A grand canonical C-field theory has been previously developed for modelling finite temperature Bose gases, the stochastic projected Gross-Pitaesvkii equation (SPGPE) [1]. Previous investigations have shown quantitative agreement between experiments and the SPGPE, with a fitted growth rate [2,3] or no fitted parameters at all [4]. These and other works [5,6] have used the number-damping SPGPE, a sub-theory of the SPGPE neglecting a scattering process between the coherent and incoherent regions that conserves the particle number of each region, known as energy-damping. The systems in these works were in quasi-equilibirium and as such a growth process, also known as number-damping, is thought to be dominant. Evidence suggests that energy-damping is significant when the system is far from equilibrium [7]; we may also postulate systems where energy-damping is the only allowed process. In this thesis we use the full SPGPE including the energy-damping reservoir interaction in systems where this process plays an important role in the dissipative evolution.\nWe model quenches of chemical potential across the Bose-Einstein condensation transition in a one dimensional Bose gas confined to a toroid. We use two different models; the full SPGPE and the number-damping SPGPE. The purpose of this is to test the results of our simulations against the predictions of the Kibble-Zurek mechanism (KZM), a theory of defect formation in second order phase transitions. We find that both models give results consistent with KZM, in that various measurable quantities obey a power law with respect to the quench time. The power law exponents are determined by critical exponents, which depend on the universality class of the phase transition. We find the number-damping SPGPE results are consistent with the critical exponents predicted by mean field theory. We are unable to find a universality class with critical exponents consistent with the results of the full theory, and in particular the dynamical critical exponent differs from that predicted by mean field theory.\nWe also use the SPGPE to simulate the motion of a bright soliton in a one dimensional attractive Bose gas confined to a toroid and in contact with a thermal cloud of a second component. The bright soliton is an analytical solution of the one-dimensional Gross-Pitaevskii equation for an attractive Bose-Einstein condensate, which can propagate in space without changing its functional form. We derive a stochastic differential equation for the soliton velocity, finding that the energy-damping reservoir interaction manifests as an Ornstein-Uhlenbeck process for velocity decay, affording a complete analytic solution for the damping and diffusion rates of the bright soliton. The results of simulating the bright soliton using the SPGPE are compared against the analytic solutions of the velocity stochastic differential equation, including the mean, variance, two-time correlations, and power spectra of the velocity. We find that the numerical and analytical solutions show excellent agreement for all these quantities, validating our procedure for obtaining the velocity equation of motion.
A grand canonical C-field theory has been previously developed for modelling finite temperature Bose gases, the stochastic projected Gross-Pitaesvkii equation (SPGPE) [1]. Previous investigations have shown quantitative agreement between experiments and the SPGPE, with a fitted growth rate [2,3] or no fitted parameters at all [4]. These and other works [5,6] have used the number-damping SPGPE, a sub-theory of the SPGPE neglecting a scattering process between the coherent and incoherent regions that conserves the particle number of each region, known as energy-damping. The systems in these works were in quasi-equilibirium and as such a growth process, also known as number-damping, is thought to be dominant. Evidence suggests that energy-damping is significant when the system is far from equilibrium [7]; we may also postulate systems where energy-damping is the only allowed process. In this thesis we use the full SPGPE including the energy-damping reservoir interaction in systems where this process plays an important role in the dissipative evolution.\nWe model quenches of chemical potential across the Bose-Einstein condensation transition in a one dimensional Bose gas confined to a toroid. We use two different models; the full SPGPE and the number-damping SPGPE. The purpose of this is to test the results of our simulations against the predictions of the Kibble-Zurek mechanism (KZM), a theory of defect formation in second order phase transitions. We find that both models give results consistent with KZM, in that various measurable quantities obey a power law with respect to the quench time. The power law exponents are determined by critical exponents, which depend on the universality class of the phase transition. We find the number-damping SPGPE results are consistent with the critical exponents predicted by mean field theory. We are unable to find a universality class with critical exponents consistent with the results of the full theory, and in particular the dynamical critical exponent differs from that predicted by mean field theory.\nWe also use the SPGPE to simulate the motion of a bright soliton in a one dimensional attractive Bose gas confined to a toroid and in contact with a thermal cloud of a second component. The bright soliton is an analytical solution of the one-dimensional Gross-Pitaevskii equation for an attractive Bose-Einstein condensate, which can propagate in space without changing its functional form. We derive a stochastic differential equation for the soliton velocity, finding that the energy-damping reservoir interaction manifests as an Ornstein-Uhlenbeck process for velocity decay, affording a complete analytic solution for the damping and diffusion rates of the bright soliton. The results of simulating the bright soliton using the SPGPE are compared against the analytic solutions of the velocity stochastic differential equation, including the mean, variance, two-time correlations, and power spectra of the velocity. We find that the numerical and analytical solutions show excellent agreement for all these quantities, validating our procedure for obtaining the velocity equation of motion.
Quantum many-body systems offer rich physics, their theoretical description, however, faces challenges. An experimentally controllable and well-characterised system is provided by Bose-Einstein condensates (BECs) motivating a complete open {quantum} system treatment of dissipative mechanisms. The stochastic projected Gross-Pitaevskii equation (SPGPE) offers such a treatment at finite temperature, capturing several relaxation mechanisms. This thesis completes the SPGPE under its usual assumptions, so that all possible two-body collisions are finally considered. The respective relevance of the distinct processes are established. The SPGPE models ultracold Bose gases by dividing them into a low-energy coherent region containing the condensate and low-energy excitations and a high-energy incoherent region acting as a reservoir. In its framework, a coupling between coherent and incoherent region has been derived via two distinct mechanisms: number damping, stemming from scattering events between incoherent region particles in which one of them loses enough energy to enter the coherent region. And energy damping, stemming form number conserving scattering events between a particle in the coherent and incoherent region each. However, in practical applications of the SPGPE, often only the first mechanism, number damping, is accounted for, despite evidence that other damping channels can play a dominant role in dynamical relaxation. In this thesis, we develop an SPGPE theory of the remaining two-body reservoir interaction. It describes scattering events between two coherent region particles leading to one gaining enough energy to leave the coherent and enter the incoherent region. I demonstrate that, despite its previous neglect, this evaporative damping is comparable in strength to number damping. Evaporative damping completes the SPGPE description of ultracold Bose gases, giving a first-principle picture containing all possible two-body collisions in their evolution at finite temperature I further perform an analysis of the linearised SPGPE, valid at sufficiently high phase space density. It allows the identification of an ideal energy cut off, dividing the coherent and incoherent region. It moreover demonstrates the importance of energy damping. The analysis shows that with rising phase space density it increasingly dominates over both, number and evaporative damping. Studying the Jones-Roberts soliton (JRS) extends the proven dominance of energy damping at high phase-space density to the decay of nonlinear structures. Energy damping is revealed to determine the decay for slow JRSs, corresponding to vortex dipoles, as well as fast JRSs, forming rarefaction pulses. The predicted decay behaviour enables an experimentally feasible quantitative test of the SPGPE reservoir theory. The results of this thesis highlight the importance of damping mechanisms in the SPGPE that have so far gained little attention. The demonstrated dominance of energy damping at high phase-space density establishes the need of including this process in any study of dynamics in the high phase-space density regime. At higher temperatures closer to the transition, the inclusion of evaporative damping is also found to become crucial in obtaining an appropriate first-principle picture of the dynamics of ultracold Bose gases. These findings have implications for finite-temperature dynamical properties such as decay rates, excitation linewidths and equilibration times.
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 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.
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 review c-field methods for simulating the non-equilibrium dynamics of degenerate Bose gases beyond the mean-field Gross-Pitaevskii approximation. We describe three separate approaches that utilise similar numerical methods, but have distinct regimes of validity. Systems at finite temperature can be treated with either the closed-system projected Gross-Pitaevskii equation (PGPE), or the open-system stochastic projected Gross-Pitaevskii equation (SPGPE). These are both applicable in quantum degenerate regimes in which thermal fluctuations are significant. At low or zero temperature, the truncated Wigner projected Gross-Pitaevskii equation (TWPGPE) allows for the simulation of systems in which spontaneous collision processes seeded by quantum fluctuations are important. We describe the regimes of validity of each of these methods, and discuss their relationships to one another, and to other simulation techniques for the dynamics of Bose gases. The utility of the SPGPE formalism in modelling non-equilibrium Bose gases is illustrated by its application to the dynamics of spontaneous vortex formation in the growth of a Bose-Einstein condensate.