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arXiv:0812.1926 · Laser Physics

The stochastic Gross-Pitaevskii equation and some applications

S. P. Cockburn, Nick P. Proukakis

Published Apr 1, 2009• 74 citations
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Abstract

The stochastic Gross-Pitaevskii equation represents a versatile approach for studying the dynamics of trapped degenerate ultracold Bose gases in the presence of large phase and density fluctuations. Following a brief review of the original formulation of Stoof, which highlights the benefits of this approach and its relation to alternative theories, we present selected applications for the dynamics of effectively one-dimensional systems, and briefly discuss the generalization to two-dimensional systems, highlighting certain potential pitfalls in their numerical implementations.

DOI
10.1134/s1054660x09040057

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The stochastic Gross-Pitaevskii equation represents a versatile approach for studying the dynamics of trapped degenerate ultracold Bose gases in the presence of large phase and density fluctuations. Following a brief review of the original formulation of Stoof, which highlights the benefits of this approach and its relation to alternative theories, we present selected applications for the dynamics of effectively one-dimensional systems, and briefly discuss the generalization to two-dimensional systems, highlighting certain potential pitfalls in their numerical implementations.
The stochastic Gross-Pitaevskii equation represents a versatile approach for studying the dynamics of trapped degenerate ultracold Bose gases in the presence of large phase and density fluctuations. Following a brief review of the original formulation of Stoof, which highlights the benefits of this approach and its relation to alternative theories, we present selected applications for the dynamics of effectively one-dimensional systems, and briefly discuss the generalization to two-dimensional systems, highlighting certain potential pitfalls in their numerical implementations.
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.
In this thesis, we apply the stochastic Gross-Pitaevskii equation, using the formulation of Stoof, to the study of weakly interacting, finite temperature atomic Bose gases, both in and out of equilibrium. Explicitly maintaining the effects of phase and density fluctuations, this state-of-the-art method is well suited to the study of low dimensional Bose systems. As such, we consider the non-equilibrium dynamics of dark matter wave solitons, oscillating within harmonically trapped, one-dimensional phase-fluctuating condensates, through numerical simulations of the stochastic Gross-Pitaevskii equation. In treating the damping due to higher lying thermal modes of the system, we include the HartreeFock contribution of the equilibrium atomic density, which is a novel feature of our work. A stochastic approach is found to be essential in capturing the shot-to-shot variation in soliton trajectories observed experimentally. Neglecting stochastic effects, we find the well-known dissipative Gross-Pitaevskii equation to recover the average dynamics of the stochastic model, on retaining the microscopically justified form for the damping of this approach. Such a coupling is found to be necessary in modelling the dissipative nature of finite temperature soliton dynamics. Furthermore, we undertake a comparison between the finite temperature equilibrium state of the stochastic Gross-Pitaevskii equation, and that predicted within a number conserving Bogoliubov approach. Through a series of numerical experiments, we compare the equilibrium properties of a harmonically trapped Bose gas, considering our results in light of those of independent theoretical treatments, and additionally assessing the sensitivity of the numerical implementation of the former method to the numerical grid spacing. In the regime in which thermal effects dominate, the stochastic Gross-Pitaevskii approach proves to be a more consistent means of generating such equilibrium states.
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.
The stochastic Gross-Pitaevskii equation is shown to be an excellent model for quasi-one-dimensional Bose gas experiments, accurately reproducing the in situ density profiles recently obtained in the experiments of Trebbia et al. [Phys. Rev. Lett. 97, 250403 (2006)] and van Amerongen et al. [Phys. Rev. Lett. 100, 090402 (2008)] and the density fluctuation data reported by Armijo et al. [Phys. Rev. Lett. 105, 230402 (2010)]. To facilitate such agreement, we propose and implement a quasi-one-dimensional extension to the one-dimensional stochastic Gross-Pitaevskii equation for the low-energy, axial modes, while atoms in excited transverse modes are treated as independent ideal Bose gases.
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.