BOSE-EINSTEIN CONDENSATION WITH INTERNAL DEGREES OF FREEDOM IN ALKALIATOM GASES

The Bogoliubov theory is extended to a Bose-Einstein condensation with internal degrees of freedom, realized recently in Na-23 gases where s everal hyperfine states are simultaneously cooled optically. Starting with a Hamiltonian constructed from general gauge and spin rotation sy mmetry principles, fundamental equations for condensate are derived. T he ground state where time reversal symmetry is broken in some cases a nd low-lying collective modes, e.g. spin and density wave modes, are d iscussed. Novel vortex as a topological defect can be created experime ntally.