We construct a nonequilibrium theory for the dynamics of two interacting finite-temperature atomic Bose-Einstein condensates and use it to numerically estimate the relative rates of the arising collisional processes near equilbrium. The condensates are described by dissipative Gross-Pitaevskii equations, coupled to quantum Boltzmann equations for the thermal atoms. The density-density interactions between atoms in different components facilitate a number of transport processes of relevance to sympathetic cooling: in particular, considering realistic miscible and immiscible trapped atomic $^{87}\mathrm{Rb}\text{\ensuremath{-}}^{41}\mathrm{K}$ and $^{87}\mathrm{Rb}\text{\ensuremath{-}}^{85}\mathrm{Rb}$ condensate mixtures, we highlight the dominance of an intercomponent scattering process associated with collisional ``exchange'' of condensed and thermal atoms between the components close to equilibrium.
We construct a nonequilibrium theory for the dynamics of two interacting finite-temperature atomic Bose-Einstein condensates and use it to numerically estimate the relative rates of the arising collisional processes near equilbrium. The condensates are described by dissipative Gross-Pitaevskii equations, coupled to quantum Boltzmann equations for the thermal atoms. The density-density interactions between atoms in different components facilitate a number of transport processes of relevance to sympathetic cooling: in particular, considering realistic miscible and immiscible trapped atomic 87Rb\ensuremath-41K and 87Rb\ensuremath-85Rb condensate mixtures, we highlight the dominance of an intercomponent scattering process associated with collisional ``exchange'' of condensed and thermal atoms between the components close to equilibrium.
We derive a nonequilibrium finite-temperature kinetic theory for a binary mixture of two interacting atomic Bose-Einstein condensates and use it to explore the degree of hydrodynamicity attainable in realistic experimental geometries. Based on the standard separation-of-time-scales argument of kinetic theory, the dynamics of the condensates of the multicomponent system are shown to be described by dissipative Gross-Pitaevskii equations self-consistently coupled to corresponding quantum Boltzmann equations for the noncondensate atoms: On top of the usual mean-field contributions, our scheme identifies a total of eight distinct collisional processes, whose dynamical interplay is expected to be responsible for the system's equilibration. In order to provide their first characterization, we perform a detailed numerical analysis of the role of trap frequency and geometry on collisional rates for experimentally accessible mixtures of $^{87}\mathrm{Rb}\ensuremath{-}^{41}\mathrm{K}$ and $^{87}\mathrm{Rb}\ensuremath{-}^{85}\mathrm{Rb}$, discussing the extent to which the system may approach the hydrodynamic regime with regard to some of those processes as a guide for future experimental investigations of ultracold Bose gas mixtures.
In this Thesis we study steady state solutions and dynamical evolutions of two– component atomic Bose–Einstein Condensates. We initially investigate the equilibrium properties of condensate mixtures in harmonic trapping potentials at zero temperature. Subsequently we simulate the coupled growth of these condensates by inclusion of damping terms. Finally, we investigate the evolution of coupled Bose gases via the so-called classical–field method. A recent experiment [D. J. McCarron et al., Phys. Rev. A, 84, 011603(R) (2011)] achieved Bose–Einstein Condensation of a two–species 87Rb–133Cs phase segregated mixture in harmonic trapping potentials. Depending on relative atom numbers of the two species, three distinct regimes of density distributions were observed. For these experimental parameters, we investigate the corresponding time–independent ground state solutions through numerical simulations of the coupled Gross–Pitaevskii equations. By including experimentally relevant shifts between the traps, we observe a range of structures including ‘ball and shell’ formations and axially/radially separated states. These are found to be very sensitive to the trap shifts. For all three experimental regimes, our numerical simulations reveal good qualitative agreement. The observed experimental profiles cannot be guaranteed to be fully equilibrated. This, coupled with the rapid sympathetic cooling of the experimental system, leads to a situation where growth may play a determining factor in the density structures formed. To investigate this further, we introduce phenomenological damping to describe the associated condensate growth/decay, revealing a range of transient structures. However, such a model always predicts the predominance of one condensate species over longer evolution times. Work undertaken by collaborators with the more elaborate Stochastic Projected Gross–Pitaevskii equations, which can describe condensate formation by coupling to a heat bath, predicts the spontaneous formation of dark–bright solitons. Motivated by this, we show how the presence of solitons can affect the condensate distribution, thus highlighting the overall dynamical role in the emerging patterns. Finally, we use classical field methods to analyse the evolution of non trapped Bose gases from strongly nonequilibrium initial distributions. The contrast between miscible (overlapping) and immiscible (phase segregated) components gives rise to important distinctions for condensate fractions and the formation of domains and vortices. In addition, splitting the particles of a single component thermalised state into two components is investigated. We then study the effects of suddenly quenching the strength of the interspecies interactions. Under suitable conditions, this quench generates isotropic vortex tangles. While this tangle subsequently decays over time, we propose how a repeat sequence of quenches at regular intervals could be employed to drive the tangle, thereby potentially providing a novel route to the generation of quantum turbulence.
The dynamics of a trapped Bose-condensed gas at finite temperatures is described by a generalized Gross-Pitaevskii equation for the condensate order parameter and a semiclassical kinetic equation for the thermal cloud, solved using N-body simulations. The two components are coupled by mean fields as well as collisional processes that transfer atoms between the two. We use this scheme to investigate scissors modes in anisotropic traps as a function of temperature. Frequency shifts and damping rates of the condensate mode are extracted, and are found to be in good agreement with recent experiments.
In this paper, we develop a numerical procedure for investigating the dynamics of trapped Bose gases based on the ZGN theory. The dynamical equations used consist of a generalized Gross-Pitaevskii equation for the condensate order parameter and a semiclassical kinetic equation for the thermal cloud. The former is solved using a fast Fourier transform split-operator technique while the Boltzmann equation is treated by means of N-body simulations. The two components are coupled by mean fields as well as collisional processes that transfer atoms between the two. This scheme has been applied to a model equilibration problem, dipole oscillations in isotropic traps and scissors modes in anisotropic traps. In the case of the latter, the frequencies and damping rates of the condensate mode have been extracted from the simulations for a wide range of temperatures. Good agreement with recent experiments has been found.
In this thesis we study the dynamics of dark solitons, vortices and Josephson effects, in trapped atomic Bose-Einstein condensates. We firstly demonstrate a novel, sound mediated, long-range interaction mechanism between dark solitons using the Gross-Pitaevskii equation. Then we present the effect of finite temperatures on the dynamics of vortices and Josephson effects using a generalised form of this equation, which additionally includes a source term modelling the coupling of the condensate to a fully dynamical thermal cloud, described by a quantum Boltzmann equation. The latter formalism is known as the Zaremba-Nikuni-Griffin scheme (ZNG). The sound-mediated interaction between dark solitons is such that the speed and trajectory of one soliton in a condensate confined by a harmonic trap can be significantly modified by the presence of a second soliton in the same trap. By confining the two solitons to spatially separated subregions of a double well trap, we show how this effect can be magnified. In particular we find it can be large, and therefore detectable, in experimentally relevant geometries, including high-periodicity optical lattices in which the ability of solitons to act as both absorbers and emitters of energy becomes apparent. At finite temperatures, we model the dynamics via the ZNG scheme. Firstly, we give a detailed description of the scheme for numerically solving these coupled equations including all collisional terms. Then we study the effect of a dynamical thermal cloud on vortex dynamics, in particular, focussing on the experimentally-relevant quantities of precession frequencies, decay rates and vortex core brightness. The changes in these are found to increase with increasing temperature for a pancake-shaped geometry, in particular for trapping parameters of a recent experiment of Freilich et al. (Science 329, 1182 (2010)). Particle exchanging collisions between the condensate and thermal cloud are found to be crucial in determining the rate of decay of a precessing vortex. We further show that, rotation of the thermal cloud can be a mechanism for radially translating the position of a vortex, a scenario under current investigation. Finally, we analyse the effect of finite temperatures on population dynamics in a double well atom chip experiment by LeBlanc et al. (Phys. Rev. Lett. 106, 025302 (2011)). Focussing on the coupled evolution in the absence of collisions, we find that it generates significant damping which is, however, less than that observed in the experiment. A detailed comparison in the presence of collisions is beyond the scope of this thesis.
In the first part of this thesis, we present a unified kinetic theory that describes the finite-temperature, non-equilibrium dynamics of a Bose-Einstein condensed gas interacting with a thermal cloud in a trap. This theory includes binary interactions to second order in the interaction potential and reduces to a diagonal quantum Boltzmann equation for Bogoliubov quasiparticles. The Hartree-Fock-Bogoliubov interactions include the pairing field and are expressed as many-body T matrices to second order. The interactions thus include the correct renormalized scattering physics. This renormalized theory is automatically gapless. Thus, the excited Bogoliubov modes are naturally orthogonal to the condensate ground state. This kinetic theory is a complete second-order theory that reduces to the Gross-Pitaevskii equation and the quantum Boltzmann equation in the respective limits and thus is capable of describing the system over a wide temperature range. In the second part, we consider a many-body theory of a dilute Fermi gas near a Feshbach resonance. Experiments explore the crossover physics between the Bardeen-Cooper-Schrieffer (BCS) superfluidity of a two-spin Fermi gas, and the Bose-Einstein condensation (BEC) of composite bosons. We consider correlations between a composite boson and a fermion pair and show that such correlations are the minimal ingredients needed in a many-body theory to generate the correct boson-boson scattering length in the Bose-Einstein limit of the crossover. We also use imaginary-time propagation to find zero-temperature ground states in the BCS/BEC crossover. A cumulant expansion allows us to systematically include higher-order interactions between bosons and fermions. In particular, we calculate the Hartree term across the resonance. We further apply the cumulant-expansion method to thermal fermions and composite bosons interacting above the transition temperature in the normal phase. We numerically calculate the full time dependence in ramps across the resonance in this regime and find different two-body and many-body time scales in the system. We calculate molecular conversion efficiencies as a function of temperature and phase-space density, and find good agreement with results from JILA potassium experiments.