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

7. Quantum simulations of evaporatively cooled Bose condensatesFigure 7.3: Simulation of (a) a two-dimensional and (b) a three-dimensional Bose condensate,showing the ensemble average (55 paths and 15 paths, respectively) atom density 〈 n(k) 〉 along onedimension in Fourier space versus time. Time has been normalised by t 0 =0.79ms and momentumby k 0 =1.32 × 10 6 m −1 .21.510.50050t/t 0100 −2−1(a)0k/k 012not have to form in the ground state, the Bose-condensed peaks that occur at differentmomentum values in single runs are averaged out in the overall ensemble.This effectis particularly pronounced in three dimensions, with the extra degree of freedom.more useful indication of condensation is given by the following measure of phase-spaceconfinement:C =∫d 3 k〈ψ 1 (k)ψ 1 (k)ψ ∗ 2 (k)ψ∗ 2 (k)〉(∫d 3 k〈ψ 1 (k)ψ ∗ 2 (k)〉) 2 x30. (7.14)This higher-order correlation function is the quantum analogue of the participation ratiodefined by Hall[75]. It offers an advantage over the variance in that for a multipeakeddistribution, it is largely insensitive to the relative positions of the peaks. For example,with a one-dimensional distribution comprised of a sum of nonoverlapping GaussiansAP (x) = ∑ iA i e − (x−x i )22σ 2 i , (7.15)the unnormalised C parameter˜C =∫dxP 2 (x)(∫dxP (x)) 2≃∑i A2 i σ i2 √ π ( ∑ i A iσ i ) 2 , (7.16)does not depend on x i , to leading order, whereas the variance does:∑σP 2 = i,j A iA j σ i σ j (σi 2 + x2 i − x ix j )( ∑ i A iσ i ) 2 . (7.17)157