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BEC is the Holy-Grail of atomic physics! Why has it taken so long to make BEC? 3dB 2.612 Liquid helium Derivation of Gross-Pitaevskii equation The grand-canonical Hamiltonian for an interacting many-body system with total particle number N is given by Kˆ Hˆ Nˆ 2 2 ˆ † ˆ dx x Vext x x 2m 1 ˆ † y ˆ † x V x, y ˆ x ˆ y dxdy 2 where ˆ x : quantum field for the particle : chemical potential and ˆ † x ˆ x N d x Bogoliubov decomposition for bosons ˆ x x ˆ x ˆ x : macroscopic wave function x ˆ x : field operator for noncondensed particle Mean-field approximations ˆ † y ˆ † x ˆ x ˆ y ˆ † y ˆ † x ˆ x ˆ y ˆ † y ˆ x ˆ † x ˆ y ˆ † y ˆ y ˆ † x ˆ x ˆ † x ˆ x ˆ † y ˆ y ˆ † x ˆ y ˆ † y ˆ x ˆ x ˆ y ˆ † y ˆ † x ˆ † y ˆ † x x ˆ y x ˆ † y ˆ † x ˆ y ˆ † y ˆ y ˆ † x ˆ † x ˆ y ˆ † y † ˆ ˆ and other terms cubic in and Consequently, we have the simplified grand-canonical Hamiltonian Kˆ eff K0 Kˆ1 Kˆ 1† Kˆ 2 where the energy functional for the condensate 2 2 K 0 dx* x Vext x x 2m 1 dxdy* y * x V x, y x y 2 is independent of ˆ , and K̂1 and K̂2 are linear and quadratic in ˆ respectively. Now consider the short-range -interaction 4 2 a V x, y x y m where a is the s-wave scattering length. After minimizing K0 with respect to the ground state wave function, one obtains the time-independent Gross-Pitaevskii equation (GPE) 2 2 2 4 2 a Vext x x x x m 2m The time-dependent GPE can be obtained straightforwardly 2 2 2 4 2 a i x, t Vext x x, t x, t t m 2m Particle-in-a-box picture of the scattering length Eg 2m L 2 Eg 2m L A 2 Consider a pair of atoms with reduced mass m m / 2, whose relative motion is confined to a box of lengh L. 2 2 Eg A mL3 For N pairs Eg 2m L A 2 2 Eg A mL3 total energy difference AN A n 3 mL m