A stochastic Gross-Pitaevskii equation is derived for partially condensed Bose gas systems subject to binary contact interactions. The theory we present provides a classical-field theory suitable for describing dissipative dynamics and phase transitions of spinor and multicomponent Bose gas systems composed of an arbitrary number of distinct interacting Bose fields. A class of dissipative processes involving distinguishable particle interchange between coherent and incoherent regions of phase space is identified. The formalism and its implications are illustrated for two-component mixtures and spin-1 Bose-Einstein condensates. For systems composed of atoms of equal mass, with thermal reservoirs that are close to equilibrium, the dissipation rates of the theory are reduced to analytical expressions that may be readily evaluated. The unified treatment of binary contact interactions presented here provides a theory with broad relevance for quasiequilibrium and far-from-equilibrium Bose-Einstein condensates.
A stochastic Gross-Pitaevskii equation is derived for partially condensed Bose gas systems subject to binary contact interactions. The theory we present provides a classical-field theory suitable for describing dissipative dynamics and phase transitions of spinor and multicomponent Bose gas systems composed of an arbitrary number of distinct interacting Bose fields. A class of dissipative processes involving distinguishable particle interchange between coherent and incoherent regions of phase space is identified. The formalism and its implications are illustrated for two-component mixtures and spin-1 Bose-Einstein condensates. For systems composed of atoms of equal mass, with thermal reservoirs that are close to equilibrium, the dissipation rates of the theory are reduced to analytical expressions that may be readily evaluated. The unified treatment of binary contact interactions presented here provides a theory with broad relevance for quasiequilibrium and far-from-equilibrium Bose-Einstein condensates.
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.
This paper develops and implements the stochastic projected Gross-Pitaevskii equation for spin-1 Bose gases, addressing key considerations for numerical simulations. As an application of the theory we explore equilibrium phases in a two-dimensional spin-1 gas, where quasi-long-range order emerges via a Berezinskii-Kosterlitz-Thouless transition. Our analysis includes the definition of superfluid densities for both mass and spin degrees of freedom in a manner suitable for implementation within a stochastic projected Gross-Pitaevskii equation simulation. We present a finite-temperature phase diagram for the ferromagnetic spin-1 Bose gas and identify three distinct superfluid phases: two exhibiting conventional Berezinskii-Kosterlitz-Thouless-like behavior and an additional phase that simultaneously supports independent mass and spin superflows. As temperature increases, the stability region of this additional phase shrinks. We provide a characterization of the phase transitions through consideration of the spin-component densities and the unbinding of multiple types of vortices. This work provides a foundation for further studies of nonequilibrium and finite-temperature phenomena in spinor Bose gases.
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.
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.
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.