Excitations in superfluids of atoms and polaritons

This thesis is devoted to the study of excitations in atomic and polariton Bose-Einstein condensates (BEC). These two specimens are prime examples for equilibrium and non equilibrium BEC. The corresponding condensate wave function of each system satisfies a particular partial differential equation (PDE). These PDEs are discussed in the beginning of this thesis and justified in the context of the quantum many-body problem. For high occupation numbers and when neglecting quantum fluctuations the quantum field operator simplifies to a semiclassical wave. It turns out that the interparticle interactions can be simplified to a single parameter, the scattering length, which gives rise to an effective potential and introduces a nonlinearity to the PDE. In both cases, i.e. equilibrium and non equilibrium, the main model corresponding to the semiclassical wave is the Gross-Pitaevskii equation (GPE), which includes certain mathematical adaptions depending on the physical context of the consideration and the nature of particles/quasiparticles, such as additional complex pumping and growth terms or terms due to motion. In the course of this work I apply a variety of state-of-the-art analytical and numerical tools to gain information about these semiclassical waves. The analytical tools allow e.g. to determine the position of the maximum density of the condensate wave function or to find the critical velocities at which excitations are expected to be generated within the condensate. In addition to analytical considerations I approximate the GPE numerically. This allows to gain the condensate wave function explicitly and is often a convenient tool to study the emergence of excitations in BEC. It is in particular shown that the form of the possible excitations significantly depends on the dimensionality of the considered system. The generated excitations within the BEC include quantum vortices, quantum vortex rings or solitons. In addition multicomponent systems are considered, which enable more complex dynamical scenarios. Under certain conditions imposed on the condensate one obtains dark-bright soliton trains within the condensate wave function. This is shown numerically and analytical expressions are found as well. In the end of this thesis I present results as part of an collaborative effort with a group of experimenters. Here it is shown that the wave function due to a complex GPE fits well with experiments made on polariton condensates, statically and dynamically.F.P. acknowledges financial support by the UK Engineering and Physical Sciences Re- search Council (EPSRC) grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis and a KAUST gran

A Fast Rotating Bose-Einstein Condensate on a Disc

Nicht angegebe

pp-adic quotient sets

For $A \subseteq \mathbb{N}$, the question of when $R(A) = \{a/a' : a, a' \in A\}$ is dense in the positive real numbers $\mathbb{R}_+$ has been examined by many authors over the years. In contrast, the $p$-adic setting is largely unexplored. We investigate conditions under which $R(A)$ is dense in the $p$-adic numbers. Techniques from elementary, algebraic, and analytic number theory are employed in this endeavor. We also pose many open questions that should be of general interest.Comment: 24 page