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

5. Weak force detection using a double Bose-Einstein condensateWe will see that with such an initial state the precision of the measurement can be controlledby the total number of atoms in the condensate. This state is easily representedon the Bloch sphere (Fig. 5.1(a)) by two diametrically opposed points on the equator. Amethod for preparing such a state has been proposed in [31] for overlapping two-speciescondensates, through control of the analogous Josephson coupling term. In principle, thetwo species of this superposition state could be coherently transfered to spatially separatedwells in order to produce the state we require here.The second step in the measurement process, after having achieved the ground statedescribed by Eq. (5.4), and before the tunnelling is turned on, is to allow the weak forceto act for a certain time τ. This force may be due to a varying gravitational field, forexample, and has the effect of adding a linear ramp to the potential:ˆ H F = ∆Ĵx, (5.5)where ∆ is the frequency shift induced by the weak force (see Fig. 5.1(b)) and j = N/2.The initial ground state (Eq. (5.4)) then evolves into the superposition:∣∣Ψ(τ) 〉 = 1 √2(ei∆jτ ∣ ∣j, −j 〉 x + e−i∆jτ∣ ∣j, j 〉 x). (5.6)The state now contains the phase shift induced by the weak force. This phase shift is“amplified” 2j times the single-particle value due to the specially prepared initial state.The next step involves a technique analogous to Ramsey interferometry. In order todetect the induced phase shift, we rotate the state (Eq. (5.6)) around the Bloch sphere by90 0 . Such a rotation can be achieved by turning on the tunnelling between the two wellsof the trap for a time t π = π/(2Ω). If the interatomic interaction is strong, the rotation2operation will be affected by the nonlinear term in Eq. (2.49), which will distort the finalstate from the y-axis on the Bloch sphere and affect the precision of the measurement.To avoid this, we could use a Feshbach resonance[89, 164] to suppress the collisions. Thestate is then rotated to∣∣Ψ(τ + t π/2 ) 〉 = √ 1 (e i∆jτ∣ ∣j, −j 〉2y + e−i∆jτ∣ ∣j, j 〉 ). (5.7)yFinally, a number measurement can be performed on one of the condensates. Thiscorresponds to a projection onto the Ĵx eigenstates. The resultant probability distributionisP x (m) = 1 2∣∣e i∆τj 〈 〉x j, m|j, −jy + 〈 〉e−i∆τj x j, m|j, jy∣ 2 . (5.8)109