Formatted abstract

When a gas of bosonic atoms is cooled sufficiently near to absolute zero, it undergoes a startling transition from a collection of randomlybehaved individual atoms to a phase in which thousands, even millions, of atoms act in collective unison. In this new phase, known as a BoseEinstein condensate (BEC), the atoms all occupy the exact same quantum mechanical ground state and can be collectively described by a single phasecoherent wavefunction of macroscopic size. But how exactly does this harmonious behaviour arise within a collection of incoherent atoms as their temperature decreases? Does the condensate wavefunction grow from a single central nucleus, or does it emerge simultaneously throughout the system? And how does the situation differ when the gas is cooled suddenly, away from thermodynamic equilibrium? Can nonequilibrium behaviour be inherited by the condensate that forms?
In this thesis we aim to answer these and other questions by studying the formation dynamics of BoseEinstein condensates and the development of phase coherence. The research described herein makes use of several theoretical tools, which we lay out on the workbench in Chapter 2: Semiclassical HartreeFock theory provides a meanfield description of a condensate and thermal cloud at equilibrium; Quantum kinetic theory is used to calculate the dynamics of condensate growth into an equilibrium state, from a nonequilibrium thermal cloud; The stochastic projected GrossPitaevskii equation (SPGPE) employs a classicalfield description to encapsulate the nonequilibrium dynamics of the condensate and lowenergy region of the atom cloud, coupled to an equilibrium thermal cloud. This thesis is composed of four principal studies, each constituting a separate chapter.
In Chapter 3 we study the formation of a BoseEinstein condensate in a cigarshaped 3D harmonic trap, induced by the controlled addition of an attractive ``dimple'' potential along the weak axis in a recent experiment [Garrett et al., Phys. Rev. A 83, 013630 (2011)]. In this manner condensation is induced without cooling due to a localized increase in the phase space density. We perform a quantitative analysis of the thermodynamic transformation in both the sudden and adiabatic regimes for a range of dimple widths and depths. We find good agreement with equilibrium calculations based on selfconsistent semiclassical HartreeFock theory describing the condensate and thermal cloud. We observe there is an optimal dimple depth that results in a maximum in the condensate fraction. We also study the nonequilibrium dynamics of condensate formation in the sudden turnon regime, finding good agreement for the observed time dependence of the condensate fraction with calculations based on quantum kinetic theory.
In Chapter 4 we study the onset of phase coherence during the formation of a BEC following a sudden quench to below the transition temperature. Using the SPGPE, we simulate a recent experiment performed at ANU in which a sudden RF quench is applied to a dilute gas of metastable Helium atoms confined in an elongated 3D harmonic magnetic trap. At regularlyspaced intervals following the quench, broadband RF pulses are applied to homogeneously outcouple small (<5%) pulses of atoms, which expand and fall to a microchannel plate detector. The secondorder correlation function at the detector, postulated to provide a measure of phase coherence of the insitu cloud from which they were outcoupled, is measured versus time and compared with the condensate growth curve. A delay was observed in the relaxation of the secondorder correlation function, relative to the peak in the number of condensate atoms. In our simulations we incorporate the effects of periodic outcoupling both the condensate and thermal cloud, and monitor the timedependence of first and secondorder correlation functions in both position and momentum space. We observe longrange phase coherence in the final equilibrium state, though mild persistent phase fluctuations are present. We also observe the curious phenomenon of ``superbunching''  an enhancement of density fluctuations  at specific values of the momentum, which we relate to anomalous correlations of fluctuations. Despite obtaining a good quantitative fit to the experimentallymeasured condensate growth curve in our SPGPE simulations, we do not observe any evidence of the delay in coherence observed in the experiment, which remains unexplained.
In Chapter 5 we use the SPGPE to study the crossover from a true BEC with longrange phase coherence to a quasicondensate dominated by phase fluctuations, for the case of a Bose gas in an highlyelongated 3D harmonic trap. We use parameters corresponding to an experiment in which a sudden quench resulted in the formation of a nonequilibrium condensate, the dynamics of which we study in the chapter that follows. Sweeping through a range of temperatures and chemical potentials within the crossover region, we reveal the behaviour of phase fluctuations through analysis of correlation functions. We compare the idealgas approximation for the transition temperature with the more accurate value obtained via the Binder cumulant. By comparing the phase coherence length with the spatial extent across which density fluctuations are suppressed, we identify the BEC/quasicondensate crossover point. We compare directly with predictions of meanfield theory [Petrov et al., Phys. Rev. Lett. 87, 050404 (2001)] and find that phase fluctuations persist to lower temperatures in our SPGPE simulations. We calculate anomalous correlations of the fluctuation field and show that the resulting correction to the meanfield condensate selfinteraction energy is significant within the quasicondensate regime.
In Chapter 6 we study the formation of a BEC into a nonequilibrium state by simulating an experiment in which a dilute gas of Rubidium87 atoms in a highlyelongated 3D harmonic magnetic trap is rapidly quenched to below the transition temperature. Following the quench, a condensate is observed to form with an excessively elongated shape, and subsequently undergo damped quadrupole shape oscillations [Shvarchuck et al., Phys. Rev. Lett. 89, 270404 (2002)]. We simulate this experiment by modifying the SPGPE to incorporate a hydrodynamic thermal cloud, parametrized by axially and temporallydependent temperature and chemical potential. Comparison against simulations without this axial dependence suggests the hydrodynamic behaviour of the thermal cloud plays a crucial role in the formation of a nonequilibrium condensate. Analyses of the firstorder correlation function and the condensate wavefunction reveal the spontaneous formation of quantum vortices, as well as strong phase fluctuations at equilibrium indicating quasicondensate behaviour.
