Ce manuscrit présente les travaux que j'ai effectués au Laboratoire de Physique Théorique durant ma thèse. Ils portent sur l'interaction d'ondes de matière avec des réseaux optiques modulables en temps et en espace. L'utilisation de ces réseaux a permis de contrôler de manière cohérente les propriétés dynamiques d'un gaz d 'atomes ultra-froids. Cette étude théorique a été réalisée en collaboration avec le groupe Atomes Froids du Laboratoire LCAR. Les variations spatiales de l'enveloppe d u réseau créent, localement, des gaps spatiaux créant une cavité de Bragg pour onde de matière, dont nous avons étudié en détail les propriétés et qui a fait l'objet d'une réalisation expérimentale impliquant la propagation d'un condensat de Bose-Einstein de rubidium 85 dans un guide d'onde. Nous avons également étudié la propagation d'un nuage d 'atomes dans un réseau bichromatique qui permet de réaliser un simulateur quantique du modèle de Harper. Le spectre du hamiltonien de ce système a une dimension fractale pouvant être caractérisée nu­ mériquement. Nous avons montré, par ailleurs, qu'il est possible d'exploiter les interactions inter-atomiques répulsives d'un condensat de Bose-Einstein afin d'amplifier les corrélations position-vitesse lors de sa pro­ pagation dans un guide. Notre étude montre qu'une mesure des grandeurs dynamiques locales du nuage atomique permet de sonder expérimentalement les résonances d'un potentiel optique jusqu'à l'échelle du picoKelvin. Enfin, un nuage d'atomes en interaction attractive admet une solution d'équilibre : le soliton. Nous avons démontré, numériquement, que celui-ci peut être utilisé pour sonder des états liés d'un poten­ tiel de taille finie, en peuplant ces états lors d'une expérience de diffusion comme, par exemple, des états de surface. / This thesis presents the studies that I did at the Laboratoire de Physique Théorique. It concerns the interaction between matter waves and time and space depandant optical lattices. Using such lattices allows one to manipulate coherently the dynamical properties of ultra cold atoms. This theoretical study has been done in collaboration with the Cold Atoms group at the LCAR laboratory. The spatial variations of the lattice envelope locally create spatial gaps which create a Bragg cavity for matter waves. We have st udied in detail their properties and the cavity has been realized experimentally by using a Ru bid ium 85 Bose-Einstein condensate in a wave guide. We have also studied the propagation of an atomic cloud in a bichromatic optical lattice which allows us to make a quantum simulator of the Harper madel. The spectrum of the system Hamiltonian· posseses a fractal dimension which can be numerically characterized. We have also shawn that it is possible to use the repulsive interatomic interaction of a Bose-Einstein condensate in arder to amplify the momentum-position correlation during propagation in a guide. Our st udy shows that a mesure of local dynamical quantities of the atomic cloud enables one to experimentally probe resonances of an optical potential down to the picoKelvin scale. At last, an atomic cloud with attractive interactions admit a stable solution, the soliton. We have numerically demonstrated that this soliton can be used to probe bound states of a potential by populating those states through a scattering experiment, for example surface states.

An atom interferometer is a sensitive device that has potential for many useful applications. Atoms are sensitive to electromagnetic fields due to their electric and magnetic moments and their mass allows them to be deflected in a gravitational field, thereby making them attractive for measuring inertial forces. The narrow momentum distribution of Bose-Einstein condensate (BEC) is a great asset in realizing portable atom interferometers. An example is a guided-wave atom interferometer that uses a confining potential to guide the motion of the condensate. Despite the promise of guided-wave atom interferometry with BEC, spatial phase and phase diffusion limit the contrast of the interference fringes. The control of these phases is required for successful development of a BEC-based guided-wave atom interferometer. This thesis analyses the guided-wave atom interferometer, where an atomic BEC cloud at the center of a confining potential is split into two clouds that move along different arms of the interferometer. The clouds accumulate relative phase due to the environment, spatially inhomogeneous trapping potential and atom-atom interactions within the condensate. At the end of the interferometric cycle, the clouds are recombined producing a cloud at rest and moving clouds. The number of atoms in the clouds that emerge depends on the relative phase accumulated by the clouds during propagation. This is investigated by deriving an expression for the probability of finding any given number of atoms in the clouds that emerge after recombination. Characteristic features like mean, standard deviation and cross-correlation function of the probability density distribution are calculated and the contrast of the interference fringes is optimized. This thesis found that optimum contrast is achieved through the control of total population of atoms in the condensate, trap frequencies, s-wave scattering length, and the duration of the interferometric cycle.

A new technique for the interferometric measurement of an atomic wavefunction is introduced theoretically, which is able to extract phase and amplitude information in a single measurement. I focus on the application of this technique to the single-particle wavefunction of a Bose condensed cloud of rubidium atoms. The technique differs from existing techniques mainly in its simplicity, as it requires only a single laser beam to be added to a typical Bose-Einstein condensation apparatus. A second novel aspect is the consideration of condensate collisions with an optical potential in the low-intensity limit where the potential barrier may be viewed as a phase mask. The technique is then demonstrated experimentally. A related effect, the transient enhancement of momentum during a collision, first predicted by JG Muga et al., has also been demonstrated. Finally, significant redesign and construction of an apparatus to produce condensates of 87Rb is documented. The main result of this work is the production of pure condensates of up to 150k atoms which can be repeated every 45s. A calibration technique is devised and demonstrated, whereby copies of the condensate are made, and the copies are used to reduce the centre-of-mass momentum uncertainty of the interacting cloud by a factor of five.

A new technique for the interferometric measurement of an atomic wavefunction is introduced theoretically, which is able to extract phase and amplitude information in a single measurement. I focus on the application of this technique to the single-particle wavefunction of a Bose condensed cloud of rubidium atoms. The technique differs from existing techniques mainly in its simplicity, as it requires only a single laser beam to be added to a typical Bose-Einstein condensation apparatus. A second novel aspect is the consideration of condensate collisions with an optical potential in the low-intensity limit where the potential barrier may be viewed as a phase mask. The technique is then demonstrated experimentally. A related effect, the transient enhancement of momentum during a collision, first predicted by JG Muga et al., has also been demonstrated. Finally, significant redesign and construction of an apparatus to produce condensates of 87Rb is documented. The main result of this work is the production of pure condensates of up to 150k atoms which can be repeated every 45s. A calibration technique is devised and demonstrated, whereby copies of the condensate are made, and the copies are used to reduce the centre-of-mass momentum uncertainty of the interacting cloud by a factor of five.

Neste trabalho, estudamos o efeito da turbulência quântica na expansão livre do condensado de Bose-Einstein de 87Rb. O BEC é produzido em uma armadilha magnética tipo QUIC, em seguida o condensado é perturbado através da aplicação de um campo magnético. A superposição de campos magnéticos excita os modos coletivos em condições de baixa amplitude de excitação. No entanto para altas amplitudes de excitação, foi possível observar também outros efeitos, tais como: a variação do perfil de densidade e a diminuição na taxa de expansão dos átomos. A distribuição de momento dos átomos perturbados, indica a presença de turbulência quântica no superfluido caracterizada por uma cascata de energia dentro da faixa inercial. Os resultados da expansão livre do BEC mostram a variação do perfil de distribuição de densidade dos átomos evoluindo de um perfil Gaussiano a um perfil exponencial para altas amplitudes de excitação. O mesmo efeito foi observado ao aumentar o tempo de excitação na condição de baixas amplitudes de excitação. Além da variação do perfil de distribuição de densidade, a taxa da expansão dos átomos não perturbados é maior do que os átomos perturbados, apresentando a ocorrência de uma diminuição anisotrópica, significativa, da velocidade dos átomos durante a expansão livre. A diminuição da taxa de expansão pode estar relacionado com o fenômeno de localização durante a expansão livre dos átomos. / In this work, we study the effect of quantum turbulence on the free expansion of the Bose-Einstein condensate of 87Rb. The BEC is produced in a quic magnetic trap, then the condensate is disturbed by the application of a magnetic field. Superposition of magnetic fields excites collective modes under conditions of low amplitude of excitation. However, for high amplitudes of excitation, it was possible to observe other effects, such as: the variation of the density profile and the decrease in the rate of expansion of the atoms. The momentum distribution of the perturbed atoms indicates the presence of quantum turbulence in the superfluid characterized by a cascade of energy within the inertial band. The results of the free expansion of the BEC show the variation of the density distribution profile of the atoms evolving from a Gaussian profile to an exponential profile for high excitation amplitudes. The same effect was observed by increasing the excitation time in the condition of low excitation amplitudes. In addition to the variation of the density distribution profile, the rate of expansion of the undisturbed atoms is greater than the perturbed atoms, with the occurrence of a significant anisotropic decrease in the velocity of the atoms during free expansion. The decrease in the rate of expansion may be related to the localization phenomenon during the free expansion of the atoms.

This dissertation discusses a number of theoretical models of coupled bosonic modes, all closely related to the Bose-Hubbard dimer. In studying these models, we will repeatedly return to two unifying themes: the classical structure underlying quantum dynamics and the impact of weakly coupling a system to an environment. Or, more succinctly, semiclassical methods and open quantum systems. Our primary motivation for studying models such as the Bose-Hubbard is their relevance to ongoing ultracold atom experiments. We review these experiments, derive the Bose-Hubbard model in their context and briefly discuss its limitations in the first half of Chapter 1. In its second half, we review the theory of open quantum systems and the master equation description of the dissipative Bose-Hubbard model. This opening chapter constitutes a survey of existing results, rather than original work. In Chapter 2, we turn to the mean-field limit of the Bose-Hubbard model. After reviewing the striking localization phenomena predicted by the mean-field (and confirmed by experiment), we identify the first corrections to this picture for the dimer. The most interesting of these is the dynamical tunneling between the self-trapping points of the mean-field. We derive an accurate analytical expression for the tunneling rate using semiclassical techniques. We continue studying the dynamics near the self-trapping fixed points in Chapter 3, focusing on corrections to the mean-field that arise at larger nonlinearities and on shorter time scales than dynamical tunneling. We study the impact of dissipation on coherence and entanglement near the fixed points, and explain it in terms of the structure of the classical phase space. The last chapter of the dissertation is also devoted to a dissipative bosonic dimer model, but one arising in a very different physical context. Abandoning optical lattices, we consider the problem of formulating a quantum model of operation of the cylindrical anode magnetron, a vacuum tube crossed-field microwave amplifier. We derive an effective dissipative dimer model and study its relationship to the classical description. Our dimer model is a first step towards the analysis of solid-state analogs of such devices.

An apparatus for producing atomic-gas Bose-Einstein condensates (BECs) of 87-Rb atoms is described. The apparatus produces 87-Rb BECs in a dual-chamber vacuumsystem that incorporates magnetic transport of trapped atoms from the magneto-optical trapping cell to the BEC production cell via the operation of a series of overlapping magnet coils. The design, construction, and operation of the apparatus are described in detail.The apparatus is used to study the creation of quantized vortices in BECs by the merging and interference of multiple trapped condensates. In this experiment, a single harmonic potential well is partitioned into three sections by an optical barrier,enabling the simultaneous formation of three independent, uncorrelated BECs. The BECs may either merge together during their growth, or, for high-energy barriers, the BECs can be merged together by barrier removal after their formation. Either process may instigate vortex formation in the resulting BEC, depending on the initially indeterminate relative phases of the condensates and the merging rate.

Cooperative effects in many-particle systems can be exploited to achieve measurement outcomes not possible with independent probe particles. We explore two measurement applications based on the cooperative phenomenon of superradiance or on correlated quantum states closely related to superradiance. In the first application we study the off-resonant superradiant Raman scattering of light from an ultracold Bose atomic vapor. We investigate the temperature dependence of superradiance for a trapped vapor and show that in the regime where superradiance occurs on a timescale comparable to a trap frequency, scattering takes place preferentially from atoms in the lowest trap levels due to Doppler dephasing. As a consequence, below the critical temperature for Bose condensation, absorption images of transmitted light serve as a direct probe of the condensed state. Subsequently, we consider a pure condensate and study the time-dependent spatial features of transmitted light, obtaining good qualitative agreement with recent imaging experiments. Inclusion of quantum fluctuations in the initial stages of the superradiant emission accounts well for shot-to-shot fluctuations. Secondly, we have used simulated annealing, a global optimization strategy, to systematically search for correlated quantum interferometer input states that approach the Heisenberg limited uncertainty in estimates of the interferometer phase shift. That limit improves over the standard quantum limit to the phase sensitivity of interferometric measurements by a factor of 1√N, where N is the number of interfering particles. We compare the performance of these states to that of other non-classical states already known to yield Heisenberg limited uncertainty.

Neste trabalho, estudamos o efeito da turbulência quântica na expansão livre do condensado de Bose-Einstein de 87Rb. O BEC é produzido em uma armadilha magnética tipo QUIC, em seguida o condensado é perturbado através da aplicação de um campo magnético. A superposição de campos magnéticos excita os modos coletivos em condições de baixa amplitude de excitação. No entanto para altas amplitudes de excitação, foi possível observar também outros efeitos, tais como: a variação do perfil de densidade e a diminuição na taxa de expansão dos átomos. A distribuição de momento dos átomos perturbados, indica a presença de turbulência quântica no superfluido caracterizada por uma cascata de energia dentro da faixa inercial. Os resultados da expansão livre do BEC mostram a variação do perfil de distribuição de densidade dos átomos evoluindo de um perfil Gaussiano a um perfil exponencial para altas amplitudes de excitação. O mesmo efeito foi observado ao aumentar o tempo de excitação na condição de baixas amplitudes de excitação. Além da variação do perfil de distribuição de densidade, a taxa da expansão dos átomos não perturbados é maior do que os átomos perturbados, apresentando a ocorrência de uma diminuição anisotrópica, significativa, da velocidade dos átomos durante a expansão livre. A diminuição da taxa de expansão pode estar relacionado com o fenômeno de localização durante a expansão livre dos átomos. / In this work, we study the effect of quantum turbulence on the free expansion of the Bose-Einstein condensate of 87Rb. The BEC is produced in a quic magnetic trap, then the condensate is disturbed by the application of a magnetic field. Superposition of magnetic fields excites collective modes under conditions of low amplitude of excitation. However, for high amplitudes of excitation, it was possible to observe other effects, such as: the variation of the density profile and the decrease in the rate of expansion of the atoms. The momentum distribution of the perturbed atoms indicates the presence of quantum turbulence in the superfluid characterized by a cascade of energy within the inertial band. The results of the free expansion of the BEC show the variation of the density distribution profile of the atoms evolving from a Gaussian profile to an exponential profile for high excitation amplitudes. The same effect was observed by increasing the excitation time in the condition of low excitation amplitudes. In addition to the variation of the density distribution profile, the rate of expansion of the undisturbed atoms is greater than the perturbed atoms, with the occurrence of a significant anisotropic decrease in the velocity of the atoms during free expansion. The decrease in the rate of expansion may be related to the localization phenomenon during the free expansion of the atoms.

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