Cold atoms Lecture 5. 15^th November, 2006 Preliminary plan/reality in the fall term Previous class: Interacting atoms L4: Scattering length, pseudopotential Previous class: Mean-field treatment of interacting atoms L4: Many-body Hamiltonian and the Hartree approximation L4: Gross-Pitaevskii equation at zero temperature Gross-Pitaevskii equation – homogeneous gas Field theoretic reformulation (second quantization) Field operator for spin-less bosons Field operator for spin-less bosons – cont'd Operators Hamiltonian Hamiltonian of the homogeneous gas Action of the field operators in the Fock space Bogolyubov method Basic idea Approximate Hamiltonian Bogolyubov transformation Bogolyubov transformation – result More about the sound part of the dispersion law Particles and quasi-particles Coherent ground state Reformulation of the Bogolyubov requirements The end On the way to the mean-field Hamiltonian On the way to the mean-field Hamiltonian On the way to the mean-field Hamiltonian On the way to the mean-field Hamiltonian Variational approach to the condensate ground state Variational estimate of the condensate properties Variational estimate of the condensate properties Variational estimate of the condensate properties Variational estimate of the condensate properties Variational estimate of the condensate properties Variational estimate of the condensate properties Variational estimate of the condensate properties