade dinâmica dos estados de equilíbrio vimos que para um sistema invariante por inversão de cor essa resposta depende apenas da razão entre as intensidades de tunelamento intra e inter-especies de modo que se JAB/J > 1 a observação e impossível e se JAB/J < 1 é possível, supondo existir uma rota adiabática ate a bifurcação. / In this thesis we used the two-component Bose-Hubbard Model (BHM) to investigate the dynamical stability of the superfluid phase of a binary mixture of ultra-cold bosonic atoms confined in a ring-shaped periodic optical lattice with M sites. In the first part we considered the BHM Hamiltonian without the presence of interspecies tunnelling. We deduced and solved the Gross-Pitaevskii equations for the equilibrium states of the BHM and showed that they are binary mixtures of condensates where the atoms of each species occupy a state of well defined quasi-momentum q. The elementary excitations were determined solving the Bogoliubov-de-Gennes equations which was possible thanks to the coupling structure of the quasi-momenta that reduced the Effective Hamiltonian to a direct sum of a doublet and quadriplets. Through the analysis of the behavior of the excitation energies as a function of the control parameters of the system, we investigated the dynamical stability of two cases of mixtures of condensates where, in one case, the atoms of each specie occupy the same state of quasi-momentum, qA = qB, and, in the other, opposite quasi-momentum, qA = qB. In both cases the stability conditions depend of the quasi-momentum q to be in the central or lateral quarters of the first Brillouin zone. In the case qA = qB, we see that the form of the stability diagram is not dependent of the quasi-momentum of the condensate. However, the same does not occur in the counter-propagating condensates qA = qB. This difference is accentuated in the thermodynamic limit where the stability diagrams in the center and in the extremities of the first Brillouin zone are identical in both cases. In the borders that separate the central and lateral quarters the behavior is different because the presence of a slightly non vanishing inter-species interaction completely destabilize the mixture with qA = qB. In all these cases it was evident the destabilizing role of the inter-species interaction. In the second part we considered the effect of a inter-species tunnelling term. The solutions of the Gross-Pitaevskii equations reveal a bi-stable structure of equilibrium states that is essential for the occurrence of the bifurcation in the system and, therefore, the presence of catastrophe. We investigated if the catastrophe is accessible to a experimental observation. According to our criteria, this observation is impossible if the bifurcation plane is the frontier of a dynamical instability domain. Through the analysis of the dynamical stability of the equilibrium states we saw that for a system invariant by color inversion this answer depends only on the ratio between the intra and inter-species tunnelling intensities in a way that if JAB/J > 1 the observation is impossible and if JAB/J < 1 it is possible, supposing that it exists an adiabatic route until the bifurcation.

O presente trabalho apresenta o estudo das propriedades da condensação de Bose-Einstein e da superfluidez em um sistema bosônico disposto em um arranjo unidimensional de potenciais periódicos em formato de anel. O Hamiltoniano efetivo usual em termos dos operadores de campo é implementado na representação construída em termos das funções de Bloch da primeira banda e o problema é resolvido por meio da sua diagonalização através de métodos numéricos. No limite de hopping pequeno, este modelo é essencialmente equivalente à representação usual do modelo de Bose-Hubbard, mas incorpora efeitos adicionais através das energias de Bloch de partícula independente e dos elementos da matriz de dois corpos na situação em que o hopping é grande [19]. Através da inclusão de rotação no sistema, as energias de partícula independente são forçadas a depender da velocidade angular. Isto implica, correspondentemente, uma dependência da velocidade angular nas funções de onda de partícula independente e nos resultados de muitos corpos obtidos através da diagonalização do Hamiltoniano. Com o objetivo de estudar a superfluidez, o critério de dois fluidos é empregado e através de resultados numéricos obtêm-se a variação da fração de superfluido com o quadrado da velocidade angular. Ainda, considera-se aqui uma expressão perturbativa para o parâmetro inercial do sistema expresso em termos das excitações do sistema sem rotação, o que permite relacionar as energias do sistema com rotação com aquelas do sistema sem rotação. Isto é particularmente interessante para obter a fração de superfluido em termos da informação espectral do sistema sem rotação. Resultados semelhantes podem ser encontrados através da definição de superfluido baseada na resposta do sistema a uma variação de fase, imposta através de condições de contorno torcidas [30, 33], mas com a diferença de que os desenvolvimentos aqui não fazem uso da hipótese do modo condensado. De maneira geral, os resultados numéricos obtidos indicam, que pelo menos para este sistema, as frações de superfluido e condensado são quantidades sem relação direta, sugerindo então que mesmo para sistemas gasosos diluídos a idéia de que a superfluidez é uma consequência da condensação de Bose-Einstein deve ser considerada com mais cuidado. / In this work we study the properties of Bose-Einstein condensation and superfluidity in a finite bosonic system in a 1-dimensional ring with a periodic potential under rotation. The usual field effective Hamiltonian is implemented in a representation constructed in terms of the first band Bloch functions and the problem is solved by numeric diagonalization. In the limit of small hopping, this model is essentially equivalent to the quasi-momentum representation of the usual Bose-Hubbard model but incorporates additional effects via Bloch single particle energies and two-body matrix elements in the case of large hopping [19]. By including rotation in the system we force the single particle energies to be a function of the angular velocity. This implies a corresponding angular velocity dependence of the single particle wavefunctions and many-body diagonalization results. In order to study superfluidity, we consider the two fluid criterion. Numerical results for the superfluid fraction involving the change of in rinsic ground state energy with the square of the angular velocity are obtained. We also consider a perturbative expression for the system inertial parameter expressed in terms of the excitation spectrum of the non rotating system, which enables us to relate the energies in the rotating system to the ones in the system without rotation. This is particularly interesting for obtaining superfluid fraction in terms of spectral information of the non rotating system. Similar results can be found by using the definition of superfluid fraction based on the response of the system to a phase variation imposed by means of twisted boundary conditions [30, 33], but with the difference that our developments do not assume the hypothesis of a condensate mode. Our numerical results indicate that in this system condensate and superfluid fractions are quite unrelated in terms of parameter values, indicating that even for dilute gases the concept that superfluidity is a consequence of Bose-Einstein condensation should be considered more carefully.

This thesis details the design, construction and characterisation of an ultracold atoms system, developed in conjunction with a flexible optical trapping scheme which utilises a Liquid Crystal Spatial Light Modulator (LC SLM). The ultracold atoms system uses a hybrid trap formed of a quadrupole magnetic field and a focused far-detuned laser beam to form a Bose-Einstein Condensate of 2×105 87Rb atoms. Cold atoms confined in several arbitrary optical trapping geometries are created by overlaying the LC SLM trap on to the hybrid trap, where a simple feedback process using the atomic distribution as a metric is shown to be capable of compensating for optical aberrations. Two novel methods for creating flexible optical traps with the LC SLM are also detailed, the first of which is a multi-wavelength technique which allows several wavelengths of light to be smoothly shaped and applied to the atoms. The second method uses a computationally-efficient minimisation algorithm to create light patterns which are constrained in both amplitude and phase, where the extra phase constraint was shown to be crucial for controlling propagation effects of the LC SLM trapping beam.

Condensados de Bose-Einstein (BEC) são sistemas macroscópicos excelentes para a observação do comportamento quântico da matéria. Desde sua obtenção experimental em gases atômicos alcalinos diluídos aprisionados por campos magnéticos, há importantes aspectos relacionados a esse sistema que foram intensamente explorados, como os modos coletivos do BEC harmonicamente aprisionado, seu tunelamento através de barreiras de potencial e os estados excitados desse sistema, incluindo vórtice e sóliton. O último consiste de pacote de onda localizado, que propaga sem mudança de forma. Nesse trabalho, investigamos os novos aspectos que surgem da dinâmica de um sistema composto (condensado aprisionado contendo um sóliton). Há muitos estudos tratando cada parte separadamente: estado fundamental do BEC ou um sóliton em um BEC infinito uniforme estacionário. Estamos nos baseando nessas análises prévias, além da simulação numérica de campo médio do nosso sistema submetido a diferentes condições iniciais (BEC aprisionado no mínimo do potencial harmônico ou BEC deslocado na armadilha contendo um sóliton, além de uma deformação no potencial) para caracterizar a dinâmica desse sistema. Alguns dos nossos resultados puderam ser explicados por meio de predições analítica da chamada aproximação de Thomas-Fermi. Ao final, comparamos as simulações de campo médio (equação de Gross-Pitaevskii) com as advindas da teoria de múltiplos orbitais a fim de justificar o regime de validade da nossa teoria. / Bose-Einstein Condensates (BEC) are excellent macroscopic systems to observe the quantum behavior of matter. Since it experimental production in dilute atomic alkali gases trapped by magnetic fields, there are important aspects related to this system that have been intensely explored, like the collective modes of the harmonically trapped BEC, its tunneling through a potential barrier and the excited states of this system, that include the vortex and soliton. The latter consist of localized disturbances, which propagate without change of form. In this work, we investigate the singular aspects that coming from the dynamics of a composite system (trapped BEC containing a soliton). There are many studies that treat each part separately, that include a fundamental state BEC or a soliton inside a uniform infinite extent stationary BEC. We are basing on these previous analyses, besides mean-field numeric simulating our particular system submitted to diferent initial conditions (minimum harmonic potential trapped BEC or dislocated trapped BEC plus a soliton, in addition to a deformation in the potential) to characterize the tunneling dynamics. Some of our results could be explained using analytical predictions of the so called Thomas-Fermi approximation. At the end, we compar the meanfield simulations (Gross-Pitavskii equation) with the simulations from the multiple orbitals theory to justify the validity regime of our theory.

Estudamos as excitações de baixa energia, presentes em um gás de bosons homogêneo, de spin nulo, sujeitos a uma interação de dois corpos repulsiva e a temperatura zero, utilizando a aproximação gaussiana, que consiste num caso particular de aproximação de campo médio. As equações dinâmicas resultantes foram linearizadas ao redor da solução estática de Hartree-Fock-Bogoliubov. Obtivemos uma banda contínua e limitada inferiormente, além de um segundo ramo discreto, que define um limite inferior para as excitações e que, ao contrário do resultado proveniente do tratamento de Hartree-Fock-Bogoliubov, possui um comportamento linear sem gap com respeito ao momento da excitação no limite de grandes comprimentos de onda, ou seja, possui uma equação de dispersão do tipo fônon. Discutimos também a forma através da qual é possível gerar desvios do equilíbrio, vinculados aos estados excitados, e concluímos haver restrições sobre os possíveis desvios das grandezas características em campo médio gaussiano, quando tais desvios são gerados por transformações infinitesimais unitárias de um corpo tomadas até primeira ordem. / We study low-lying excitations of a spinless, homogeneous bose gas, with repulsive interaction, at zero temperature, in terms of a gaussian mean field approximation. The dynamical equations of this approximation have been linearized in small displacements from the well known static Hartree-Fock-Bogoliubov solution. We obtain a gapped continous band of excitations above a discrete branch with phonon behavior at large wavelengths. We also discuss the allowed forms of excitations and conclude that restrictions exist for the allowed deviations of the general set of gaussian mean field parameters, when they are generated in first orders by infinitesimal unitary transformations.

L'interférométrie atomique a démontré sa capacité à effectuer des mesures de grande précision, notamment pour la réalisation de capteurs inertiels, les tests de physique fondamentale ou la mesure de constantes fondamentales. Une piste pour l'amélioration de la sensibilité des interféromètres atomiques est la réduction de la dispersion en vitesse de la source en utilisant un ensemble d'atomes ultra-froids pour augmenter le temps d'interrogation des atomes et accroitre la séparation spatiale entre les bras de l'interféromètre. Un nouvel interféromètre atomique à bras séparés est en construction au Laboratoire Collisions Agrégats et Réactivité de Toulouse. Ce dispositif répond à deux objectifs. Premièrement sa conception a pour but l'étude et le développement de nouveaux types de sources de condensat de Bose-Einstein (C.B.E.) double espèce de rubidium 85 et 87 adaptées à l'interférométrie. Cette source de C.B.E. repose sur l'utilisation de puces pour la manipulation et le refroidissement des atomes. Cette technologie est compacte et consomment peu d'énergie, ce qui est adaptée aux applications spatiales. L'autre objectif est d'utiliser cet interféromètre pour tester la neutralité de la matière via l'effet Aharonov-Bohm scalaire. Dans ce manuscrit je commence par exposer et justifer les choix techniques fait lors du dimensionnement et de la construction de la source de C.B.E. double isotopes. Par la suite, je présente les premiers résultats expérimentaux accompagnés de simulations numériques et d'explications théoriques. Lors de la première étape de refroidissement laser nous produisons un nuage de rubidium 87 et 85 contenant 4 × 10^10 atomes à une température de 10 µK avec un taux de cycle de 1 s. A la suite du refroidissement laser 8 × 10^9 atomes sont chargés dans le piège magnétique millimétrique de surface. Différentes expériences de caractérisation sont réalisées et expliquées à la lumières de simulations numériques. L'étude des fréquences de piégeage et de la profondeur a révélé les limites du premier prototype de piège millimétrique que nous avons réalisé au laboratoire. Cependant ces développements expérimentaux et théoriques servent à développer et implémenter dans le dispositif une nouvelle génération de puce à échelle micrométrique. / Atom interferometry has shown its interest for high precision measurements, such as inertial sensors, tests of fundamental physics or fundamental constant measurements. A way to improve sensitivity of such device is to reduce speed dispersion of the atomic cloud. The use of ultra-cold atoms allows increasing the interogation time of atoms and the spatial separation between the interferometer arms. The building of a new atom interferometer with separated arms is ongoing in the laboratory "Collisions Agrégats et Réactivité" at Toulouse. This new setup must meet two objectives. One aim of its conception is to study and develop a new kind of double species Bose-Einstein condensate (B.E.C.) source for atom interferometry with rubidium 87 and 85. This B.E.C. source relies on atom chip technology to cool down and manipulate atoms. This technology is compact and low power consuming, therefore suitable for transportable applications in space. A second aim is to use this interferometer to fix new boundary on the experimental value of atom neutrality thanks to the scalar Aharonov-Bohm effect. In this manuscript I start by exposing and justifying technical choices made for the design of the double isotope B.E.C. source. Then I present the first experimental results compared with numerical simulations and theoretical explanations. During the first laser cooling stage we produce a cloud including 4 × 10^10 rubidium atoms of both isotopes (87 and 85) at 10 µK. This operation can be repeated every second. Following the laser cooling 8×10^9 atoms are loaded into a millimeter sized magnetic trap. Various experiments were performed to characterize the trap. Studies of the trap frequency and depth revealed the limitations of this first prototype. However these theoretical and experimental developments led to design and future implementation of a new generation of micro-chip in our apparatus.

Estudamos as excitações de baixa energia, presentes em um gás de bosons homogêneo, de spin nulo, sujeitos a uma interação de dois corpos repulsiva e a temperatura zero, utilizando a aproximação gaussiana, que consiste num caso particular de aproximação de campo médio. As equações dinâmicas resultantes foram linearizadas ao redor da solução estática de Hartree-Fock-Bogoliubov. Obtivemos uma banda contínua e limitada inferiormente, além de um segundo ramo discreto, que define um limite inferior para as excitações e que, ao contrário do resultado proveniente do tratamento de Hartree-Fock-Bogoliubov, possui um comportamento linear sem gap com respeito ao momento da excitação no limite de grandes comprimentos de onda, ou seja, possui uma equação de dispersão do tipo fônon. Discutimos também a forma através da qual é possível gerar desvios do equilíbrio, vinculados aos estados excitados, e concluímos haver restrições sobre os possíveis desvios das grandezas características em campo médio gaussiano, quando tais desvios são gerados por transformações infinitesimais unitárias de um corpo tomadas até primeira ordem. / We study low-lying excitations of a spinless, homogeneous bose gas, with repulsive interaction, at zero temperature, in terms of a gaussian mean field approximation. The dynamical equations of this approximation have been linearized in small displacements from the well known static Hartree-Fock-Bogoliubov solution. We obtain a gapped continous band of excitations above a discrete branch with phonon behavior at large wavelengths. We also discuss the allowed forms of excitations and conclude that restrictions exist for the allowed deviations of the general set of gaussian mean field parameters, when they are generated in first orders by infinitesimal unitary transformations.