Theoretical Investigations of Trapped Ultra-Cold Bose Gases: Rotating Bose-Einstein Condensation and Anomalous Segregation

Abstract

In this thesis we present two theoretical investigations of ultra-cold Bose gases. In Part I we extend the stochastic Gross-Pitaevskii theory of Bose-Einstein condensates (BECs) developed by Gardiner et al. [1,2] to describe a partially condensed Bose gas in contact with a rotating thermal cloud. We derive a stochastic Gross-Pitaevskii equation (SGPE) that describes the condensate and a set of low energy non-condensed states. The SGPE is derived within the truncated Wigner approximation, leading to a diffusion equation for the effects of collisions with the thermal cloud. The formalism involves orthogonal projection into a low energy subspace of the confining potential. We derive some useful consistency conditions satisfied by ensemble averaged observables of stochastic simulations. These generalised Ehrenfest relations are also used as non-perturbative criteria for assessing the influence of the projectors. We then develop an efficient and accurate computational algorithm to evolve the SGPE in a rotating frame of reference. The projectors are explicitly included which resolves problems associated with aliasing and periodic boundary conditions that arise from using Fourier grid methods in this regime. Using this formalism we examine the effect of vacuum and thermal fluctuations on vortex lattice formation. Using an idealized model, we find that the fluctuations lead to faster lattice formation than the ordinary Gross-Pitaevskii equation seeded by weak initial noise. In particular, the vacuum noise initiates Landau-Beliaev damping processes required for the system to proceed to a new equilibrium. It is shown that thermal fluctuations do not significantly alter the rate of this process, but they do change the resulting equilibrium state. We then study the process of Bose-Einstein condensation from a rotating thermal cloud. We model a rapid quench below threshold by evolving samples from the non-interacting Wigner distribution according to the SGPE. Each sample contains phase defects which are inherited by the condensate, and individual trajectories evolve into a BEC containing a regular vortex lattice in equilibrium with the rotating thermal cloud. We are thus able to qualitatively verify the rapid quench limit of the Kibble-Zurek mechanism [3,4] of defect pinning for Bose-Einstein condensation from a rotating thermal cloud. The simulations indicate that this mechanism is central to the condensation process in the rotating Bose-Einstein condensation experiments of Haljan et al. [5]. In Part II we investigate the experiment of Lewandowski et al. [6] who studied non-condensed ultra-cold 87Rb using Ramsey spectroscopy. In the experiment the gas is placed into a coherent superposition of two hyperfine states by driving a particular two photon transition. After switching off the coupling laser, the two hyperfine states exhibit transient spatial separation in the trap, the so-called 'anomalous segregation'. We derive a set of coupled equations of motion for the phase space amplitudes of the two states and the correlation between them, and then simulate the evolution subject to a relaxation time approximation for the coherence of the superposition. The simulations accurately reproduce the dynamics observed in the experiment. To assess the damping effect of collisions we derive a modified Quantum Boltzmann equation for the damping which includes the effect of coherence and the collisions between the two internal states. The leading contribution to the damping time is much longer than the timescale of segregation - a consequence of the near degeneracy of the S-wave scattering lengths of the hyperfine states. We also consider the effect of interactions on the Ramsey frequency measured in the experiment, and show that there is a transient contribution to the interaction energy which depends on the coherence of the superposition. The standard interpretation of Ramsey experiments must be modified since the Ramsey frequency is shown to be insensitive to this energy shift.