We present reduced-dimensional stochastic projected Gross-Pitaevskii equations describing regimes of confinement and temperature where a one-dimensional (1D) or 2D superfluid is immersed in a 3D thermal cloud. The projection formalism provides both a formally rigorous and physically natural way to effect the dimensional reduction. The 3D form of the number-damping (growth) terms is unchanged by the dimensional reduction. Projection of the energy-damping (scattering) terms leads to modified stochastic equations of motion describing energy exchange with the thermal reservoir. The regime of validity of the dimensional reduction is investigated via variational analysis. For the 1D case, we validate our variational treatment by comparing numerical simulations of a trapped prolate system in three dimensions with the 1D theory and establish a consistent choice of cutoff for the 1D theory. We briefly discuss the scenario involving two components with different degeneracy, suggesting that a wider regime of validity exists for systems in contact with a buffer-gas reservoir.
We present reduced-dimensional stochastic projected Gross-Pitaevskii equations describing regimes of confinement and temperature where a one-dimensional (1D) or 2D superfluid is immersed in a 3D thermal cloud. The projection formalism provides both a formally rigorous and physically natural way to effect the dimensional reduction. The 3D form of the number-damping (growth) terms is unchanged by the dimensional reduction. Projection of the energy-damping (scattering) terms leads to modified stochastic equations of motion describing energy exchange with the thermal reservoir. The regime of validity of the dimensional reduction is investigated via variational analysis. For the 1D case, we validate our variational treatment by comparing numerical simulations of a trapped prolate system in three dimensions with the 1D theory and establish a consistent choice of cutoff for the 1D theory. We briefly discuss the scenario involving two components with different degeneracy, suggesting that a wider regime of validity exists for systems in contact with a buffer-gas reservoir.
The stochastic projected Gross-Pitaevskii equation (SPGPE) is a finite-temperature theory of Bose-Einstein condensate dynamics, utilizing a classical-field approach to describing the dynamics of a low-energy coherent subspace in contact with an incoherent thermal reservoir. Interactions with the reservoir are encapsulated in two distinct processes, known as number damping and energy damping. Historically the energy-damping process has received little attention, at least in part due to difficulty in finding a sufficiently efficient and accurate numerical algorithm for solving the SPGPE. Rooney et al. [Phys. Rev. E 89, 013302 (2014)] developed such an algorithm, providing new opportunities to investigate the energy-damping process. In this work we provide an analytic treatment of the energy-damping process for a range of systems using functional Ito calculus.\n\nWe perform a linear fluctuation analysis on the SPGPE for a homogeneous system. We find the dispersion relation to be the usual Bogoliubov dispersion with an additional momentum dependent damping rate. The damping rate is a combination of number- and energy-damping terms with distinct momentum dependence, revealing that each damping process is dominant for excitations of different length scales. We also find the spectra of density and phase fluctuations in momentum space, again with distinct momentum dependence associated with the two damping processes. These results demonstrate that there always exists a regime of length scales where one can expect energy-damping to be the dominant process.\n\nWe derive stochastic Ehrenfest relations (SER) from the SPGPE using projected functional change of variables. The SER take the form of stochastic differential equations for matter wave moments of the system and include drift and diffusion terms corresponding to the number- and energy-damping processes. We find that for a well-chosen cutoff the SER can often admit approximate analytic solutions. Analytic solutions of the SER for a quasi-1D Thomas-Fermi system show good agreement with simulations of the 1D SPGPE.\n\nNext, we characterise the centre of mass equilibrium properties of a 3D harmonically trapped finite temperature system in the Thomas-Fermi regime using the SER for position, momentum, and particle number. We find that the energy-damping process is generally dominant over the number-damping process when considering position and momentum fluctuations. Considering the stochastic evolution of particle number in this system, we find that the steady-state fluctuation properties depend only on the number-damping process. This suggests that the effects of each reservoir interaction can be measured individually in the same system, allowing for experimental detection of the energy-damping reservoir interaction.\n\nWe finish by considering the motion of a singly-charged quantum vortex confined to a disc trap in an energy-damped system. We find a stochastic differential equation for the dissipative motion of the vortex. It is revealed that the presence of noise can cause a stabilizing effect on the vortex, extending its lifetime. We also consider the presence of a rotating thermal cloud, a scenario that leads to the vortex becoming the energetically favourable state. This allows for analysis of steady-state stochastic properties of the vortex motion.
We have achieved a full implementation of the stochastic projected Gross-Pitaevskii equation for a three-dimensional trapped Bose gas at finite temperature. Our work advances previous applications of this theory, which have only included growth processes, by implementing number-conserving scattering processes. We evaluate an analytic expression for the coefficient of the scattering term and compare it to that of the growth term in the experimental regime, showing that the two coefficients are comparable in size. We give an overview of the numerical implementation of the deterministic and stochastic terms for the scattering process, and use simulations of a condensate excited into a large amplitude breathing mode oscillation to characterize the importance of scattering and growth processes in an experimentally accessible regime. We find that in such nonequilibrium regimes the scattering can dominate over the growth, leading to qualitatively different system dynamics. In particular, the scattering causes the system to rapidly reach thermal equilibrium without greatly depleting the condensate, suggesting that it provides a highly coherent energy-transfer mechanism.
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.
The stochastic projected Gross-Pitaevskii equation (SPGPE) is a finite temperature theory of Bose-Einstein condensate (BEC) dynamics. It utilizes C-field techniques, which describe the dynamics of a low-energy coherent subspace in contact with a thermal reservoir [1]. There are two distinct mechanisms when interactions between the coherent field and the incoherent field are taken into account, known as number-damping and energy-damping. Number-damping has been treated extensively, while energy-damping has had less attention due to the lack of an accurate and efficient numerical method for solving it. However, Rooney et al. [2] developed a numerical technique allowing opportunities for the investigation of the energy-damping process. Bradley and McDonald examined the analytic treatment of the energy-damping process using functional Ito calculus for a range of systems, except for dark soliton. They found there is usually a regime where energy-damping dominates number-damping. Wright [3] has done analytical work on the dissipation of dark soliton in a homogenous system via a Lagrangian approach (where the number-damping term is treated as the perturbation to GPE). Still, the effect of the energy-damping process was not considered. In this thesis, we present an analytic investigation of the dissipation effects of number-damping and energy-damping on dark solitons. This is done through the framework of the one-dimensional Damped Gross-Pitaevskii equation (1D-DGPE). We take a Lagrangian variational approach and treat the number-damping and energy-damping as perturbations to the GPE. Dynamics of the soliton position are obtained, and anti-damping of the dark soliton in a harmonically trapped prolate BEC is observed in the perturbation regime. Our analytic work shows that dissipation effects of energy- damping becomes more dominant as the chemical potential of the system increases; for even modest chemical potential, energy-damping dominates number-damping. We compare our analytic results against numerical simulations and find good agreement for dark soliton position dynamics, kinetic energy, potential energy, and interaction energy dynamics in the regime where the dark soliton is near to the trap center and moving slowly relative to the speed of sound.
We consider many-body effects on particle scattering in one-, two-, and three-dimensional (3D) Bose gases. We show that at T=0 these effects can be modeled by the simpler two-body T matrix evaluated off the energy shell. This is important in 1D and 2D because the two-body T matrix vanishes at zero energy and so mean-field effects on particle energies must be taken into account to obtain a self-consistent treatment of low-energy collisions. Using the off-shell two-body T matrix we obtain the energy and density dependence of the effective interaction in 1D and 2D and the appropriate Gross-Pitaevskii equations for these dimensions. Our results provide an alternative derivation of those of Kolomeisky and co-workers. We present numerical solutions of the Gross-Pitaevskii equation for a 2D condensate of hard-sphere bosons in a trap. We find that the interaction strength is much greater in 2D than for a 3D gas with the same hard-sphere radius. The Thomas-Fermi regime is, therefore, approached at lower condensate populations and the energy required to create vortices is lowered compared to the 3D case.
We present a method for solving the stochastic projected Gross-Pitaevskii equation (SPGPE) for a three-dimensional weakly interacting Bose gas in a harmonic-oscillator trapping potential. The SPGPE contains the challenge of both accurately evolving all modes in the low-energy classical region of the system, and evaluating terms from the number-conserving scattering reservoir process. We give an accurate and efficient procedure for evaluating the scattering terms using a Hermite-polynomial based spectral-Galerkin representation, which allows us to precisely implement the low-energy mode restriction. Stochastic integration is performed using the weak semi-implicit Euler method. We extensively characterize the accuracy of our method, finding a faster-than-expected rate of stochastic convergence. Physical consistency of the algorithm is demonstrated by considering thermalization of initially random states.