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

5. Weak force detection using a double Bose-Einstein condensatewhich is unaffected by ĤStr.If the Ω is not exactly zero then, for a strong atom-light interaction, a back actionmay develop over time which will induce tunnelling[33] through momentum fluctuations.This would directly affect the phase of the cavity field, and it also may open a way for theatom-atom interaction to have an effect. In order to prevent this, the interwell couplingmust be suppressed by switching on a high barrier, either magnetic or with a far detunedlaser pulse.5.6 Mean-field limitThe scheme outlined above depended on starting in a state which was a quantum superpositionof two condensates and on the resulting entanglement. For comparison, we willanalyse the mean-field analogue to discover which features of this scheme remain in theabsence of quantum entanglement.In the mean-field limit, the system is described by a Gross-Pitaevskii (GP) equation[106,154, 185]:i ˙Φ(x, t) =(− 2 ∂ 2)2M ∂x 2 + Rx + V (x)+U 0N|Φ(x, t)| 2 Φ(x, t), (5.36)where M is the atomic mass, the constant R is the gradient of the force and U 0 is thestrength of the interparticle interactions. Suppose that the interatomic collisions are negligible.Then, when the overlap between the wells is small, we may expand the mean fieldin terms of the local wave functions of each well:Φ(x, t) =b 1 (t)e −iE0t/ u 1 (x)+b 2 (t)e −iE0t/ u 2 (x), (5.37)whereandu j (x) =√1e (x−(−1)j q 0 ) 2 /4r 2(2πr 0 ) 1 0, r0 = , (5.38)42Mω 0∫b j (t) =u ∗ j (x)Φ(x, t)dx. (5.39)The ground-state energy of each of the local modes is E 0 = (ν + ω 0 )/2, as in Ch. 2. TheGP equation (Eq. (5.36)), then generates the following equations of motion:ḃ j (t) = (−1)j+1 iRq 0b j (t)+ iΩ 2 b 3−j(t). (5.40)116