6 research outputs found

### Intrinsic Riemannian Functional Data Analysis for Sparse Longitudinal Observations

A novel framework is developed to intrinsically analyze sparsely observed Riemannian functional data. It features four innovative components: a frame-independent covariance function, a smooth vector bundle termed covariance vector bundle, a parallel transport and a smooth bundle metric on the covariance vector bundle. The introduced intrinsic covariance function links estimation of covariance structure to smoothing problems that involve raw covariance observations derived from sparsely observed Riemannian functional data, while the covariance vector bundle provides a rigorous mathematical foundation for formulating the smoothing problems. The parallel transport and the bundle metric together make it possible to measure fidelity of fit to the covariance function. They also plays a critical role in quantifying the quality of estimators for the covariance function. As an illustration, based on the proposed framework, we develop a local linear smoothing estimator for the covariance function, analyze its theoretical properties, and provide numerical demonstration via simulated and real datasets. The intrinsic feature of the framework makes it applicable to not only Euclidean submanifolds but also manifolds without a canonical ambient space.Comment: 36 pages, 8 figure

### Spin fragmentation of Bose-Einstein condensates with antiferromagnetic interactions

We study spin fragmentation of an antiferromagnetic spin 1 condensate in the presence of a quadratic Zeeman (QZ) effect breaking spin rotational symmetry. We describe how the QZ effect turns a fragmented spin state, with large fluctuations of the Zeemans populations, into a regular polar condensate, where atoms all condense in the $m=0$ state along the field direction. We calculate the average value and variance of the Zeeman state $m=0$ to illustrate clearly the crossover from a fragmented to an unfragmented state. The typical width of this crossover is $q \sim k_B T/N$, where $q$ is the QZ energy, $T$ the spin temperature and $N$ the atom number. This shows that spin fluctuations are a mesoscopic effect that will not survive in the thermodynamic limit $N\rightarrow \infty$, but are observable for sufficiently small atom number.Comment: submitted to NJ

In this thesis, we study theoretically and in experimentally the properties of spin-1 Bose-Einstein condensates with antiferromagnetic interactions realized in the ultra-cold sodium gases confined in optical dipole traps. We present in detail how to realize, diagnose and control the spinor Bose-Einstein condensates in our experiment. In order to describe the condensate, a mean-field theory is first adopted. This theory predicts a phase transition when changing the magnetic field $B$. The predictions of the mean-field theory agree very well with most of our experimental results, including the values of the critical magnetic field $B_c$, asymptotic values of $n_0$ (relative atom number in $m_F=0$ state) at large $B$ and the phase diagram of $n_0$. However, for small magnetization and small magnetic fields, we find abnormally large fluctuations (super-Poissonian) of $n_0$. In order to describe these large fluctuations, we develop a full quantum statistical description for the spinor condensate at finite temperature. To describe uncondensed thermal atoms (also present in the same trap), we use the semi-ideal" Hartree-Fock approximation to deal with the interactions between the condensate atoms and the thermal ones. The experimental results lead us to introduce two kinds of temperatures, the spin temperature" $T_s$ and the kinetic temperature" $T_k$ with $T_s\ll T_k$, characterizing respectively the fluctuaitons of the condensate spin and the thermal gas. We conclude that the system is reaching a quasi-equilibrium states, where different degrees of freedom reach equilibrium by separate mechanisms but where the mutual thermalization does not occur over the lifetime of the cloud.Nous prÃ©sentons en dÃ©tail comment rÃ©aliser, analyser et contrÃ´ler de tels condensats spineurs. Afin de dÃ©crire le condensat, nous adoptons d'abord une thÃ©orie du champ moyen. Cette thÃ©orie prÃ©dit une transition de phase quand on change le champ magnÃ©tique $B$. Dans la plupart des cas, les prÃ©dictions par cette thÃ©orie du champ moyen s'accordent trÃ¨s bien avec la plupart des nos rÃ©sultats expÃ©rimentaux, incluant les valeurs de champ magnÃ©tique critique Bc, les valeurs asymptotiques de $n_0$ (nombre d'atome relative en Ã©tat $m_F=0$) en grand B et le diagramme de phase de n0. Cependant, pour des magnÃ©tisations faibles et des champs magnÃ©tiques petits, nous constatons des fluctuations anormalement grandes (super-Poissoniennes) des populations des Ã©tats Zeeman individuels. Pour comprendre leur origine et dÃ©crire ces systÃ¨mes fluctuants, nous dÃ©veloppons une description statistique quantique des condensat spineur Ã  tempÃ©rature finie. Pour dÃ©crire le nuage thermique prÃ©sent Ã©galement dans le piÃ¨ge, nous utilisons l'approximation Hartree-Fock semi-idÃ©ale" pour traiter les interactions entre les condensats et les nuages thermiques. Les rÃ©sultats expÃ©rimentaux nous amÃ¨nent Ã  introduire deux types des tempÃ©ratures: la tempÃ©rature de spin" T_s et la tempÃ©rature cinÃ©tique" T_k$, avec$T_s\ll T_k\$, qui caractÃ©risent les fluctuations du spin du condensat et le nuage thermique, respectivement. On en conclut que le systÃ¨me se trouve dans un Ã©tat de quasi-Ã©quilibre , ou diffÃ©rents degrÃ©s de libertÃ© s'Ã©quilibrent sÃ©parÃ©ment mais leur thermalisation mutuelle ne s'opÃ¨re pas sur la durÃ©e de vie du systÃ¨me