Cold atoms Lecture 3. 17. October 2007 BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational properties of the GP equation Are the interactions important? Fortunate properties of the interactions Fortunate properties of the interactions Interatomic interactions Interatomic interactions Interatomic interactions Interatomic interactions Interatomic interactions Scattering length, pseudopotential Experimental data Experimental data Experimental data Mean-field treatment of interacting atoms Many-body Hamiltonian and the Hartree approximation Many-body Hamiltonian and the Hartree approximation Hartree approximation at zero temperature Gross-Pitaevskii equation at zero temperature Gross-Pitaevskii equation at zero temperature Gross-Pitaevskii equation at zero temperature Gross-Pitaevskii equation at zero temperature Gross-Pitaevskii equation at zero temperature Gross-Pitaevskii equation – "Bohmian" form Gross-Pitaevskii equation – "Bohmian" form Gross-Pitaevskii equation – "Bohmian" form Gross-Pitaevskii equation – variational interpretation Gross-Pitaevskii equation – chemical potential Gross-Pitaevskii equation – chemical potential Gross-Pitaevskii equation – chemical potential Interacting atoms in a constant potential The simplest case of all: a homogeneous gas The simplest case of all: a homogeneous gas The simplest case of all: a homogeneous gas The simplest case of all: a homogeneous gas The simplest case of all: a homogeneous gas The simplest case of all: a homogeneous gas The simplest case of all: a homogeneous gas Interacting atoms in a parabolic trap Reminescence: The trap potential and the ground state Parabolic trap with interactions Parabolic trap with interactions Parabolic trap with interactions Importance of the interaction – synopsis Importance of the interaction Importance of the interaction Importance of the interaction Importance of the interaction 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 The end