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

The A-B transition in superfluid helium-3 under confinement in a thin slab geometry

The influence of confinement on the topological phases of superfluid 3He is studied using the torsional pendulum method. We focus on the phase transition between the chiral A-phase and the time-reversal-invariant B-phase, motivated by the prediction of a spatiallymodulated (stripe) phase at the A-B phase boundary. We confine superfluid 3He to a single 1.08 {\mu}m thick nanofluidic cavity incorporated into a high-precision torsion pendulum, and map the phase diagram between 0.1 and 5.6 bar. We observe only small supercooling of the A-phase, in comparison to bulk or when confined in aerogel. This has a non-monotonic pressure dependence, suggesting that a new intrinsic B-phase nucleation mechanism operates under confinement, mediated by the putative stripe phase. Both the phase diagram and the relative superfluid fraction of the A and B phases, show that strong coupling is present at all pressures, with implications for the stability of the stripe phase.Comment: 6 figures, 1 table + supplemental informatio