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We show how to adapt the ideas of local energy and momentum conservation in order to derive modifications to the Gross-Pitaevskii equation which can be used phenomenologically to describe irreversible effects in a Bose-Einstein condensate. Our approach involves the derivation of a simplified quantum kinetic theory, in which all processes are treated locally. It is shown that this kinetic theory can then be transformed into a number of phase-space representations, of which the Wigner function description, although approximate, is shown to be the most advantageous. In this description, the quantum kinetic master equation takes the form of a GPE with noise and damping added according to a well-defined prescription--an equation we call the stochastic GPE. From this, a very simplified description we call the phenomenological growth equation can be derived. We use this equation to study i) the nucleation and growth of vortex lattices, and ii) nonlinear losses in a hydrogen condensate, which it is shown can lead to a curious instability phenomenon.
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We provide a derivation of a more accurate version of the stochastic Gross-Pitaevskii equation, as introduced by Gardiner et al. (J. Phys. B 35,1555,(2002). The derivation does not rely on the concept of local energy and momentum conservation, and is based on a quasi-classical Wigner function representation of a high temperature master equation for a Bose gas, which includes only modes below an energy cutoff E_R that are sufficiently highly occupied (the condensate band). The modes above this cutoff (the non-condensate band) are treated as being essentially thermalized. The interaction between these two bands, known as growth and scattering processes, provide noise and damping terms in the equation of motion for the condensate band, which we call the stochastic Gross-Pitaevskii equation. This approach is distinguished by the control of the approximations made in its derivation, and by the feasibility of its numerical implementation.
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.
We study the Cauchy problem for the 3D Gross-Pitaevskii equation. The global well-posedness in the natural energy space was proved by Gerard cite{Gerard}. In this paper we prove scattering for small data in the same space with some additional angular regularity, and in particular in the radial case we obtain small energy scattering.
We study the evolution of 3d weakly interacting bosons at finite chemical potential with the stochastic Gross-Pitaevskii equation. We fully characterise the vortex network in an out of equilibrium. At high temperature the filament statistics are the ones of fully-packed loop models. The vortex tangle undergoes a geometric percolation transition within the thermodynamically ordered phase. After infinitely fast quenches across the thermodynamic critical point deep into the ordered phase, we identify a first approach towards the critical percolation state, a later coarsening process that does not alter the fractal properties of the long vortex loops, and a final approach to equilibrium. Our results are also relevant to the statistics of linear defects in type II superconductors, magnetic materials and cosmological models.
In this paper, we consider the dynamical evolution of dark vortex states in the two-dimensional defocusing discrete nonlinear Schroedinger model, a model of interest both to atomic physics and to nonlinear optics. We find that in a way reminiscent of their 1d analogs, i.e., of discrete dark solitons, the discrete defocusing vortices become unstable past a critical coupling strength and, in the infinite lattice, they apparently remain unstable up to the continuum limit where they are restabilized. In any infinite lattice, stabilization windows of the structures may be observed. Systematic tools are offered for the continuation of the states both from the continuum and, especially, from the anti-continuum limit. Although the results are mainly geared towards the uniform case, we also consider the effect of harmonic trapping potentials.
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