Beyond Gross-Pitaevskii Mean Field Theory

A large number of effects related to the phenomenon of Bose-Einstein Condensation (BEC) can be understood in terms of lowest order mean field theory, whereby the entire system is assumed to be condensed, with thermal and quantum fluctuations completely ignored. Such a treatment leads to the Gross-Pitaevskii Equation (GPE) used extensively throughout this book. Although this theory works remarkably well for a broad range of experimental parameters, a more complete treatment is required for understanding various experiments, including experiments with solitons and vortices. Such treatments should include the dynamical coupling of the condensate to the thermal cloud, the effect of dimensionality, the role of quantum fluctuations, and should also describe the critical regime, including the process of condensate formation. The aim of this Chapter is to give a brief but insightful overview of various recent theories, which extend beyond the GPE. To keep the discussion brief, only the main notions and conclusions will be presented. This Chapter generalizes the presentation of Chapter 1, by explicitly maintaining fluctuations around the condensate order parameter. While the theoretical arguments outlined here are generic, the emphasis is on approaches suitable for describing single weakly-interacting atomic Bose gases in harmonic traps. Interesting effects arising when condensates are trapped in double-well potentials and optical lattices, as well as the cases of spinor condensates, and atomic-molecular coupling, along with the modified or alternative theories needed to describe them, will not be covered here.Comment: Review Article (19 Pages) - To appear in 'Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment', Edited by P.G. Kevrekidis, D.J. Frantzeskakis and R. Carretero-Gonzalez (Springer Verlag

Laser spectroscopic characterization of the nuclear-clock isomer 229m^{229m}Th

The isotope $^{229}$Th is the only nucleus known to possess an excited state $^{229m}$Th in the energy range of a few electron volts, a transition energy typical for electrons in the valence shell of atoms, but about four orders of magnitude lower than common nuclear excitation energies. A number of applications of this unique nuclear system, which is accessible by optical methods, have been proposed. Most promising among them appears a highly precise nuclear clock that outperforms existing atomic timekeepers. Here we present the laser spectroscopic investigation of the hyperfine structure of $^{229m}$Th$^{2+}$, yielding values of fundamental nuclear properties, namely the magnetic dipole and electric quadrupole moments as well as the nuclear charge radius. After the recent direct detection of this long-searched-for isomer, our results now provide detailed insight into its nuclear structure and present a method for its non-destructive optical detection.Comment: 18 page