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

2. Properties of an atomic Bose condensate in a double-well potentialIn the two-mode approximation, we expand the field operators in terms of the localmodes and introduce the Heisenberg picture annihilation and creation operators∫ĉ j (t) = d 3 ru ∗ j (r) ˆψ(r,t) (2.12)so that [ĉ j , ĉ † k ] ≈ δ jk. The validity of this expansion is ensured when the overlap is small:RE 0= Ω ω 0≪ 1. (2.13)The ratio of the separation of the minima of the global potential V (r) tothepositionuncertainty in the states u j (r) can be as small as 2q 0 /r 0 = 6 (as in some of the simulationspresented in this thesis), and yet still satisfy this condition. The many-body Hamiltonianthen reduces to the following two-mode approximation:Ĥ 2a (t) = E 0 (ĉ † 1ĉ1 +ĉ † 2ĉ2)+ Ω ()2 (ĉ 1ĉ † 2 +ĉ† 1ĉ2)+κ (ĉ † 1 )2 ĉ 2 1 +(ĉ † 2 )2 ĉ 2 2 , (2.14)where κ = U 0 /2V eff ,andV −1eff= ∫ d 3 r|u 0 (r)| 4 is the effective mode volume of each well.Here we have retained only the self-interaction contribution to the nonlinearity, as thecross-interaction contributions are of order ɛ 2 .The two-mode approximation is valid when many-body interactions produce only smallmodifications of the ground state properties of the individual potentials. This is true whenω 0 =22mr 2 0≫ N|U 0|V eff. (2.15)Using V eff ≈ 8π 3/2 r 3 0atoms:for this case, we obtain the following condition on the number ofN ≪ r 0|a 0 | . (2.16)If we choose values of r 0 =5µm and a 0 =5nm, thenN = 100 satisfies this criterion.Thus the two-mode approximation is valid for small number of atoms compared to currentexperiments with N =10 3 ↔ 10 6 .In studying the semiclassical and quantum dynamics of the two-mode condensate, wewill only be concerned with the case where κ>0, corresponding to repulsive interactionbetween atoms. A negative κ is also possible, although gasses with this attractive interactionare rarely used in experiments. Because we have chosen to study condensates inspatially separated wells, the nonlinear cross-interaction between the condensates is negligible.However, one may also study overlapping two-species condensates, as is commonin the literature, where such terms are important.23