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

7. Quantum simulations of evaporatively cooled Bose condensatesWe have now only one phase-space equation in which the quantum effects enter as vacuumfluctuations in the initial conditions:〈∆ψ(0, x)∆ψ ∗ (0, y) 〉 = 1 2 δd (x − y) . (7.11)The effect of the stochastic field ξ(t, x) is to maintain the vacuum noise that would otherwisebe damped by the absorption term.Unless otherwise indicated, all results and figures in this chapter pertain to the positive-P calculations.7.4 Computational methodIn order to generate numerical solutions to the phase-space equations, scaled variablescan be introduced: τ = t/t 0 , z = x/x 0 , a j = ψ j x d/20 and ζ j = ξ j t 1/20 x d/20 . The positive-Pequations, in scaled form, become[∂i∂τ a j(τ,z) =2 ∇2 − iu 0 a j (τ,z)a ∗ 3−j (τ,z) − iv(τ,z) − γ(τ,z)−i√ iu 0 ξ j (t, z)]a j (τ,z) ,(7.12)where v(z) =t 0 V (x)/, γ(z) =t 0 γ(x)/2 andu 0 = U 0 t 0 /(x d 0 ). Here x 0 is set to x 0 =(t 0 /m) 1/2 .In the simulations, these equations were discretised and propagated forward using asplit-step Fourier method on a transverse lattice z i of l d sites. For most of the simulations,l = 32. The deterministic dispersion and potential/absorption terms in Eq. (7.12) wereincorporated into exponential propagators, U k−prop (k i , ∆τ k )andU x−prop (z i , ∆τ k ) respectively,with the dispersion term applied in Fourier space. The nonlinear and stochasticcontribution to the increment ∆a NL (a(z i ,τ k ), ∆τ k ) were applied in co-ordinate space usingan iterative semi-implicit method, which ensures stability and convergence for stiffnonlinear equations with stochastic terms[49]. In this algorithm, the field value at τ k+1 is150