In this thesis we develop a fully quantum-mechanical treatment of collisions between distinct atomic Bose-Einstein condensate wavepackets, with particular emphasis on the incoherently scattered atoms that form s-wave haloes around the condensate wavepackets. Previous theoretical treatments of these systems have been unable to account simultaneously for both the evolution of the halo and the depletion of the condensates, and were therefore restricted to the small scattering limit. Our approach uses the truncated Wigner method, a particular example of the classical field methods familiar from quantum optics. The atomic field is restricted to a low-energy subspace of single-particle states, and the method is applicable even to highly-scattered systems. We present a comprehensive derivation of the truncated Wigner method for ultracold bosonic fields, and discuss in detail the validity regime of the Wigner truncation for inhomogeneous multimode systems. The method gives rise to a set of coupled stochastic differential equations that describe the evolution of a single realisation of the atomic field, and have a form similar to that of the well known Gross-Pitaevskii equation, but with the important difference that the stochastic differential equations include well prescribed quantum fluctuations. To propagate our systems we develop algorithms that allow for highly efficient numerical evolution of realistic experimental collisions. By investigating individual trajectories of the colliding system, we find that the scattering halo is composed of many distinct highly-populated phase grains separated by large numbers of vortices, a behaviour we label quantum turbulence. We develop a spatial averaging method for approximately calculating quantum correlation functions from a single trajectory, and calculate various properties of the halo. Based on these results, we propose a mechanism to explain the observed features of scattering halo formation. We find by using an appropriately extended truncated Wigner approach that three-body recombination events have negligible effect on the collisions. Using an ensemble of trajectories we calculate correlation functions of a particular collisional system to give a rigorous characterisation of the quantum statistics of the field, and obtain results that are remarkably similar to those obtained using single trajectory spatial averaging. For global field quantities, such as the total coherent population, we find that accurate estimates can be achieved using just two trajectories, a result we use to efficiently explore the dependence of the system on key physical parameters. Finally, we apply the truncated Wigner method to collisions between condensates in differing hyperfine states, whose (single-trajectory and ensemble) behaviour we find is qualitatively similar to that of single-component collisions.

The mesoscopic effects in the quantum trapped gases of the Bose atoms constitute the main subject of the present thesis. These effects are the most difficult for the theoretical analysis in the quantum statistical physics since they can’t be seen by neither a standard quantum mechanics of the simple microscopic systems of one or very few atoms nor a standard statistical physics of the macroscopic systems that are infinite in the bulk (thermodynamic) limit. Most of the experiments on the cold quantum gases performed in the last decade, starting from the first demonstration of BEC in 1995, involve the mesoscopic systems of a finite number of atoms. The mesoscopic effects should manifest themselves most clearly and easily near a critical temperature of BEC; however, they could be observed also above and below the critical temperature. Here I study the quantum and thermal fluctuations of the Bose-Einstein condensate (BEC) in a box with the periodic boundary conditions under a particle-number constraint. The above constraint is the only reason for the BEC and is crucial for the mesoscopic effects in the BEC fluctuations, especially in the vicinity of the critical temperature in the Bose gas. I employ the particle-number conserving operator formalism of Girardeau and Arnowitt introduced in 1959 to analyze the canonical ensemble fluctuations. I present analytical formulas and numerical calculations for the central moments of the ground state occupation fluctuations in an ideal Bose gas in a box with a mesoscopic number of particles. I present the analysis of the BEC statistics both on a temperature at a fixed number of particles and on a number of particles at a fixed temperature. Both analyses are valid for the purpose of understanding the important mesoscopic effects near the critical temperature. I emphasize the non-Gaussian nature of the fluctuations. The presented formalism can be generalized to the case of a weakly interacting Bose gas in a box in the framework of the Bogoliubov approximation. The work in this direction is in progress but is not included in the present thesis.

Diss.--Hamburg Universität, 2000. / Bibliogr. p.169-178.

Thesis (M. Sc.)--University of Queensland, 2002. / Includes bibliographical references.

Thesis (M.S.)--Worcester Polytechnic Institute. / Keywords: interferometry; bose-einstein condensation. Includes bibliographical references (p. 49-50).

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