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

2. Properties of an atomic Bose condensate in a double-well potentialIn terms of these operators, the total atom number operator is∫ˆN = d 3 r ˆψ † (t, r), ˆψ(t, r). (2.3)The trapping potential is a double-well potential in which the interwell coupling occursalong the x-axis:V (r) =b ( x 2 − q02 ) 2 1 +2 mω2 t (y 2 + z 2 ), (2.4)where ω t is the trap frequency in the y − z plane and where b gives the strength of theconfinement in the x direction. This potential has elliptic fixed points at r 1 =+q 0 x,r 2 = −q 0 x, at which the linearised motion is harmonic with frequency ω 0 = q 0 (8b/m) 1/2 .For simplicity, we set ω t = ω 0 and scale the length by r 0 = √ /2mω 0 ,whichistheposition uncertainty in a harmonic oscillator ground state.The barrier height is thengiven by B =(ω 0 /8)(q 0 /r 0 ) 2 .A full analysis of the quantum dynamics resulting from the many-body HamiltonianEq. (2.1) is not tractable, however considerable insight can be gained within a two-modeapproximation or with a semiclassical picture. For a suitable choice of B, thetwolowestsingle-particle energy eigenstates will lie beneath the barrier, with the third eigenstateseparated by a relatively large energy gap. We will assume that the atoms interact sufficientlyweakly such that, near zero temperature, they have condensed into the two lowestsingle-particle states. This enables a treatment of the many-body problem with a twomodeapproximation, as Fig. 2.1 illustrates. The resulting model is sufficiently simple toenable an analytic solution to be found for the semiclassical equations, and to permit atractable numerical comparison of the semiclassical description with the full quantum dynamics.We will first derive the two-mode Hamiltonian before discussing the semiclassicaldynamics in Sec. 2.2.We can expand the potential around each minimum into local harmonic potentials:V (r) =Ṽ (2) (r − r j )+ ... , (2.5)where j =1, 2andṼ (2) (r) =4bq 2 0|r| 2 . (2.6)The local-mode solutions of the individual wells u j (r) arethenu j (r) = −(−1)j(2πr0 2) e − 1 3 4 (x−q 0) 2 +y 2 +z 2 )/r0 2 (2.7)421