Open Quantum Dynamics of Mesoscopic Bose-Einstein ... - Physics

7. Quantum simulations of evaporatively cooled Bose condensatesinvestigation into the quantum dynamics of Bose condensates. The methods used here canand must be further developed. Hence we will also discuss the limitations of the methodand also possible ways in which it may be improved to extend the results presented here.In Sec. 7.2 we will overview different approaches to tackling this problem and introducethe phase-space techniques that will be used. Section 7.3 gives the details of the positive-Pand Wigner methods, and also describes the physical system modelled in the simulations.Details of the numerical technique are given in Sec. 7.4, as well as the computationalconstraints that are placed on the range of physical situations that can be modelled. Thelargest section of this chapter (Sec. 7.5) contains the numerical results, and the thesisconcludes in Sec. 7.6 with a discussion of possible future directions.7.2 Quantum simulationsMany calculations of cooling dynamics have treated the cooling process classically[17,36, 101], often using the classical transport equation for either a truncated Boltzmanndistribution[8] or the more accurate truncated Bose distribution[183, 184]. This leads tothe question of how to handle the transition to the final quantum-dominated condensate.It is often assumed to be a canonical ensemble at a temperature estimated from theclassical theory with the final ensemble behaviour calculated from the mean-field Gross-Pitaevskii (GP) equation[11, 34, 96, 141]. Mean-field theory can only give semiclassical results,although some authors have included quantum corrections to the mean-field groundstate[15, 146, 189] to account for nonclassical features. Others have developed quantumkinetic theories, either through quantum corrections[175] to the GP equations or directlyfrom a master equation[64, 65, 66, 83, 92, 93]. These master-equation approaches are derivedin the weak-interaction limit in which the condensate states are taken to be low-lyingeigenstates of the trapping potential, populated from a thermal reservoir of excited atoms.The theory of Gardiner et al[63] enables a quantitative prediction of condensate growththat agrees well with experiment. In an alternative approach, Stoof[168] has derived aFokker-Planck equation for a Wigner function, which can be used to describe the kinetic,coherent and transition regimes of evaporative cooling, for the limits of both weak andstrong coupling. All of these quantum approaches make assumptions about how the statesevolve in order to achieve tractable calculations of measurable properties.143