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

5. Weak force detection using a double Bose-Einstein condensateNow the inner product of the J x and J y eigenstates will be a binomial function of mpeaked around m =0,with〈 〉x j, m|j, −jy = 〈 〉e−imπ x j, m|j, jy . (5.9)This leads toP x (m) = 2cos 2 〈 (∆jτ + mπ/2) ∣ x j, m|j, j〉y∣ 2 (5.10)⎧⎨ 2cos 2 〈 ∆jτ ∣ x j, m|j, j〉y∣ 2 m even=⎩2sin 2 〈 ∆jτ ∣ x j, m|j, j〉y∣ 2 , (5.11)m oddwhich describes how the output fringes are shifted by the presence of the force ∆. In theabsence of the force, all the odd fringes are absent, but for ∆ ≠ 0, the probability that aparticular measurement will fall on an odd fringe isPr{m odd} =sin 2 ∆jτ ≃ (∆jτ) 2 (5.12)for small ∆.Note that Eq. (5.11) is only correct when j is an integer (i.e. an even number of atoms).For half-integral values of j, Eq. (5.10) becomesP x (m) = 2(1+(−1) n 〈 sin(2∆jτ)) ∣ x j, m|j, j〉y∣ 2 , (5.13)where n = m + 1 2, an integer. In the absence of the force, odd and even fringes are equallyprobable, but for ∆ ≠ 0, the odd fringes reduce and the even fringes grow:Pr{n odd} ≃ 1 − ∆jτ (5.14)2for small ∆. This does not offer the same advantage as Eq. (5.12), because the probabilitydoes not grow from zero, and the change is only linear in the size of the force.5.4 Measurement readoutThe measurement of atom number is effected through the homodyne scheme[33] illustratedin Fig. 3.1. The condensate is placed in an optical cavity, which at the time of the readoutstage of the measurement contains a light field which is highly driven and damped. Theoptical field is thus in a coherent state with amplitude α 0 . It is also detuned from anyatomic resonance so that the condensate merely imposes a phase shift on the light. This110