The theory of open quantum systems is central to describing the dynamics and equilibration of dilute-gas Bose-Einstein condensates (BECs). We present an analysis of the linearized stochastic projected Gross-Pitaevskii equation (SPGPE) describing finite-temperature BECs. Our treatment provides an optimal choice for the cutoff that divides the Bose gas into the low-energy coherent region forming a classical wave and the high-energy thermal cloud treated as a reservoir. Moreover, it highlights the relevance of energy damping, the number-conserving scattering between thermal and coherent atoms. We analyze the equilibrium properties and near-equilibrium relaxation of a homogeneous BEC in one, two, and three dimensions at high phase-space density and calculate the autocorrelation function and power spectrum of the density and phase fluctuations. Simulations of the full nonlinear SPGPE are in close agreement and extend our arguments beyond the linear regime. Our work suggests the need for a reexamination of decay processes in BECs studied under the neglect of energy damping.
The theory of open quantum systems is central to describing the dynamics and equilibration of dilute-gas Bose-Einstein condensates (BECs). We present an analysis of the linearized stochastic projected Gross-Pitaevskii equation (SPGPE) describing finite-temperature BECs. Our treatment provides an optimal choice for the cutoff that divides the Bose gas into the low-energy coherent region forming a classical wave and the high-energy thermal cloud treated as a reservoir. Moreover, it highlights the relevance of energy damping, the number-conserving scattering between thermal and coherent atoms. We analyze the equilibrium properties and near-equilibrium relaxation of a homogeneous BEC in one, two, and three dimensions at high phase-space density and calculate the autocorrelation function and power spectrum of the density and phase fluctuations. Simulations of the full nonlinear SPGPE are in close agreement and extend our arguments beyond the linear regime. Our work suggests the need for a reexamination of decay processes in BECs studied under the neglect of energy damping.
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.
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.
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.
When a gas of bosonic atoms is cooled sufficiently near to absolute zero, it undergoes a startling transition from a collection of randomly-behaved individual atoms to a phase in which thousands, even millions, of atoms act in collective unison. In this new phase, known as a Bose-Einstein condensate (BEC), the atoms all occupy the exact same quantum mechanical ground state and can be collectively described by a single phase-coherent 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 non-equilibrium 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 Bose-Einstein 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: Semi-classical Hartree-Fock theory provides a mean-field 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 non-equilibrium thermal cloud; The stochastic projected Gross-Pitaevskii equation (SPGPE) employs a classical-field description to encapsulate the non-equilibrium dynamics of the condensate and low-energy 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 Bose-Einstein condensate in a cigar-shaped 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 self-consistent semi-classical Hartree-Fock 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 non-equilibrium dynamics of condensate formation in the sudden turn-on 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 regularly-spaced 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 second-order correlation function at the detector, postulated to provide a measure of phase coherence of the in-situ 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 second-order 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 time-dependence of first- and second-order correlation functions in both position and momentum space. We observe long-range 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 experimentally-measured 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 long-range phase coherence to a quasicondensate dominated by phase fluctuations, for the case of a Bose gas in an highly-elongated 3D harmonic trap. We use parameters corresponding to an experiment in which a sudden quench resulted in the formation of a non-equilibrium 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 ideal-gas 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 mean-field 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 mean-field condensate self-interaction energy is significant within the quasicondensate regime. In Chapter 6 we study the formation of a BEC into a non-equilibrium state by simulating an experiment in which a dilute gas of Rubidium-87 atoms in a highly-elongated 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 temporally-dependent 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 non-equilibrium condensate. Analyses of the first-order correlation function and the condensate wavefunction reveal the spontaneous formation of quantum vortices, as well as strong phase fluctuations at equilibrium indicating quasicondensate behaviour.
Describing partially-condensed Bose gases poses a long-standing theoretical challenge. We present exact stochastic Ehrenfest relations for the stochastic projected Gross-Pitaevskii equation, including both number and energy damping mechanisms, and all projector terms that arise from the energy cutoff separating system from reservoir. We test the theory by applying it to the center of mass fluctuations of a harmonically trapped prolate system, finding close agreement between c-field simulations and analytical results. The formalism lays the foundation to analytically explore experimentally accessible hot Bose-Einstein condensates.
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.