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

Chapter 2Properties of an atomic Bose condensatein a double-well potentialThis chapter introduces the model of the condensate that will be used throughout thefirst half of this thesis. We will overview some of the semiclassical and quantum propertiesof the double-well condensate in isolation, before considering coupling to the environmentin Sec. 2.3.4. Dynamical properties, including Josephson tunnelling and macroscopicself-trapping, will be analysed, as well as equilibrium properties, such as the transitiontemperature, energy eigenstates and spatial correlations.2.1 Condensate modelWe consider a dilute gas of atoms moving in a symmetric double-well tapping potential.This is described by the normally-ordered many-body Hamiltonian[72]:∫ [ ]Ĥ(t) = d 3 2r2m ∇ ˆψ † ·∇ˆψ + V ˆψ † U 0 ˆψ +2 ˆψ † ˆψ† ˆψ ˆψ , (2.1)where m is the atomic mass, U 0 =4π 2 a 0 /m measures the strength of the two-bodyinteraction, and a 0 is the s-wave scattering length. The binary hard-sphere collision modelused here to describe interparticle interactions presupposes that all relevant distance scalesare larger then a 0 ∼ 1nm. The Heisenberg picture field operators, ˆψ(r,t)and ˆψ† (r,t),annihilate and create atoms at position r and commute at equal times according to[ˆψ(r), ˆψ† (r )]′ = δ 3 (r − r ′ ). (2.2)20