Experimental Investigation of Three-Dimensional Single and Double optical Lattices

A complete laser cooling setup was built, with focus on threedimensional near-resonant optical lattices for cesium. These consist of regularly ordered micropotentials, created by the interference of four laser beams. One key feature of optical lattices is an inherent ”Sisyphus cooling” process. It efficiently extracts kinetic energy from the atoms, leading to equilibrium temperatures of a few µK. The corresponding kinetic energy is lower than the depth of the potential wells, so that atoms can be trapped. We performed detailed studies of the cooling processes in optical lattices by using the time-of-flight and absorption-imaging techniques. We investigated the dependence of the equilibrium temperature on the optical lattice parameters, such as detuning, optical potential and lattice geometry. The presence of neighbouring transitions in the cesium hyperfine level structure was used to break symmetries in order to identify, which role “red” and “blue” transitions play in the cooling. We also examined the limits for the cooling process in optical lattices, and the possible difference in steady-state velocity distributions for different directions. Moreover, in collaboration with ´Ecole Normale Sup´erieure in Paris, numerical simulations were performed in order to get more insight in the cooling dynamics of optical lattices. Optical lattices can keep atoms almost perfectly isolated from the environment and have therefore been suggested as a platform for a host of possible experiments aimed at coherent quantum manipulations, such as spin-squeezing and the implementation of quantum logic-gates. We developed a novel way to trap two different cesium ground states in two distinct, interpenetrating optical lattices, and to change the distance between sites of one lattice relative to sites of the other lattice. This is a first step towards the implementation of quantum simulation schemes in optical lattices.

Topics in geometry, analysis and inverse problems

The thesis consists of three independent parts. Part I: Polynomial amoebas We study the amoeba of a polynomial, as de ned by Gelfand, Kapranov and Zelevinsky. A central role in the treatment is played by a certain convex function which is linear in each complement component of the amoeba, which we call the Ronkin function. This function is used in two di erent ways. First, we use it to construct a polyhedral complex, which we call a spine, approximating the amoeba. Second, the Monge-Ampere measure of the Ronkin function has interesting properties which we explore. This measure can be used to derive an upper bound on the area of an amoeba in two dimensions. We also obtain results on the number of complement components of an amoeba, and consider possible extensions of the theory to varieties of codimension higher than 1. Part II: Differential equations in the complex plane We consider polynomials in one complex variable arising as eigenfunctions of certain differential operators, and obtain results on the distribution of their zeros. We show that in the limit when the degree of the polynomial approaches innity, its zeros are distributed according to a certain probability measure. This measure has its support on the union of nitely many curve segments, and can be characterized by a simple condition on its Cauchy transform. Part III: Radon transforms and tomography This part is concerned with different weighted Radon transforms in two dimensions, in particular the problem of inverting such transforms. We obtain stability results of this inverse problem for rather general classes of weights, including weights of attenuation type with data acquisition limited to a 180 degrees range of angles. We also derive an inversion formula for the exponential Radon transform, with the same restriction on the angle.

Type Ia Supernova Cosmology : Quantitative Spectral Analysis

Type Ia supernovae have been successfully used as standardized candles to study the expansion history of the Universe. In the past few years, these studies led to the exciting result of an accelerated expansion caused by the repelling action of some sort of dark energy. This result has been confirmed by measurements of cosmic microwave background radiation, the large-scale structure, and the dynamics of galaxy clusters. The combination of all these experiments points to a “concordance model” of the Universe with flat large-scale geometry and a dominant component of dark energy. However, there are several points related to supernova measurements which need careful analysis in order to doubtlessly establish the validity of the concordance model. As the amount and quality of data increases, the need of controlling possible systematic effects which may bias the results becomes crucial. Also important is the improvement of our knowledge of the physics of supernovae events to assure and possibly refine their calibration as standardized candle. This thesis addresses some of those issues through the quantitative analysis of supernova spectra. The stress is put on a careful treatment of the data and on the definition of spectral measurement methods. The comparison of measurements for a large set of spectra from nearby supernovae is used to study the homogeneity and to search for spectral parameters which may further refine the calibration of the standardized candle. One such parameter is found to reduce the dispersion in the distance estimation of a sample of supernovae to below 6%, a precision which is comparable with the current lightcurve-based calibration, and is obtained in an independent manner. Finally, the comparison of spectral measurements from nearby and distant objects is used to test the possibility of evolution with cosmic time of the intrinsic brightness of type Ia supernovae.