Bose Condensation and Superfluidity in Finite Rotating Bose Systems

A long standing problem in the theory of superfluidity concerns the question of how close is the connection between Bose condensation and superfluidity[1]. For the case of superfluid He, the Bose condensate fraction is quite small (if not virtually zero), yet the superfluid fraction goes to unity in the limit of low temperatures T → 0. For the superfluid He case, the observation of superfluidity is much more simple than the observation of Bose condensation. For the case of dilute Bose gases in atomic traps, it has been the case that the observation of Bose condensation appears more simple than the observation of superfluidity. Let us pause to review the differences in theoretical definition between the Bose condensate fraction and superfluid fraction. For a Bose fluid in equilibrium, the reduced one particle density matrix is defined as