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

7. Quantum simulations of evaporatively cooled Bose condensates(real) time domain to allow dynamical calculations has not yet been achieved, althoughprogress to this end has been made with coherent-state operator expansions[21].Any simulation of evaporative cooling must include all the relevant modes for thetrapped atoms, up to the energy levels required for hot atoms to escape. The threedimensionalcalculations presented here include more than 3 × 10 4 relevant modes, withup to 1.0 × 10 4 atoms present. The quantum state vector therefore has over 10 40000 components.Keeping track of all these components and calculating the evolution matrix toperform quantum number-state calculations in the time domain is an enormous computationalproblem, as Feynman emphasised.But we have seen in Ch. 6 how a many-body multimode quantum system can besimulated, using the phase-space methods that have already proved successful in lasertheory[67, 111, 169]. These techniques can handle large numbers of particles, but can alsosystematically treat departures from classical behaviour. For example, generalised phasespacerepresentations were used to correctly predict quadrature squeezed quantum solitondynamics in optical fibres[24, 50], which as we saw in Ch. 6 are described by nearly identicalquantum equations (Eq. (6.3)) to those used in atom-atom interaction studies[163]. Thecoherent-state (positive-P ) phase-space equations are exactly equivalent to the relevantquantum equations, provided that phase-space boundary terms 1 vanish. They have theadvantage that they are computationally tractable for the large Hilbert spaces typical ofBose condensation experiments. Techniques of this sort may provide a way of extendingQMC methods[28, 29] into the time domain.7.3 Phase-space equationsThe model that we will use includes the usual nonrelativistic Hamiltonian for neutralatoms, in a trap V (x), interacting via a potential U(x), in d =2ord = 3 dimensions:∫Ĥ =[ d d 2x2m ∇ ˆψ † (x)∇ ˆψ(x)+V (x) ˆψ † (x) ˆψ(x)+ ˆψ † (x) ˆR(x)+ ˆψ(x) ˆR † (x)+ 1 2∫d d yU(x − y) ˆψ † (x) ˆψ † (y) ˆψ(y) ˆψ(x)]. (7.1)Where ˆψ † and ˆψ are the bosonic many-body creation and annihilation operators for atomsat position x. HereˆR(x) represents a localised absorber that removes the neutral atoms;1 A careful treatment of this problem of boundary terms appears in [60]. See also [163].145