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

5. Weak force detection using a double Bose-Einstein condensateFigure 5.3: The shift in the mean population difference for an initial Bloch state, as a functionof the Bloch-state parameters (θ, φ). The condensate contains 100 atoms and has been subjectedto a phase shift of τ∆ =0.1.proportion of detection events falling on odd fringes (Eq. (5.12)). The uncertainty in thisbinomial distribution with probability P =sin 2 ∆jτ is√P (1 − P )δP =, (5.49)N Dwhere N D is the number of detection events. The relative uncertainty in the phase is thusδφφ = 1 dP−11φ ∣ dφ ∣ δP =∆τN √ . (5.50)N DWhen the initial state is the Bloch coherent state ∣ ∣ j, −j〉, the uncertainty in the mean ofthe distribution is√ 〈m2 〉 − 〈 m 〉 2δ 〈 m 〉 =This gives a relative uncertainty in φ ofδφφ = 1 d 〈 m 〉φ ∣ dφ ∣−1δ 〈 m 〉 =N D. (5.51)1∆τ √ NN D. (5.52)Thus the precision of the measurement grows in proportion to the number of condensedatoms when the entangled state (Eq. (5.4)) is used, but only as the square root of the119