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

7. Quantum simulations of evaporatively cooled Bose condensateswhere α(t) =α min{t, t max } and L =min{L j }. The advantage of this method is thatthe spatial density of atoms is kept high, which aids the thermalisation necessary forevaporative cooling.In deterministic limit, the positive-P equations (Eq. (7.4)) correspond precisely to thewell-known Gross-Pitaevskii (GP) equation, with the addition of a coefficient Γ(x) forthe absorption of atoms by the reservoirs. Quantum effects come from the spontaneouscollision terms ξ j , which are real Gaussian stochastic fields, with correlations:〈ξ1 (s, x)ξ 2 (t, y) 〉 = δ ij δ(s − t)δ d (x − y) . (7.8)Unlike the Wigner equation for phase-space evolution (Eq. (6.11)), no explicit vacuumnoise in the initial conditions is required. This is because the positive-P variables correspondto normally ordered operator products, in which vacuum contributions do notexplicitly appear. The quantum correlations that can be calculated include n(k) =〈ψ1 (k)ψ ∗ 2 (k)〉 , which gives the observed momentum distribution.7.3.2 Wigner techniqueAn alternative strategy to generate phase-space equations is to use the Wigner function,as in Ch. 6. The Wigner technique allows the direct calculation of symmetrically orderedproducts, in which a vacuum contribution explicitly appears. The disadvantage of usingthe Wigner function is that it is not guaranteed to be positive, with higher-order termsappearing in the Fokker-Planck equation.Once these terms have been removed, theWigner function is constrained to be positive and Langevin equations can be generated.Thus the Wigner technique is not exact, and may produce incorrect results at long times,especially for low occupation numbers.positive-P results.However, it does provide a good check on theWith the higher-order terms removed, the resultant phase-space evolution is given byi ∂ [ −2∂t ψ(t, x) = 2m ∇2 + V (t, x) − i ]2 Γ(t, x)+U 0ψ(t, x)ψ ∗ (t, x) ψ(t, x)+ i √2√Γ(x)ξ(t, x) , (7.9)where ξ(t, x) is a complex delta-correlated stochastic field:〈ξ(s, x)ξ ∗ (t, y) 〉 = δ(s − t)δ d (x − y) . (7.10)149