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

5. Weak force detection using a double Bose-Einstein condensateAs before the force acts for a time τ and then the tunnelling for a time t π2 :b j (τ + t π/2 ) = e (−1)j+1 iRq 0 τ/ b j (0) cos Ωt π/22+ ie (−1)j iRq 0 τ/ b 3−j (0) sin Ωt π/2. (5.41)2Suppose we start off with an equal occupation in each well, such that b 1 (0) = b 2 (0) = 1 √2.Then the mean population difference is shifted by the presence of the weak force:〈m〉=N2(|b2 (τ + t π/2 )| 2 −|b 1 (τ + t π/2 )| 2)= − N 2 sin 2Rq 0τ= − N sin ∆τ. (5.42)2This did not occur in the previous quantum treatment, in which 〈 m 〉 ≡ 0.Evenwhen the system is simply in a number state (not a superposition) with an equal numberof atoms in each well, i.e. ∣ j, 0 , the mean population difference is unaffected by the〉xpresence of the force. This demonstrates that, as a classical treatment, the mean fieldsituation cannot be regarded as the large number limit of a quantum number state.5.7 Coherent statesIn quantum optics, it is the minimum uncertainty coherent state ∣ ∣ α〉which is most like amean field with amplitude α. The analogue in our case is the atomic coherent state, orBloch state, first introduced in Sec. 4.3.1:⎛〉 ∑∣ β = ⎝m2jm + j⎞⎠βm+j ∣ 〉 ∣j, m(1 + |β| 2 ) j z , (5.43)where β can be described in terms of the angular coordinates of a point on the sphereβ =tanθe iψ .Suppose the system starts in the state given by β = 0, which is symmetric withrespect to the two wells. This state is also the ground state ∣ ∣j, −j 〉 of the system whenzΩ ≫|κ|N. Hence the coherence between the two condensates has been well establishedthrough tunnelling. To work out the mean population difference, we can use the Heisenbergequations of motion. Under the weak-force Hamiltonian ĤF (Eq. (5.5)), the annihilation117