# Stanford APPPHYS 389 - Lecture Notes (2 pages)

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- Appphys 389 - Boise-einstein Condensation And Lasers

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Bose Einstein Condensation and Lasers APPPHYS389 Winter 2010 Instructor Professor Yoshihisa Yamamoto Problem Set 6 Problem I 50 points An interacting Bose gas in uniform 2D systems I 1 15 points According to Landau s theorem of superfluidity the normal component mass density 2n mn2n can be calculated in terms of the elementary excitation spectrum Z 1 dNp 2 d2 p 2n p 2 d 2 2 where Np is the quasi particle distribution function at vs vn 0 Obtain the analytical expression for 2n when the excitation spectrum is given by q p p2 c2 p2 2m 2 and kB T mc2 Hint The above integral is mainly contributed from the momentum regime mc p mkB T I 2 10 points Obtain the critical temperature T c for quasi BEC from the above result Hint When T Tc the normal component 2n must be smaller than the total mass density mn2D This means the appearance of superfluide component 2s I 3 10 points Discuss the above result for T c in the limit of c 0 I 4 15 points Show that at temperatures TBKT T T c free vortices are thermally excited but at T TBKT the number of vortices decreases Discuss the behavior the superfluid mass density 2s as a function of continuously decreasing temperature 1 2 Problem II 50 points Phase locking between condensate and excitations II 1 15 points Let us consider the Bogoliubov interaction Hamiltonian 1 X 2 2 H B Vq a 0 a q a q a a a q q 0 2V q If the variation state for the condensate is expressed as 0 i e a 0 P q q a q a q 0i calculate the interaction energy EB associated with the Bogoliubov Hamiltonian II 2 20 points Estimate the order of the above energy Is it macroscopic or microscopic in terms of the energy per one quasi particle Find the condition to minimize the above energy II 3 10 points Discuss the impact of the above result in terms of unlimited growth or limited value for the excitation population II 4 Extra point Can you point out a different physical system which features a similar behavior

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