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

2. Properties of an atomic Bose condensate in a double-well potentialFigure 2.4: Bifurcation diagrams for fixed points of the semiclassical system, showing the Θdependence of (a) z ∞ and (b) x ∞ . The continuous line corresponds to elliptic fixed points whilethe dashed line corresponds to the saddle fixed points. A pitchfork bifurcation occurs at Θ= 1 2 .0.51(a)0.53(b)z ∞03,4x ∞01,2−0.52ellipticsaddle−0.54ellipticsaddle0 0.5 1 1.5 2Θ0 0.5 1 1.5 2ΘThe resultant eigenvalues for each fixed point are:λ 1 = ± √ 2Θ − 1 (2.44a)λ 2 = ±i √ 2Θ + 1 (2.44b)λ 3,4 = ±i √ 4Θ 2 − 1 (2.44c)Thus we have a pitchfork bifurcation at Θ = 1 2 . For Θ < 1 2, there are two elliptic pointsat the top and bottom of the Bloch sphere. Above the bifurcation, i.e. Θ > 1 2, the ellipticpoint at the top of the sphere turns into a saddle point, and two elliptic points split offfrom the north pole and move towards the equator at x = ± 1 2. All of this is illustrated inthe bifurcation diagrams in Fig. 2.4.Phase-space portraitsWe can analyse the motion away from the fixed points by plotting the phase-space portraits.This is most easily done by using polar co-ordinates (θ, φ) = (arccos 2z,arctan y/x),so that x = 1 2 cos φ sin θ, y = 1 2 sin φ sin θ and z = 1 2cos θ. The equations of motion in polarco-ordinates become:˙θ = −κN sin 2φ sin θ (2.45a)˙φ = Ω− κN (1 + cos 2φ)cosθ. (2.45b)31