Condensate Oscillations, Kinetic Equations and Two-fluid Hydrodynamics in a Bose Gas

Trapped Bose-condensed atomic gases 1,2 are remarkable because, in spite of the fact that these are very dilute systems, they exhibit robust coherent dynamic behaviour when perturbed. These quantum “wisps of matter” are a new phase of highly coherent matter. While binary collisions are very infrequent, the coherent mean field associated with the Bose condensate ensures that interactions play a crucial role in determining the collective response of these superfluid gases. In our discussion of the theory of collective oscillations of atomic condensates, which is the main theme of these lectures, the macroscopic Bose wavefunction Φ(r, t) will play a central role. This wavefunction is the BEC order parameter. The initial attempts at defining this order parameter began with the pioneering work of Fritz London 3 in 1938, was further developed by Bogoliubov 4 in 1947 and finally formalized in the general quantum field theoretic formalism of Beliaev 5 in 1957. The first extension of these ideas to inhomogeneous Bose condensates was by Pitaevskii and, independently, by Gross in 1961, which led to the now famous Gross-Pitaevskii (GP) equation of motion for Φ(r, t). Most of this early work was limited to T = 0 where, in a dilute Bose gas, one can assume all of the atoms are in the condensate. In Section 2, I will first review the dynamics of a pure condensate at T = 0, based on solving the linearized GP equation. This limit is especially appealing since one can ignore all the complications which arise from the presence of non-condensate atoms (the thermal cloud). We will discuss the normal mode solutions of the T = 0 GP equation of motion using the “quantum hydrodynamic” formalism, which works in terms of the local condensate density nc(r, t) and superfluid velocity vc(r, t). Within the Thomas-Fermi approximation, Stringari 6 has shown that the equations of motion for these two variables can be combined to give a wave equation for oscillations of the condensate.