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

7. Quantum simulations of evaporatively cooled Bose condensates7.4.1 Constraints and limitationsIn practical computations, it is necessary to consider rather small traps. This is becausethe numerical lattice spacing used to sample the stochastic fields in x-space must be oforder ∆x = 1/k max , where k max is the largest ordinary momentum considered in theproblem. However, the value of the corresponding kinetic energy, E K =(k max ) 2 /2m,must be large enough to allow energetic atoms to escape over the potential barrier of thetrap; otherwise no cooling can take place. This sets an upper bound on the lattice spacingand hence on the maximum trap size, which depends on the number of lattice points thatcan be computed.The available lattice sizes used here were 32 3 points in three dimensions and 64 2points in two dimensions. With this limit, and parameter values similar to those used inthe experiments, the available trap sizes that can be treated are of the order of microndimensions. These are smaller than those used currently, although traps of this type arequite feasible. The other possibility within the constraints is to use a trap which is of largerdimensions but lower in potential height. For this type of trap, which was simulated here,the width was L j =10µm, with a maximum potential height of V max /k B =1.9 × 10 −7 K,and an initial temperature of T 0 =2.4 × 10 −7 K.A further limitation is that the initial density must be such that 〈 n(k) 〉 1; otherwisethe gas would already be Bose condensed, with many particles in particular modes. Thisplaces a limit on the number of atoms which can simulated, if we assume an initiallynoncondensed grand canonical ensemble of (approximately) noninteracting atoms. Therewere initially around 500 atoms in the two-dimensional simulations presented here, and10, 000 in the three-dimensional case. These correspond to atomic densities of n 0 =5.0 ×10 12 /m 2 and n 0 =1.0 × 10 19 /m 3 respectively.However not all the lattice points that atoms can occupy are contained within the trap.Necessarily, some of the lattice points correspond to untrapped sites. The proportion oflattice sites that lie within the trap depends on the trapping potential. For the sinusoidaltrap in three dimensions, up to half the sites lie outside the trap, a result of embedding anapproximately spherical potential within a cubic lattice structure. The nonspherical trapV 3 defined in Eq. (7.5) overcomes this problem and confines more atoms. But in this casethe larger nonlinear terms, coupled with the stochastic terms, drive large excursions intophase space at long times. This causes the fields a j to vary significantly from one trajectory152