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

2. Properties of an atomic Bose condensate in a double-well potential2.2 Semiclassical dynamicsWe begin the semiclassical analysis with the mean-field approximation. For this we employthe Hartree approximation [185] for a fixed number of atoms N, and write the atomic statevector as|Ψ N (t)〉 = 1 √N![∫d 3 rΦ N (r,t) ˆψ † (r, 0)] N|0〉, (2.17)where |0〉 is the vacuum.The self-consistent Gross-Pitaevskii (GP) equation 1 for thecondensate wave function Φ N (r,t) follows from the Schrödinger equation i| ˙Ψ N (t)〉 =Ĥ(0)|Ψ N (t)〉, and is given by [106, 154, 185]i ∂Φ N∂t=[− 22m ∇2 + V (r)+NU 0 |Φ N | 2 ]Φ N . (2.18)For a particular choice of the global potential V (r), Eq. (2.18) can be solved numericallyfor a given initial condition. We will assume a factorised solution Φ N (r,t) =φ N (x, t)Φ N (y,z,t) and only consider the nonharmonic motion in the x-direction.resulting equation isi ∂φ N∂t=[− 2The∂ 2]2m ∂x 2 + V (x)+NU 0|φ N | 2 φ N . (2.19)Before proceeding to a numerical solution of this equation we will first derive an approximatetwo-mode model for which the Gross-Pitaevskii equation is analytically solvable.2.2.1 Two-mode semiclassical analysisDeriving the equations of motionWe begin by expanding the mean field φ N in terms of the local modes as followsφ N (x, t) =u(x) · A(t), (2.20)whereu(x) = (u 1 (x),u 2 (x))()= (2πr0 2 )− 1 4 e −(x−q 0) 2 /4r0 2 , −e−(x+q 0 ) 2 /4r02(2.21)A(t) = (A 1 (t),A 2 (t)) . (2.22)1 Also known as the nonlinear Schrödinger equation24