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

2. Properties of an atomic Bose condensate in a double-well potential⎛Using M ⃗ = ⎝ 0 11 0⎞⎠, Eq. (2.28) can be written in the formiȦ = ν2 A + Ω 2 ⃗ MA +2Nκdiag(|A 11 | 2 + |A 22 | 2 )A, (2.29)This is the discrete self-trapping equation considered in [10], which describes a system ofcoupled anharmonic oscillators. Finally if we let b j = e iνt/2 A j , we get the semiclassicalequations of motion:ḃ 1 = −iΩ2 b 2 − 2iκN|b 1 | 2 b 1 (2.30a)ḃ 2 = −iΩ2 b 1 − 2iκN|b 1 | 2 b 2 . (2.30b)These equations share the same form with the Heisenberg equations of motion for theannihilation and creation operators, ĉ j and ĉ † j .The total particle number constraint |b 1 | 2 +|b 2 | 2 = 1 provides a constant of the motion,which enables us to rewrite Eq. (2.30) in terms of three real variables, x, y, z defined byThe equations of motion then becomewhich trace out motion on the surface of a sphere defined byAnalytic solutionx = 1 2 (|b 2| 2 −|b 1 | 2 ) (2.31a)y = − i 2 (b∗ 1b 2 − b ∗ 2b 1 ) (2.31b)z = 1 2 (b∗ 1 b 2 + b 1 b ∗ 2 ). (2.31c)ẋ = −Ωy (2.32a)ẏ = Ωx − 4κNzx (2.32b)ż = 4κNyx, (2.32c)14 = x2 + y 2 + z 2 . (2.33)The semiclassical equations for the first order moments (Eq. (2.32)) can be analysed byfinding the constants of the motion. If Eq. (2.32a) is substituted into Eq. (2.32c) and theresult integrated, we get another constant of the motion:C 1 = z + 2κNΩ x2 = z 0 + 2κNΩ x2 0. (2.34)26