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

5. Weak force detection using a double Bose-Einstein condensatewhere χ α′ mWis the characteristic function for the coherent state ∣ ∣α ′ m〉. Due to the linearityof the Fourier transform, the Wigner function will thus be a sum of Gaussians, weightedby the atom number distribution of the condensate:G(α, α ∗ )= 2 πj∑m=−jP x (m)e −2|α−α′ m| 2 (5.20)For convenience, we set the initial conditions of the light field such that x(0) = 0, i.e.α 0 = i 2y(0). This givesG(x, y) = 2 πj∑m=−jAfter integration, the marginal distribution becomesp(x) = 1 √2πP x (m)e − 1 2 (x−y(0) sin χt I m) 2 − 1 2 (y−y(0) cos χt I m) 2 . (5.21)j∑m=−jP x (m)e − 1 2 (x−y(0) sin χt I m) 2 . (5.22)Thus each m value is mapped onto a Gaussian at position y(0) sin χt I m with width equalto one. This is illustrated in Fig. 5.2. To be able to distinguish without ambiguity differentm values in the output, there should be at least 4 standard deviations between the meansof adjacent Gaussians. The resultant condition on the atom-light coupling is thenχt I > 4|y(0)| = 2|α 0 |(5.23)andχt I ≪ 1 N . (5.24)If the fringes close to m = 0 only are needed, then this last condition may be relaxedconsiderably to a condition which merely prevents aliasing:χt I < π N . (5.25)Nevertheless, there is a limit on the size of the induced phase shift.The analysis above assumes perfect detector efficiencies and an infinite time of integrationso that all the light is removed from the cavity. The results can be generalisedto hold when this is not the case[[179],p63]. For a detector efficiency of η ∞ and a totalintegration time of T , then the distribution for the integrated photocurrent isp(x) = 1 √ 2πηj∑m=−jP x (m)e − 1 2 (x−ηy(0) sin χt I m) 2 /η , (5.26)112