Bose-Einstein Condensation and LasersAPPPHYS389/ Winter 2010Instructor: Professor Yoshihisa YamamotoProblem Set 6Problem I (50 points): An interacting Bose gas in uniform 2D systemsI.1 (15 points) According to Landau’s theorem of superfluidity, the normal compo-nent (mass density ρ2n= mn2n) can be calculated in terms of the elementaryexcitation spectrum:ρ2n=12Zµ−dNp(ε)dε¶p2·d2p(2π~)2,where Np(ε) is the quasi-particle distribution function at vs−vn= 0. Obtainthe analytical expression for ρ2nwhen the excitation spectrum is given byε(p) =qp2c2+ (p2/2m)2and kBT À mc2. (Hint: The above integral ismainly contributed from the momentum regime mc ≤ p ≤√mkBT .)I.2 (10 points) Obtain the critical temperature˜Tcfor quasi-BEC from the aboveresult. (Hint: When T < Tc, the normal component ρ2nmust be smallerthan the total mass density mn2D. This means the appearance of superfluidecomponent ρ2s.)I.3 (10 points) Discuss the above result for˜Tcin the limit of c → 0.I.4 (15 points) Show that at temperatures, TBKT< T <˜Tc, free vortices arethermally excited, but at T . TBKTthe number of vortices decreases. Discussthe behavior the superfluid mass density ρ2sas a function of continuouslydecreasing temperature.12Problem II (50 points + α): Phase locking between condensate and excitationsII.1 (15 points) Let us consider the Bogoliubov interaction Hamiltonian:ˆHB=12VXqVq¡ˆa+20ˆaqˆa−q+ ˆa+−qˆa+qˆa20¢.If the variation state for the condensate is expressed as|ψ0i = eφˆa+0+Pqλqˆa+qˆa+−q|0i,calculate the interaction energy EBassociated with the Bogoliubov Hamilton-ian.II.2 (20 points) Estimate the order of the above energy. Is it macroscopic ormicroscopic in terms of the energy per one quasi-particle? Find the conditionto minimize the above energy.II.3 (10 points) Discuss the impact of the above result in terms of unlimited growthor limited value for the excitation population.II.4 (Extra-point) Can you point out a different physical system which features asimilar