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

2. Properties of an atomic Bose condensate in a double-well potentialIn these variables, regular Josephson tunnelling is described by the contours of constantφ. Numerical integration of these equations gave the phase portraits shown in Fig. 2.5. Inthese plots, the north pole corresponds to the bottom edge of the window, the south poleto the top edge, the point x = − 1 2 to the centre of the plot and the point x = 1 2to themidpoint of the vertical edges.Now the bifurcation point Θ = 1 2 is not the same as the critical point Θ c =1. WhenΘ < 1 2, all trajectories in phase space orbit around the sphere approximately parallel tothe equator, as shown in Fig. 2.5(a), and this corresponds to regular tunnelling. WhenΘ > 1 2, trajectories in the lower portion of the sphere still execute tunnelling oscillations,but in the upper portion of the sphere, the motion is trapped on either the right or theleft half of the sphere in orbits around the two new elliptic fixed points (Fig. 2.5(b)). Theseparatrices of the saddle point at the top of the sphere bound the two types of orbits (Fig.2.5(d)). The critical point is then merely when a trajectory starting at x = ± 1 2lies on theseparatrix (Figs. 2.5(c&d)). Above the critical point, this trajectory will travel around thenew elliptic point, in orbits of decreasing size as the elliptic point moves closer to x = ± 1 2(Fig. 2.5(e)). Even for Θ > 1, there will still be trajectories that can cross from one side ofthe sphere to the other in tunnelling oscillations. A trajectory that starts at x 0 = 0, willalways be able to do this (see Fig. 2.5(e)). This is consistent with the dynamics shown inFig. 2.3.2.2.2 The Gross-Pitaevskii equationWe now turn to the full numerical solution of the Gross-Pitaevskii (GP) equation (Eq.(2.18)) for the double-well system. Rewriting the GP equation using normalised variablesfacilitates the numerical computation:i ∂u(z,τ)∂τ[= − 1 ∂ 22 ∂z 2 +z432z02− z216 + z2 032 + K|u(z,τ)|2 ]u(z,τ), (2.46)where τ = t/t 0 , z = x/r 0 , z 0 = q 0 /r 0 and u = √ r 0 Φ N . The scaling constants aret 0 =1/(2ω 0 )andr 0 = √ t 0 /m = √ /(2mω 0 ). The coefficient of the nonlinear termis K =2 √ πκN/ω 0 =8 √ π 3 Θt 0 /T 0 ,whereT 0 is the period of the tunnelling oscillations.To generate numerical solutions from this scaled GP equation, a split-step Fourier semiimplicitmethod was used. The details of this algorithm may be found in the second halfof this thesis. The code was adapted from the Multi-Dimensional Simulator program of32