In this thesis, we apply the stochastic Gross-Pitaevskii equation, using the formulation of Stoof, to the study of weakly interacting, finite temperature atomic Bose gases, both in and out of equilibrium. Explicitly maintaining the effects of phase and density fluctuations, this state-of-the-art method is well suited to the study of low dimensional Bose systems. As such, we consider the non-equilibrium dynamics of dark matter wave solitons, oscillating within harmonically trapped, one-dimensional phase-fluctuating condensates, through numerical simulations of the stochastic Gross-Pitaevskii equation. In treating the damping due to higher lying thermal modes of the system, we include the HartreeFock contribution of the equilibrium atomic density, which is a novel feature of our work. A stochastic approach is found to be essential in capturing the shot-to-shot variation in soliton trajectories observed experimentally. Neglecting stochastic effects, we find the well-known dissipative Gross-Pitaevskii equation to recover the average dynamics of the stochastic model, on retaining the microscopically justified form for the damping of this approach. Such a coupling is found to be necessary in modelling the dissipative nature of finite temperature soliton dynamics. Furthermore, we undertake a comparison between the finite temperature equilibrium state of the stochastic Gross-Pitaevskii equation, and that predicted within a number conserving Bogoliubov approach. Through a series of numerical experiments, we compare the equilibrium properties of a harmonically trapped Bose gas, considering our results in light of those of independent theoretical treatments, and additionally assessing the sensitivity of the numerical implementation of the former method to the numerical grid spacing. In the regime in which thermal effects dominate, the stochastic Gross-Pitaevskii approach proves to be a more consistent means of generating such equilibrium states.
In this thesis, we apply the stochastic Gross-Pitaevskii equation, using the formulation of Stoof, to the study of weakly interacting, finite temperature atomic Bose gases, both in and out of equilibrium. Explicitly maintaining the effects of phase and density fluctuations, this state-of-the-art method is well suited to the study of low dimensional Bose systems. As such, we consider the non-equilibrium dynamics of dark matter wave solitons, oscillating within harmonically trapped, one-dimensional phase-fluctuating condensates, through numerical simulations of the stochastic Gross-Pitaevskii equation. In treating the damping due to higher lying thermal modes of the system, we include the HartreeFock contribution of the equilibrium atomic density, which is a novel feature of our work. A stochastic approach is found to be essential in capturing the shot-to-shot variation in soliton trajectories observed experimentally. Neglecting stochastic effects, we find the well-known dissipative Gross-Pitaevskii equation to recover the average dynamics of the stochastic model, on retaining the microscopically justified form for the damping of this approach. Such a coupling is found to be necessary in modelling the dissipative nature of finite temperature soliton dynamics. Furthermore, we undertake a comparison between the finite temperature equilibrium state of the stochastic Gross-Pitaevskii equation, and that predicted within a number conserving Bogoliubov approach. Through a series of numerical experiments, we compare the equilibrium properties of a harmonically trapped Bose gas, considering our results in light of those of independent theoretical treatments, and additionally assessing the sensitivity of the numerical implementation of the former method to the numerical grid spacing. In the regime in which thermal effects dominate, the stochastic Gross-Pitaevskii approach proves to be a more consistent means of generating such equilibrium states.
The fluctuating and dissipative dynamics of matter-wave dark solitons within harmonically trapped, partially condensed Bose gases is studied both numerically and analytically. A study of the stochastic Gross-Pitaevskii equation, which correctly accounts for density and phase fluctuations at finite temperatures, reveals dark-soliton decay times to be lognormally distributed at each temperature, thereby characterizing the previously predicted long-lived soliton trajectories within each ensemble of numerical realizations [S. P. Cockburn et al., Phys. Rev. Lett. 104, 174101 (2010)]. Expectation values for the average soliton lifetimes extracted from these distributions are found to agree well with both numerical and analytic predictions based upon the dissipative Gross-Pitaevskii model (with the same ab initio damping). Probing the regime for which 0.8 ${k}_{B}T<\ensuremath{\mu}<1.6$ kBT, we find average soliton lifetimes to scale with temperature as $\ensuremath{\tau}\ensuremath{\sim}{T}^{\ensuremath{-}4}$, in agreement with predictions previously made for the low-temperature regime ${k}_{B}T\ensuremath{\ll}\ensuremath{\mu}$. The model is also shown to capture the experimentally relevant decrease in the visibility of an oscillating soliton due to the presence of background fluctuations.
We analyze the equilibrium properties of a weakly interacting, trapped quasi-one-dimensional Bose gas at finite temperatures and compare different theoretical approaches. We focus in particular on two stochastic theories: a number-conserving Bogoliubov (NCB) approach and a stochastic Gross-Pitaevskii equation (SGPE) that have been extensively used in numerical simulations. Equilibrium properties like density profiles, correlation functions, and the condensate statistics are compared to predictions based upon a number of alternative theories. We find that due to thermal phase fluctuations, and the corresponding condensate depletion, the NCB approach loses its validity at relatively low temperatures. This can be attributed to the change in the Bogoliubov spectrum, as the condensate gets thermally depleted, and to large fluctuations beyond perturbation theory. Although the two stochastic theories are built on different thermodynamic ensembles (NCB, canonical; SGPE, grand-canonical), they yield the correct condensate statistics in a large Bose-Einstein condensate (BEC) (strong enough particle interactions). For smaller systems, the SGPE results are prone to anomalously large number fluctuations, well known for the grand-canonical, ideal Bose gas. Based on the comparison of the above theories to the modified Popov approach, we propose a simple procedure for approximately extracting the Penrose-Onsager condensate from first- and second-order correlation functions that is both computationally convenient and of potential use to experimentalists. This also clarifies the link between condensate and quasicondensate in the Popov theory of low-dimensional systems.
In this thesis we use the stochastic Gross-Pitaevskii equation (SGPE), a finite temperature model for weakly interacting ultracold Bose gases which fully incorporates density and phase fluctuations, to demonstrate ab initio quantitative agreement with a number of quasi-one-dimensional experiments. To achieve this, we propose and numerically solve a quasi-one-dimensional form of the SGPE, supported by a self-consistent treatment of radially-excited thermal modes. The quasi-one-dimensional stochastic Gross-Pitaevskii equation provides an accurate finite temperature description of the dynamical equilibrium of the lowenergy axial modes of a Bose gas, assumed to be highly populated and thus treated within the ‘classical field’ approximation. This treatment allows to selfconsistently account for transverse, quasi-one-dimensional effects, which makes it a valid model in the regime where the chemical potential, μ, is approximately equal to a few times the transverse excitation energy, ~ω⊥. In the regime where the thermal energy, kBT , is also comparable to or larger than ~ω⊥, the transverse excited states play an increasingly important role, and are treated here as onedimensional independent Bose gases at static equilibrium. Firstly, we demonstrate that this is an excellentmodel for ab initio investigation of equilibrium properties, such as density profiles and density fluctuations. This is shown by accurately reproducing the in situ density profiles recently obtained in the experiments of Trebbia et al. [Phys. Rev. Lett. 97, 250403 (2006)] and van Amerongen et al. [Phys. Rev. Lett. 100, 090402 (2008)], and the density fluctuation data reported by Armijo et al. [Phys. Rev. Lett. 105, 230402 (2010)]. Moreover, we perform an ab initio analysis of the temperature dependence of the phase coherence of finite temperature, quasi-one-dimensional Bose gases measured in the experiments of Richard et al. [Phys. Rev. Lett. 91, 010405 (2003)] and Hugbart et al. [Eur. Phys. J. D 35, 155 (2005)]. We find very good agreement across the entire observed temperature range in both experiments, and improve upon previous theoretical modelling of the latter.
We review the stochastic Gross-Pitaevskii approach for non-equilibrium finite temperature Bose gases, focussing on the formulation of Stoof; this method provides a unified description of condensed and thermal atoms, and can thus describe the physics of the critical fluctuation regime. We discuss simplifications of the full theory, which facilitate straightforward numerical implementation, and how the results of such stochastic simulations can be interpreted, including the procedure for extracting phase-coherent (`condensate') and density-coherent (`quasi-condensate') fractions. The power of this methodology is demonstrated by successful ab initio modelling of several recent atom chip experiments, with the important information contained in each individual realisation highlighted by analysing dark soliton decay within a phase-fluctuating condensate.
Providing a complete description of dissipative superfluid dynamics is one of the major challenges of many-body quantum field theory. In this thesis we make a fundamental step towards this goal by implementing the stochastic projected Gross-Pitaevskii equation (SPGPE) in complete form for the first time. The SPGPE is a high-temperature theory of Bose-Einstein condensate dynamics, providing a classical-field description of a low-energy subspace in contact with a thermal reservoir. The reservoir interaction terms account for dissipation and noise from thermal interactions, and arise from two distinct processes described as number-damping and energy-damping. This work advances previous applications of the SPGPE theory, which have only included number-damping processes, by implementing the energy-damping processes. We describe the properties of the deterministic and noise terms corresponding to the energy-damping process, and develop a novel algorithm to accurately and efficiently evaluate the energy-damping terms in the SPGPE. We apply the SPGPE to a range of experimentally accessible systems, considering both non-equilibrium and quasi-equilibrium dynamics. We model the experiment of Neely et al. [Phys. Rev. Lett. 111, 235301 (2013)], where stirring of a toroidally trapped Bose-Einstein condensate generates a disordered array of quantum vortices that decay, via thermal dissipation, to form a macroscopic persistent current. We perform numerical simulations of the experiment using the number-damping SPGPE and ab initio determined reservoir parameters. We quantitatively reproduce both the formation time and size of the persistent current, as measured in the experiment. In the first application of the full SPGPE, we consider the non-equilibrium dynamics of a condensate excited into a large-amplitude breathing mode. We find that in such non-equilibrium regimes, the energy-damping dominates over the number-damping process, leading to qualitatively different system dynamics. In particular, energy damping causes the system to rapidly reach thermal equilibrium without greatly depleting the condensate, showing that energy damping provides a highly coherent dissipation mechanism. Finally, we apply the SPGPE to the quasi-equilibrium dynamics of single-vortex decay. Energy-damping processes have previously been neglected for this system. SPGPE simulations show that in fact energy-damping has a dominant effect on the lifetime of a single vortex, with lifetimes less than half those predicted by the number-damping SPGPE. In contrast to the breathing mode decay, we observe little qualitative difference between the energy-damping and number-damping descriptions of vortex decay. Our findings show that while energy-damping processes are important to quantitatively describe quasi-equilibrium dynamics, the system behavior may be described by the number-damping SPGPE with a suitably modified dissipation rate.
We consider the decay of a dark soliton in a homogeneous BEC at finite temperature, with a particular emphasis on the effect of thermal noise on the stability of the soliton.We present an analytical treatment of a damped soliton in an unbounded system, comparing the predicted dynamics with numerical stochastic simulations of a spatially confined system. By varying the system temperature we study the relative importance of noise on the motion of the soliton. In the regime of low temperature the soliton is largely immune to thermal fluctuations, and is well described by the damped Gross-Pitaevskii equation. In this regime our analytical treatment is in close agreement with numerical simulations. For sufficiently high temperature, the thermal fluctuations have the interesting effect of increasing the stability of the soliton, extending its lifetime beyond the predictions of damped Gross-Pitaevskii theory.