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

2. Properties of an atomic Bose condensate in a double-well potentialFor the two-mode model, this becomesρ(x, t) = 〈 N2 − Ĵx(t) 〉 |u 1 (x)| 2 +〈 N2 + Ĵx(t)〉|u 2 (x)| 2 − 2 〈 Ĵ z (t) 〉 u 1 (x)u 2 (x), (2.71)where the u i are the local modes for each well, as defined in Eq. (2.7). Now ρ can becalculated analytically from the ground state in the two limiting cases of the collisionalstrength. To facilitate the discussion, we will use the single-particle normalised interactionstrength κ = κ/Ω, which differs by a factor of N from the multiparticle normalised interactionstrength Θ introduced in the Sec. 2.2.1. For the ∣ j, −j〉zfunction factorises intoground state, the densityρ a (x, t) =N|φ(x)| 2 ;φ(x) = (u 1(x) − u 2 (x))√2, (2.72)where φ(x) is the single-particle ground state of the global potential. Because this canbe factorised into the square of a single macroscopic wave-function, it possesses full firstordercoherence and can be characterised by a single long-range order parameter. For thestrongly interacting case, the ground state ∣ ∣ j, 0〉xleads toρ b (x, t) = N 2(u1 (x) 2 + u 2 (x) 2) , (2.73)which is just the sum of the density functions for each of the local modes. Both of thesecases are illustrated in Fig. 2.16, for a double-well potential in which the two minima areat q 0 ± 3x 0 . The feature that distinguishes the two extremes is the presence or absenceof the interference fringe in the centre between the two wells, whose presence reduces thedensity function at this point. Its size is a maximum for the tunnelling dominated groundstate (when κ ≪ 1) and is zero when the κ ≫ 1. In general, its relative size is proportionalto the moment | 〈 ∣ ∣ 〉Ψ ∣Ĵz∣Ψ0 0 |/N 2 ,where ∣ 〉 Ψ0 is the ground state. Figure 2.17(a) showsthat 〈 ∣ ∣ 〉Ψ ∣Ĵz∣Ψ0 0 /N changes little as N varies, and therefore the size of the interferenceterm varies approximately inversely with the size of the condensate, becoming negligiblefor large N. Figure 2.17(b) reveals how the interference term varies with κ in between thetwo extremes. There appears to be a threshold value of κ, above which the interferenceterm is close to zero.Second-order coherence features are indicated by the normally-ordered second-ordercorrelation function:G 2 (x 1 ,x 2 ,t)= 〈 ˆψ† (x 1 ) ˆψ † (x 2 ) ˆψ(x 2 ) ˆψ(x 1 ) 〉 . (2.74)50