Moment of inertia and superfluidity of a trapped Bose gas.

The temperature dependence of the moment of inertia of a dilute Bose gas confined in a harmonic trap is determined. Deviations from the rigid value, due to the occurrence of Bose-Einstein condensation, reveal the superfluid behaviour of the system. In the noninteracting gas these deviations become important at temperatures of the order of TcN −1/12. The role of interactions is also discussed. PACS numbers: 03.75.Fi,05.30.Jp,32.80.Pj,67.90.+z Typeset using REVTEX 1 Evidence of Bose-Einstein condensation in magnetically trapped gases of Rb [1] and Li [2] atoms has been recently reported. These measurements are expected to open further stimulating perspectives in the study of the phenomenon of Bose-Einstein condensation (BEC) both from the experimental and theoretical point of view [3]. In this context the understanding of the superfluid behaviour of a Bose condensed atom cloud is a question of primary importance that we plan to investigate in the present letter. A natural way to discuss superfluidity in a confined system is to focus on its rotational properties [4]. For a macroscopic system the moment of inertia is given by the rigid value unless it exhibits superfluidity. Crucial deviations from the rigid motion occur in rotating liquid helium below the lambda temperature [5]. Also finite systems, like some deformed atomic nuclei [6] and helium clusters [7], are known to exhibit important superfluid effects in the moment of inertia. The moment of inertia Θ, relative to the z-axis, can be defined as the linear response of the system to a rotational field Hext = −ωJz, according to the formula < Jz >= ωΘ (1) where Jz = ∑ i(xip y i −yipi ) is the z-component of the the angular momentum and the average is taken on the state perturbed by Hext. For a classical system the moment of inertia takes the rigid value Θrig ≡ mN < x + y > (2) where N is the number of atoms in the trap. Vice versa the quantum mechanical determination of Θ is less trivial. It involves a calculation of dynamical properties of the system and, according to perturbation theory, can be written as