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

2. Properties of an atomic Bose condensate in a double-well potentialUsing this expansion, Eq. (2.19) becomes,iu(x) · Ȧ(t) ≃(V (x) − V (2) (x)+NU 0(u21 (x)|A 1 (t)| 2 + u 2 2 (x)|A 2(t)| 2)) u(x) · A(t),(2.23)where cross-interaction terms have been neglected in the nonlinear term and where∂ 2V (2) (x)u(x) = 22m ∂x 2 u(x)= mω2 02 diag ( (x − q 0 ) 2 − 2r0 2 , (x + q 0) 2 − 2r02 )u(x). (2.24)After a little algebraic manipulation and an integration over x, wegetthefollowingequationfor the linear evolution of A:⎛iȦ = ⎜⎝R 11 −S 12 R 211−S 12 S 21R 12 −S 12 R 221−S 12 S 21R 21 −S 21 R 111−S 21 S 21R 22 −S 21 R 121−S 21 S 21⎞⎟ A, (2.25)⎠whereR jj =R ij =S ij =∫∫∫(V − V (2)jj )u2 j dx ≃ 3ω 0r0216q02 + ω 02(V − V (2)jj )u iu j dx ≃ Ω 2u i u j dx = e −q2 0 /2r2 0 = ɛ(2.26a)(2.26b)(2.26c)for i ≠ j.If the overlap between the local mode functions is sufficiently small, i.e.(q 0 /r 0 ) 2 ≫ 1, then |R jj |≫|S ij R ji | and |R ij |≫|S ij R jj |. This leads to the followingapproximate equations of motion:where ν =3ω 0 r 2 0 /(8q2 0 )+ω 0.⎛ ⎞iȦ = ⎝ R 11 R 12⎠ AR 21 R 22⎛ ⎞=Ω2⎝ ν 2Ω2Ω2ν2⎠ A, (2.27)Similar considerations of the nonlinear terms lead to the full equations of motion:⎛⎞νiȦ = ⎝ 2 + NU 0|A 11 | 2ΩV eff2 ⎠ A. (2.28)ν2 + NU 0|A 22 | 2V eff25